Multiphoton entanglement and interferometry

Optical quantum computing proposals based on the one-way model reduce the computational resource overhead by three orders of magnitude compared to the KLM scheme.

“Optical quantum computing proposals (Browne and Rudolph, 2005) based on the one-way model reduce the computational resource overhead by three orders of magnitudes compared to the KLM scheme.”

• cites Browne and Rudolph, 2005 sourcePartly supported

  The Introduction (p. 1) describes a cluster-state optical computing scheme and estimates resource savings of “many orders of magnitude over KLM.” It does not specify three orders of magnitude for resource overhead; its explicit three-orders figure concerns reduced sensitivity to phase instability, not computational resources. (LLM, conf=0.88)

  Experimental — accuracy not yet measured

Together with additional phase shifters in the input and output ports, a Mach-Zehnder interferometer can perform an arbitrary SU(2) unitary transformation on a qubit encoded in two spatial modes.

“Together with additional phase shifters in the input and output ports, the Mach-Zehnder interferometer can perform an arbitrary SU(2) unitary transformation on a qubit encoded in the two spatial modes (see, e.g., Englert et al. (2001)).”

• cites Englert et al. (2001) sourceSupported

A linear optical Bell-state analyzer suggested by Weinfurter and by Braunstein and Mann is based on two-photon interference at a 50:50 beamsplitter and via coincidence analysis can distinguish two of the four Bell states.

“A linear optical Bell-state analyzer was suggested by Weinfurter (1994) and by Braunstein and Mann (1995).It is based on the two-photon interference effect at a 50:50 BS and via a coincidence analysis can distinguish two of the four Bell states.”

• cites Weinfurter (1994) sourceSupported
• cites Braunstein and Mann (1995) sourceNot checked — no full text of the source

Because photons are produced coherently along the entire length of the crystal, one can completely compensate for the longitudinal walk-off.

“Fortunately, because the photons are produced coherently along the entire length of the crystal, one can completely compensate for the longitudinal walk-off (Rubin et al., 1994).”

• cites Rubin et al., 1994 sourceSupported

The standard method to produce polarization-entangled photons is the noncollinear type-II SPDC process.

“Currentlythe standard method to produce polarization-entangled photons is the noncollinear type-II SPDC process (Kwiat et al., 1995).”

• cites Kwiat et al., 1995 sourcePartly supported

  The paper describes producing polarization-entangled photon pairs by noncollinear type-II parametric down-conversion (Abstract; page not given). It presents the technique as novel, however, and does not establish that it is currently the standard method. It says that, at the time of publication, most down-conversion experiments had used type-I phase matching. (LLM, conf=0.85)

  Experimental — accuracy not yet measured

A pulsed-pump version of Franson interferometry uses time-bin entanglement.

“A pulsed-pump version of Franson-interferometry uses so-called time-bin entanglement (Brendel et al., 1999;Marcikic et al., 2002;Gisin et al., 2002) (see Fig. 10).”

• cites Brendel et al., 1999 sourceNot checked — no full text of the source
• cites Marcikic et al., 2002 sourceSupported
• cites Gisin et al., 2002 sourceSupported

A pulsed and narrow-band (0.15 nm), wavelength tunable entangled photon source based on such a configuration with spectral brightness up to 273000 pairs (s·mW·nm)^-1 has been reported.

“A pulsed (Kuzucu and Wong, 2008), and narrow-band (0.15nm), wavelength tunable entangled photon source based on such a configuration with a spectral brightness up to 273000 pairs (s•mW•nm) -1 has been reported (Fedrizzi et al. 2007).”

• cites Fedrizzi et al. 2007 sourcePartly supported

  The Conclusion (p. 10) reports a wavelength-tunable, narrowband entangled-photon source with spectral brightness of 273000 pairs (s·mW·nm)^-1. However, the performance section (p. 9) gives a measured FWHM bandwidth of 0.3 nm, not 0.15 nm. The citing sentence attaches “pulsed” to Kuzucu and Wong, so that descriptor is not assessed against Fedrizzi et al. (LLM, conf=0.91)

  Experimental — accuracy not yet measured

An operational method to swap entanglement of two pairs of particles has been used in many experiments, including the pioneering Innsbruck teleportation.

“We shall show in detail an operational method to swap entanglement of two pairs of particles ( Żukowski, Zeilinger, and Weinfurter, 1995), which has been used in many experiments (the pioneering one was the Innsbruck teleportation, Bouwmeester et al. 1997).”

• cites Żukowski, Zeilinger, and Weinfurter, 1995 sourceSupported
• cites Bouwmeester et al. 1997 sourcePartly supported

  The opening section (p. 1) identifies this as an experimental demonstration of quantum teleportation, supporting the citation’s identification of the Innsbruck experiment. But THE NEXT STEPS (p. 6) describes transferring entanglement by teleporting a member of an entangled pair as a possibility, not as what this experiment demonstrated. The paper therefore does not support calling the experiment a demonstrated use of entanglement swapping between two pairs. (LLM, conf=0.88)

  Experimental — accuracy not yet measured

If one chooses a non-factorizable function S(s1,s2)=1/2(1+s1+s2-s1s2), the CHSH inequality is recovered.

“If one chooses a non-factorizable function S(s 1 , s 2 ), say, 1 2 (1 + s 1 + s 2 -s 1 s 2 ), the famous CHSH (Clauser, Horne, Shimony, and Holt, 1969) inequality is recovered”

• cites Clauser, Horne, Shimony, and Holt, 1969 sourceSupported

The data from Bouwmeester et al. (1997), together with similar results for other polarizations, constitute a positive result of a test for teleportation of an arbitrary state.

“The data, together with similar results for other polarizations, constitute a positive result of a test for teleportation of an arbitrary state.(Bouwmeester et al., 1997).”

• cites Bouwmeester et al., 1997 sourceSupported

When particle 1 is a member of an entangled pair, the teleportation leads to entanglement swapping.

“This is the case when, as already remarked by Bennett et al. (1993), particle 1 itself is a member of an entangled pair, ultimately leading to entanglement swapping ( Żukowski et al., 1993;Bose et al., 1998).”

• cites Bennett et al. (1993) sourceSupported
• cites Żukowski et al., 1993 sourceSupported
• cites Bose et al., 1998 sourceNot checked — no full text of the source

In the Bouwmeester et al. experiment, if Alice obtains a coincidence count in detectors f1 and f2, implying collapse into |ψ-12, Bob knows his photon 3 is in the original polarization state of photon 1.

“In the case of receiving a classical information indicating that Alice obtained a coincidence count in detectors f1 and f2, which implies a collapse into |ψ - 12, Bob knows that his photon 3 is in the original polarization state of photon 1.”

• cites Bouwmeester et al., 1997 sourceSupported

Marcikic et al. (2003) realized teleportation of qubits at telecommunication wavelengths over a fiber length of 2 km.

“Marcikic et al. (2003) realized a teleportation of qubits at telecommunication wavelengths over a fiber length of 2 km.”

• cites Marcikic et al. (2003) sourceSupported

Entanglement swapping provides a method of entangling two particles that never interacted or even have no common past.

“Entanglement swapping ( Żukowski et al., 1993) provides a method of entangling two particles that never interacted or even have no common past.”

• cites Żukowski et al., 1993 sourceSupported

Entanglement distillation, concentration and purification are important tools to overcome noise in the quantum communication channel.

“An important tool to overcome the noise in the quantum communication channel is entanglement distillation, concentration and purification, proposed by Bennett et al. (1996aBennett et al. ( , 1996bBennett et al. ( , 1996c) ) and Deutsch et al. (1996).”

• cites Bennett et al. (1996a, 1996b, 1996c)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Deutsch et al. (1996) sourcePartly supported

  The Abstract (p. 1) describes an entanglement-purification procedure in a cryptographic scheme secure over a noisy quantum channel; the discussion on p. 4 explains how it produces nearly pure, maximally entangled pairs from mixed pairs. This supports purification as a way to address channel noise, but our copy does not establish entanglement concentration as a separate tool proposed by Deutsch et al. (LLM, conf=0.84)

  Experimental — accuracy not yet measured

The original KLM scheme was only very recently implemented in a sophisticated setup using polarization encoding, and Sagnac-interferometers for increased stability, by Okamoto et al., (2010).

“The original KLM scheme was only very recently implemented in a sophisticated setup up using polarization encoding, and Sagnac-interferometers for increased stability, by Okamoto et al., (2010).”

• cites Okamoto et al., 2010 sourceSupported

Politi et al. (2008) reported a high-fidelity silica-on-silicon integrated optical realizations of key quantum photonic circuits.

“More recently, Politi et al. (2008) reported a high-fidelity silica-on-silicon integrated optical realizations of key quantum photonic circuits.”

• cites Politi et al., 2008 sourceSupported

Consider a detection of a pair of photons at the output modes 3 and 4, only one photon at each output (the four mode case). Depending on the registered polarizations, up to a specific unitary transformation, a non-destructive CNOT gate operation is then performed on |χ 2 5.

“Consider a detection of a pair of photons at the output modes 3 and 4 , only one photon at each output (the four mode case).Depending on the registered polarizations, up to a specific unitary transformation, a non-destructive CNOT gate operation is then performed on |χ 2 5 .”

• cites Pittman et al., 2001 sourcePartly supported

  Split into 2 atoms — 0 confirmed, 2 partially supported, 0 disputed, 0 misattributed, 0 unverifiable. Expand the claim to see per-atom verdicts.

  Experimental — accuracy not yet measured

Koashi et al., 2001, and Pittman et al., 2001, showed that by using entangled pairs of photons as ancilla, the success of the gate operation can be inferred by the detection of photons in ancilla-outputs.

“In a separate development, Koashi et al., 2001, andPittman et al., 2001, showed that by using entangled pairs of photons as ancilla, the success of the gate operation can be inferred by the detection of photons in ancilla-outputs.”

• cites Koashi et al., 2001 sourceNot checked — no full text of the source
• cites Pittman et al., 2001 sourceSupported

Such gates, assisted by entangled or unentangled photon pairs, were reported using four or five photons, see Gasparoni et al., (2004), Zhao et al., (2005b), Bao et al., (2007), Tokunaga et al., (2008) and Gao et al., (2010b).

“Such gates, assisted by entangled or unentangled photon pairs, were reported using four or five photons, see Gasparoni et al., (2004), Zhao et al., (2005b), Bao et al., (2007), Tokunaga et al., (2008) and Gao et al., (2010b).”

• cites Gasparoni et al., 2004 sourceSupported
• cites Zhao et al., 2005bNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Bao et al., 2007 sourceNot checked — no full text of the source

Nielsen showed that efficiency can be greatly enhanced by building photonic cluster states using easy non-deterministic gates.

“Nielsen showed that efficiency can be greatly enhanced by building photonic cluster states using easy non-deterministic gates.”

• cites Nielsen, 2004 sourceSupported

Following proposals by Han et al. (2007) and Pachos (2007), four- and six-qubit graph states were created to mimic the ground state of the Kitaev spin-lattice model.

“Following proposals by Han et al (2007) and Pachos (2007), four-and six-qubit graph states were created to mimic the ground state of the Kitaev spin-lattice model (Kitaev, 2003).”

• cites Han et al (2007) sourceNot checked — no full text of the source
• cites Pachos (2007) sourceNot checked — no full text of the source
• cites Kitaev, 2003 sourceSupported

A compiled version of Shor's algorithm has been demonstrated using four photonic qubits by Lu et al. (2007) and Lanyon et al. (2007).

“Recently, a compiled version of Shor's algorithm has been demonstrated using four photonic qubits by Lu et al. (2007) and Lanyon et al. (2007).”

• cites Lu et al. (2007)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Lanyon et al. (2007) sourceSupported

Later optical implementations of Shor's algorithm moved into an integrated photonic chip device.

“Later optical implementations of Shor's algorithm later moved into an integrated photonic chip device (Politi et al. 2009).”

• cites Politi et al. 2009 sourceSupported

Quantum teleportation became an experimental reality with four-photon interferometries.

“Quantum teleportation-the transfer and reconstruction of quantum states over arbitrary distances became an experimental reality with four-photon interferometries (Bouwmeester et al. 1997, Marcikic et al. 2003).”

• cites Bouwmeester et al. 1997 sourcePartly supported

  The paper reports the first experimental verification of quantum teleportation (p. 1). But it describes the method as two-photon interferometry (p. 1), not four-photon interferometry; its four-fold coincidence measurements (Results, pp. 5–6) are a different detail. (LLM, conf=0.88)

  Experimental — accuracy not yet measured
• cites Marcikic et al. 2003 sourcePartly supported

  The Conclusion (p. 6) reports an experimental demonstration of quantum teleportation. But the Experimental Realization (pp. 3–4) describes removing one photon from a generated pair and counting the laser pulse in its four-fold coincidences; it does not describe the demonstration as a four-photon interferometry experiment. The experimental-teleportation part is supported, but that specific characterization is not. (LLM, conf=0.86)

  Experimental — accuracy not yet measured

New generations of single-photon sources have been developed based on solid-state devices, atoms, molecules and ions.

“New generations of single-photon sources have been developed: they are based on solid-state devices, atoms, molecules and ions etc. (see Lounis and Orrit, (2005) for a recent review).”

• cites Lounis and Orrit, (2005) sourceSupported

O'Brien et al. (2005) demonstrated a two-qubit code for correction of a Z-measurement error.

“O'Brien et al. (2005) demonstrated a two-qubit code for correction of a Zmeasurement error.”

• cites O'Brien et al. (2005) sourceSupported

The type-I fusion operation allows one to efficiently generate arbitrarily long linear cluster states.

“the type-I fusion operation allows one to efficiently generate arbitrarily long linear cluster states.”

• cites Browne and Rudolph, 2005 sourceSupported
• cites Duan and Raussendorf, 2005 sourcePartly supported

  The paper states on p. 2 that 1D cluster states of arbitrary length can be created efficiently. But its construction uses probabilistic controlled phase-flip (CPF) gates, as described on pp. 1–2; our copy does not identify these as type-I fusion operations or establish the claim specifically for type-I fusion. (LLM, conf=0.88)

  Experimental — accuracy not yet measured

The data, together with similar results for other polarizations, constitute a positive result of a test for teleportation of an arbitrary state.

“The data, together with similar results for other polarizations, constitute a positive result of a test for teleportation of an arbitrary state.(Bouwmeester et al., 1997).”

• cites Bouwmeester et al., 1997 sourceSupported

In a two-photon EPR interferometer, the coincidence detection probabilities for same detector pairs (1c,2c and 1d,2d) are 1/4[1−cos(φ1−φ2)].

“p 1c,2c (φ 1 , φ 2 ) = p 1d,2d (α, φ 2 ) = 1 4 [1 -cos(φ 1 -φ 2 )].”

• AssumptionThe authors state this as an assumption: A central source emits two photons in the entangled state |ψ12 = 1/√2(|a1|a2 + |b1|b2).

In a two-photon EPR interferometer, the coincidence detection probabilities for opposite detector pairs (1c,2d and 1d,2c) are 1/4[1+cos(φ1−φ2)].

“p 1c,2d (φ 1 , φ 2 ) = p 1d,2c (φ 1 , φ 2 ) = 1 4 [1 + cos(φ 1 -φ 2 )]”

• AssumptionThe authors state this as an assumption: A central source emits two photons in the entangled state |ψ12 = 1/√2(|a1|a2 + |b1|b2).

In a three-photon GHZ interferometer, the three-fold coincidence detection probability for output modes (1c,2c,3c) is 1/8[1+sin(φ1+φ2+φ3)].

“p 1c,2c,3c (φ 1 , φ 2 , φ 3 ) = 1 8 [1 + sin(φ 1 + φ 2 + φ 3 )]”

In a three-photon GHZ interferometer, the three-fold coincidence detection probability for output modes (1d,2c,3c) is 1/8[1−sin(φ1+φ2+φ3)].

“p 1d,2c,3c (φ 1 , φ 2 , φ 3 ) = 1 8 [1 -sin(φ 1 + φ 2 + φ 3 )]”

For two spectrally identical photons of the same polarization each entering opposite input ports of a 50:50 beamsplitter at the same moment, the terms describing each photon exiting by a different port cancel.

“We have a cancellation of the two terms, ĉ ddĉ = 0, which describe the cases in which each photon exits by a different exit port.”

• AssumptionThe authors state this as an assumption: In a simplified theory of beamsplitters, the beamsplitter acts identically for every relevant frequency.

The cancellation of the different-exit-port terms occurs if the two photons are perfectly indistinguishable in all their other degrees of freedom such as frequency, time, or polarization.

“This cancellation occurs if the two photons are perfectly indistinguishable in terms of all their other degrees of freedom such as frequency, time, or polarization.”

If two incident photons have opposite polarizations, they always go out along the same direction, so there are two photons in one output and none in the other.

“If the two incident photons have opposite polarizations, they will always go out along the same direction, so there will be two photons in one of the two outputs and none in the other.”

The probability of finding a coincidence count at exits c and d is (1/2)|β|^2, where β denotes the degree of distinguishability between photons 1 and 2.

“The probability of finding a coincidence count at exits c and d, which is given by the square of the norm of 1 2 (ĉ † 1 d † 2 -d † 1 ĉ † 2 )|Ω , is (1/2)|β| 2”

If the photons are at least partially distinguishable, the terms that contribute to cases in which each photon exits by a different port do not cancel.

“Since ĉ1 = ĉ2 and d1 = d2 , the terms that contribute to the cases in which each photon exits by a different exit port, namely 1 2 (ĉ † 1 d † 2 -d † 1 ĉ † 2 )|Ω , do not cancel with each other.”

The three-particle GHZ interferometer does not exhibit any two-particle fringes.

“Second, this three-particle interferometer does not exhibit any two-particle fringes.”

In a symmetric 50:50 beamsplitter, an outgoing particle can be found with equal probability (50%) in either output mode c or d, no matter through which single input beam it came.

“In such a case, an outgoing particle can be found with equal probability (50%) in either of the output modes c and d, no matter through which single input beam it came.”

• AssumptionThe authors state this as an assumption: In a simplified theory of beamsplitters, the beamsplitter acts identically for every relevant frequency.

For incident state 1/√2(|Ha|Hb+|Va|Vb), one always observes a coincidence either between detectors DH1 and DH2 or DV1 and DV2.

“Specifically, for the incident state 1 √ 2 (|H a |H b + |V a |V b ) we always observe a coincidence either between detectors D H1 and D H2 or D V 1 and D V 2 .”

• AssumptionThe authors state this as an assumption: Two spectrally identical photons enter the Bell-state analyzer by modes a and b and arrive at PBS ab simultaneously, so their wavefunctions overlap behind it.

The other two Bell states would lead to no coincidence at detectors in modes 1 and 2.

“The other two Bell states would lead to no coincidence at detectors in modes 1 and 2.”

• AssumptionThe authors state this as an assumption: Two spectrally identical photons enter the Bell-state analyzer by modes a and b and arrive at PBS ab simultaneously, so their wavefunctions overlap behind it.

For a color entangled state at frequencies ω1 and ω2, one can observe spatial quantum beating, where two-photon detection exhibits modulation of the form cos(ω1−ω2)τ.

“For a color (at frequency ω 1 and ω 2 ) entangled state, one can further observe”

• cites Ou and Mandel, 1988bNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

For proper path length, one has a coincidence peak instead of a dip.

“For proper path length, one has a coincidence peak (instead of a dip) (Mattle et al. 1996).”

• cites Mattle et al. 1996 sourceNot checked — no full text of the source

For incident state 1/√2(|Ha|Hb−|Va|Vb), one observes coincidence at detectors DH1 and DV2 or DV1 and DH2.

“On the other hand, if the incident state is 1 √ 2 (|H a |H b -|V a |V b ) we observe coincidence at detectors D H1 and D V 2 or D V 1 and D H2 .”

• AssumptionThe authors state this as an assumption: Two spectrally identical photons enter the Bell-state analyzer by modes a and b and arrive at PBS ab simultaneously, so their wavefunctions overlap behind it.

In the modified Bell-state analyzer, properties of a PBS and a coincidence detection in modes 1 and 2 allow one to distinguish |φ+12 and |φ−12 polarization Bell states.

“The properties of a PBS depicted in Fig. 4, and a coincidence detection in modes 1 and 2, allow one to distinguish |φ + 12 and |φ - 12 polarization Bell states.”

• AssumptionThe authors state this as an assumption: Two spectrally identical photons enter the Bell-state analyzer by modes a and b and arrive at PBS ab simultaneously, so their wavefunctions overlap behind it.

By superposing two (H+V) polarized photons at a PBS, one can observe an entangled state |Hc|Hd + |Vc|Vd in the post-selected coincidence counts.

“In a similar way, by superposing two (H + V ) polarized photons at a PBS, one can observe an entangled state |H c|H d + |V c|V d in the post-selected coincidence counts.”

The entanglement quality is sensitive to optical path length changes of the order of the photons' coherence length.

“As in the case of the Hong-Ou-Mandel interferometer, the entanglement quality is sensitive to optical path length changes of the order of the photons' coherence length.”

In a Hong-Ou-Mandel interferometer, the electromagnetic field emerging from the beamsplitter associated with the pair of photons is in an entangled NOON state [(c†)^2+(d†)^2]|Ω = |2c0d + |0c2d with N=2.

“Note first, that according to Eq. ( 30), in a Hong-Ou-Mandel interferometer, the electromagnetic field emerg-ing from the BS associated with the pair of photons is in an entangled state [(ĉ † ) 2 + ( d † ) 2 ]|Ω = |2 c 0 d + |0 c 2 d , a so-called NOON state (see Section IV.C), with N = 2.”

As the polarization state |Ψ− is antisymmetric, it results in a coincidence detection at the outputs of the beamsplitter.

“As the polarization state |Ψ -is antisymmetric, it results in a coincidence detection at the outputs of the BS (i.e., a coincidence at detectors DH1 and DV2 or at DH2 and DV1).”

For the remaining three Bell states, both photons exit via the same output port of the beamsplitter.

“For the remaining three states, both photons exit via the same output port of the BS.”

In noncollinear type-II SPDC, the state emerging through beams A and B is a superposition of |H|V and |V|H with relative phase α due to birefringence.

“The state emerging through the two beams A and B is a superposition of |H |V and |V |H , namely, 1 √ 2 (|H A |V B + e iα |V A |H B )(36) where the relative phase α is due to the birefringence.”

The success probability of the GHZ analyzer is 1/4.

“The success probability of the GHZ analyzer is thus 1/4.”

By pumping two BBO crystals with optical axes aligned in perpendicular planes, a two-photon (2⊗2)⊗(3⊗3)⊗(2⊗2)-dimensional hyperentangled state was produced, approximately described by Eq. (41).

“By pumping two BBO crystals with optical axes aligned in perpendicular planes, a twophoton (2 ⊗ 2) ⊗ (3 ⊗ 3) ⊗ (2 ⊗ 2)-dimensional hyperentangled state was produced, approximately described by (|HH + |V V ) polarization ⊗ (|1, -1 LG + α|0, 0 LG + | -1, 1 LG ) spatial ⊗ (|EE + |LL ) energy-time .(41)”

• cites Torres et al., 2003 sourceNot checked — no full text of the source

In an experiment by Barreiro et al. (2005), photon pairs from a single nonlinear crystal were entangled in polarization, energy, and orbital angular momentum.

“In an experiment by Barreiro et al (2005), besides the entanglement in polarization and in energy, photon pairs from a single nonlinear crystal were also entangled in orbital angular momentum.”

• cites Barreiro et al (2005) sourceNot checked — no full text of the source

Later, twin-beam entanglement of up to 12 photons was experimentally observed.

“Later, twin-beam entanglement of up to 12 photons (Eisenberg et al., 2004) was experimentally observed.”

• cites Eisenberg et al., 2004 sourceNot checked — no full text of the source

The state |ψ is a superposition of the states |ψ^-_n of n indistinguishable photon pairs.

“The state |ψ is a superposition of the states |ψ - n of n indistinguishable photon pairs.”

These experiments demonstrated that the effective de Broglie wavelength of the multiphoton state is by 1/N shorter than the wavelength of the single photon.

“These experiments demonstrated an interesting feature of NOON states: the effective de Broglie wavelength of the multiphoton state is by 1/N shorter than the wavelength of the single photon (Jacobson et al., 1995).”

• cites Jacobson et al., 1995 sourceNot checked — no full text of the source

It can be shown that the visibility is reduced below 50% if |z|^2 > (√17 - 3)/8 ≈ 0.140.

“It can be shown ( Żukowski et al., 1999) that the visibility is reduced below 50% if |z| 2 > ( √ 17 -3)/8 ≈ 0.140.”

• cites Żukowski et al., 1999 sourceNot checked — no full text of the source

The average photon pair number is n = 2 sinh^2 τ.

“The average photon pair number is n = 2 sinh 2 τ .”

In control experiments with the polarizer in front of detector D1 set at 0°, projecting the GHZ state into a separable state |V2>|H3>, the results show no correlation.

“In this case, the results in Fig. 16(c) show no correlation.”

The obtained data are consistent with the prediction that a result + at detector D1 projects the state of the other photons to 1/√2(|+2>|+3> - |-2>|-3>).

“The obtained data are consistent with this prediction, as can be seen from Fig. 16(b).”

The filtering process stretched the coherence time so that it was substantially larger than the pump pulse duration (~200 fs).

“The filtering process stretched the coherence time so that it was substantially larger than the pump pulse duration (∼ 200fs).”

The observed Hong-Ou-Mandel interference, conditioned on a joint detection event of two idlers, had a raw visibility of 76.1%.

“The observed interference, conditioned on a joint detection event of two idlers, had a raw visibility of 76.1%.”

The intensity ratio between the desired events (HHV and VVH) and the six other undesired ones was about 12:1.

“The results showed that the intensity ratio between the desired events (HHV and V V H) and the six other undesired ones was about 12 : 1.”

In actual SPDC experiments, due to the absence of perfect pair sources and perfect single photon detectors, both three- and four-photon entanglement are observed only under the condition that there is one and only one photon in each of the four outputs.

“Due to the absence of perfect pair sources and perfect single photon detectors, in the experiments both three-and four-photon entanglement”

• Also stated in: § 3. First proposals · plain

  “is observed only under the condition that there is one and only one photon in each of the four outputs.”

• cites Bouwmeester et al., 1999aNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Pan et al., 2001aNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Zhao et al., 2003aNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

A typical observed value of S_Bell was 2.73 ± 0.02.

“A typical observed value of S Bell , the right side of inequality (54), was as high as 2.73 ± 0.02.”

The actual orientation for local polarization analysis was determined independently by a quantum physical random number generator.

“The actual orientation for local polarization analysis was determined independently by a quantum physical random number generator (Jennewein et al. 2000).”

• cites Jennewein et al. 2000 sourceNot checked — no full text of the source

The individual measurements were finished within a time much shorter than 1.3 μs, which is the distance of the two observation stations in light seconds.

“The individual measurements were finished within a time much shorter than 1.3 µs, which is the distance of the two observation stations in light seconds.”

In 10 s, 14700 coincidence events were collected.

“In 10 s 14700 coincidence events were collected.”

In Pan et al. (2000), the measured values for x1ŷ2ŷ3, ŷ1x2ŷ3, and ŷ1ŷ2x3 followed quantum predictions in a fraction 0.85 ± 0.04 of all cases.

“In the experiment by Pan et al. (2000) the measured values for x1 ŷ2 ŷ3 , ŷ1 x2 ŷ3 , and ŷ1 ŷ2 x3 followed the values predicted by quantum physics in a fraction of 0.85 ± 0.04 of all cases.”

• cites Pan et al. (2000) sourceNot checked — no full text of the source

The data from Pan et al. (2000), in the form of imperfect GHZ correlations, violated local realism.

“The obtained data in the form of the (necessarily imperfect) GHZ correlations was shown to violate local realism.”

• cites Pan et al. (2000) sourceNot checked — no full text of the source

The experimental results of Pan et al. (2000) confirmed quantum predictions within experimental uncertainty.

“The experimental results confirmed the quantum predictions, within an experimental uncertainty.”

• cites Pan et al. (2000) sourceNot checked — no full text of the source

The Irvine et al. experiment gave results agreeing with the quantum prediction that joint detection in the right exit ports is possible.

“Using a similar realistic reasoning as before this leads to the conclusion that no joint detection in the right exit ports of the two interferometers is possible.The quantum prediction is different, and the experiment gave results agreeing with it.”

• cites Irvine et al. 2005 sourceNot checked — no full text of the source

A proposal by Hardy (1992) was implemented using a pair of Mach-Zehnder interferometers coupled via bosonic photon bunching at a beam splitter (Irvine et al. 2005).

“Another proposal suggested by Hardy (1992) was implemented using a pair of Mach-Zehnder interferometers that couple via the bosonic photon bunching effect at a beam splitter (Irvine et al. 2005).”

• cites Hardy (1992) sourceNot checked — no full text of the source
• cites Irvine et al. 2005 sourceNot checked — no full text of the source

Experimental demonstrations of the two-photon hyper-entanglement protocol were done by Yang et al. 2005 and Cinelli et al. (2005).

“Experimental demonstrations of such a protocol were done by Yang et al. 2005 andCinelli et al. (2005) 45 using a two-photon hyper-entangled state in Eq. ( 40) (for hyperentanglement see section IV.B.2).”

• cites Yang et al. 2005 sourceNot checked — no full text of the source
• cites Cinelli et al. (2005) sourceNot checked — no full text of the source

The fourth experiment measuring x1x2x3 in Pan et al. (2000) also agreed with quantum mechanics for the same fraction of events.

“The fourth experiment, measuring x1 x2 x3 , also was performed, and yielded results as shown in Fig. 18.The data again agree with quantum mechanics for the same fraction of events.”

• cites Pan et al. (2000) sourceNot checked — no full text of the source

The results of Pan et al. (2000) are in clear disagreement with a local realistic prediction range made from the first three measurements.

“The results are in a clear disagreement with a prediction range that can be made with the data of the three first measurements using a local realistic model.”

• cites Pan et al. (2000) sourceNot checked — no full text of the source

The Leggett-type inequality was experimentally tested by Gröblacher et al. (2007) using a SPDC source with visibilities ∼99.0±1.2% (H/V), ∼99.2±1.6% (±45°), and ∼98.9±1.7% (R/L).

“The Leggett-type inequality valid for nonlocal realistic theories of the discussed type was experimentally tested by Gröblacher et al. (2007).In the experiment a SPDC source of high-fidelity two-photon entangled states was used, with visibilities ∼ 99.0 ± 1.2% in the H/V basis, ∼ 99.2 ± 1.6% in the ±45 • basis and ∼ 98.9 ± 1.7% in the R/L basis.”

• cites Gröblacher et al. (2007) sourceNot checked — no full text of the source

The observed S_NLHV = 3.8521 ± 0.0227 violates inequality (68) by 3.2 standard deviations.

“The observed S N LHV = 3.8521 ± 0.0227 violates inequality (68) by 3.2 standard deviations.”

• cites Gröblacher et al. (2007) sourceNot checked — no full text of the source

A proposal by Klyachko (2008) for a single qutrit contextuality experiment was experimentally realized for photons in superposition of three modes (Lapkiewicz, 2011).

“Most recently, a proposal by Klyachko (2008) for a single qutrit contextuality experiment involving only six different measurements was experimentally realized for photons prepared in superposition of three modes (Lapkiewicz, 2011).”

• cites Klyachko (2008)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Lapkiewicz, 2011Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Inequality (69) was violated in an experiment by Amselem et al. (2009).

“This inequality was violated in an experiment by Amselem et al. (2009), where the chosen observables had the form”

• cites Amselem et al. (2009) sourceNot checked — no full text of the source

Schuck et al. (2006) realized a complete linear-optical Bell-state analyzer able to distinguish all four Bell states provided the initial pair is hyper-entangled.

“Schuck et al. (2006) realized a complete linear-optical Bell-state analyzer which is able to distinguish all four Bell states , provided the initial pair is entangled in two degrees of freedom, called hyper-entangled, here polarization entanglement plus the intrinsic time-energy correlation in SPDC (Kwiat and Weinfurter (1998)).”

• cites Schuck et al. (2006) sourceNot checked — no full text of the source
• cites Kwiat and Weinfurter (1998) sourceNot checked — no full text of the source

The first experimental realization of quantum dense coding was reported by the Innsbruck group (Mattle et al., 1996).

“The first experimental realization of quantum dense coding was reported by the Innsbruck group (Mattle et al., 1996).”

• cites Mattle et al., 1996 sourceNot checked — no full text of the source

For each initial mixed state the purification scheme worked.

“For each initial mixed state the purification scheme worked.”

• cites Pan et al. (2003b)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

An experimental demonstration of the entanglement purification scheme has been reported by Pan et al. (2003b).

“An experimental demonstration of the entanglement purification scheme has been reported by Pan et al. (2003b).”

• cites Pan et al. (2003b)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The measured entanglement fidelity after purification for the first experiment was F=0.92±0.01.

“The measured results in Fig. 30(c) and Fig. 30(d) show a significant improvement of entanglement fidelity to the value of F = 0.92 ± 0.01 for the purified subensemble.”

• cites Pan et al. (2003b)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

In the first purification experiment, a mixed state with fidelity F=0.75 was prepared.

“In the first purification experiment, a mixed state as in (82) with fidelity of F = 0.75 was prepared.”

• cites Pan et al. (2003b)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The total channel loss of 64 dB corresponded to the estimated attenuation regime for a two-photon satellite quantum communication scenario.

“The total channel loss of 64 dB corresponded to the estimated attenuation regime for a two-photon satellite quantum communication scenario.”

Another weakness of this experiment is that due to double pairs emission events in SPDC, a phase stabilization was required.

“another weakness of this experiment is that due to double pairs emission events in SPDC, a phase stabilization was required.”

• AssumptionRests on an assumption: The probabilistic nature of SPDC causes two photon pairs to be emitted into one mode pair with a probability of the same order of magnitude.

The final photon states were found to preserve excellent, noise-limited fidelity, even though they experienced extreme attenuation due to mainly turbulent atmospheric effects.

“The final photon states were found to preserve excellent, noise-limited fidelity, even though they experienced extreme attenuation due to mainly turbulent atmospheric effects.”

• cites Ursin et al., 2007 sourceNot checked — no full text of the source
• cites Fedrizzi et al., 2009 sourceNot checked — no full text of the source

The photon-pair flight time of ~0.5 ms represents the longest lifetime of photonic Bell states reported so far.

“The photon-pair flight time of ∼ 0.5 ms represents the longest lifetime of photonic Bell states reported so far.”

The entanglement of the received two-photon states was confirmed by violating the CHSH inequality by more than 5 standard deviations.

“The entanglement of the received two-photon states was confirmed by violating the CHSH inequality by more than 5 standard deviations.”

• cites Ursin et al., 2007 sourceNot checked — no full text of the source
• cites Fedrizzi et al., 2009 sourceNot checked — no full text of the source

The resulting projected state of the ensembles L and R is nearly maximally entangled.

“The resulting projected state of the ensembles L and R is nearly maximally entangled.”

• cites DLCZ, 2001, 2002Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Crespi et al. (2011) reported the first probabilistic logic gates on integrated circuits also for polarization qubits.

“Crespi et al. (2011) reported the first probabilistic logic gates on integrated circuits also for polarization qubits.”

• cites Crespi et al., 2011Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Few-photon cluster states were created in several recent experiments (Zhang et al., 2006a; Walther et al., 2005b; Kiesel et al., 2005a; Lu et al., 2007; Tokunaga et al., 2008).

“Few-photon cluster states were created in several recent experiments (Zhang et al., 2006a;Walther et al., 2005b;Kiesel et al., 2005a;Lu et al., 2007;Tokunaga et al., 2008).”

• cites Zhang et al., 2006aNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Walther et al., 2005bNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Kiesel et al., 2005aNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Biexciton radiative decay in a self-assembled quantum dot was demonstrated as a triggered entangled photon source in Stevenson et al. (2006).

“This was demonstrated in Stevenson et al. (2006).”

• cites Stevenson et al. (2006) sourceNot checked — no full text of the source

In the experiment of Walther et al. (2005b), a four-photon polarization-entangled cluster state was created via SPDC with fidelity F = 0.63 ± 0.02.

“In the experiment of Walther et al. (2005b), via a SPDC a four-photon polarization-entangled cluster state was created (with a fidelity of F = 0.63 ± 0.02)”

• cites Walther et al. (2005b)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

In long-lived quantum memory based on atomic ensembles, storage times were reported to reach up to 8 ms.

“In the case of a long-lived quantum memory based on atomic ensembles storage times were reported to reach up to 8 ms (Zhao et al. 2009), with potential improvements when using cavities.”

• cites Zhao et al. 2009Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

With continuous variable encoding, Aoki et al. (2009) realized a nine-qubit Shor's code able to correct an arbitrary single-qubit error.

“With a continuous variable encoding, Aoki et al. (2009) realized a nine-qubit Shor's code, which is able to correct an arbitrary single-qubit error.”

• cites Aoki et al. (2009) sourceNot checked — no full text of the source

Single photons and entangled photons have been generated from self-assembled quantum dots embedded in a microcavity, with extraction efficiency up to 80%.

“For instance, single photons and entangled photons have been generated from self-assembled quantum dots embedded in a microcavity, with extraction efficiency up to 80% (Dousse et al. 2010).”

• cites Dousse et al. 2010 sourceNot checked — no full text of the source

The probability of joint detection of two SPDC photons depends on the difference of the detection times.

“i.e., the probability depends on the difference of the detection times.”

• AssumptionThe authors state this as an assumption: The detectors see only photons of a specified direction of propagation, which is a good assumption if the detectors are far from the crystal and the apertures are narrow.
• AssumptionThe authors state this as an assumption: The maxima of the filter-detector response functions agree with the frequencies given by the phase-matching conditions.

For Gaussian filters, as σ → ∞ the detection probability approaches δ(t - t'), giving perfect time correlation.

“As σ → ∞ the expression (A18) approaches δ(t -t ).We have a perfect time correlation.”

• AssumptionThe authors state this as an assumption: The maxima of the filter-detector response functions agree with the frequencies given by the phase-matching conditions.

The GHZ theorem shows that the concept of EPR's elements of reality is self-contradictory.

“The Greenberger-Horne-Zeilinger (GHZ) theorem (Greenberger, Horne, and Zeilinger, 1989;Greenberger et al., 1990;Mermin, 1990a) showed that the concept of EPR's elements of reality is selfcontradictory.”

• cites Greenberger, Horne, and Zeilinger, 1989Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Greenberger et al., 1990 sourceNot checked — no full text of the source
• cites Mermin, 1990aNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Interference patterns involving single photons or light intensity do not reveal strictly non-classical phenomena.

“The interference patterns involving single photons or, equally well, the light intensity does not reveal strictly non-classical phenomena.”

Bell's inequalities reveal that two-particle correlations for the two spin-1/2 singlet disagree with any local realistic model.

“Bell (1964) derived his remarkable inequalities which revealed that two-particle correlations for the two spin-1 2 singlet disagree with any local realistic model.”

• cites Bell (1964) sourceNot checked — no full text of the source

KLM showed that scalable quantum computation is in principle possible with linear optics, single photon sources and detectors.

“Remarkably, by exploiting the nonlinearity induced by measurement, Knill, Laflamme, and Milburn (KLM) showed that scalable quantum computation is in principle possible with linear optics, single photon sources and detectors.”

• cites Knill, Laflamme, and Milburn (KLM)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

There are situations for which local realism and quantum mechanics make completely opposite predictions, even for perfectly correlated results.

“That is, there are situations for which local realism and quantum mechanics make completely opposite predictions, even for perfectly correlated results -which were the starting point of the EPR argumentation.”

• cites Greenberger, Horne, and Zeilinger, 1989Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Greenberger et al., 1990 sourceNot checked — no full text of the source
• cites Mermin, 1990aNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

A quantum repeater would allow in principle an efficient long-distance high-fidelity transmission of entanglement.

“A quantum repeater (Briegel et al. 1998) would allow in principle an efficient long distance high-fidelity transmission of entanglement.”

• cites Briegel et al. 1998 sourceNot checked — no full text of the source

For fields with fluctuating phase differences, the visibility of intensity correlations is never full, maximally 50%.

“Thus, the visibility of intensity correlations for fields with fluctuating phase differences is never full, maximally 50%.”

There is no bound on visibility in the quantum case.

“As we shall see, there is no bound on visibility in the quantum case.”

By simultaneously monitoring detectors on both sides of the two-photon EPR interferometer while varying the phase shifts, interference fringes are observed as sinusoidal terms.

“Thus, by simultaneously monitoring the detectors on both sides of the interferometer, while varying the phase shifts φ 1 andφ 2 , the interference fringes will be observed as shown by the sinusoidal terms.”

• AssumptionThe authors state this as an assumption: A central source emits two photons in the entangled state |ψ12 = 1/√2(|a1|a2 + |b1|b2).

The distinguishability was tuned in the original Hong-Ou-Mandel experiments with the temporal delay between the two photons.

“The distinguishability was tuned in the original experiments with the temporal delay between the two photons (Fig. 5(b)).”

• cites Hong, Ou and Mandel, 1987 sourceNot checked — no full text of the source

Rarity et al. observed interference between independent photons, one a triggered single photon from a down-converted pair and the other in an attenuated beam from pumping laser light.

“Rarity et al. (1996) observed the interference between independent photons, one of which was a triggered single photon from a down-converted pair, and the other one was in an attenuated beam 17 derived from the pumping laser light.”

• cites Rarity et al. (1996)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The Hong-Ou-Mandel effect depends critically on the distinguishability of photons.

“Therefore, the Hong-Ou-Mandel effect depends critically on the distinguishability of photons.”

The three-fold coincidence rates in the GHZ interferometer display sinusoidal oscillations depending on the sum φ1+φ2+φ3.

“The three-fold coincidence rates given in Eq. ( 20) display sinusoidal oscillations depending on the sum φ 1 +φ 2 +φ 3 .”

No single-particle fringes can be observed in the three-particle GHZ interferometer.

“Finally, a similar argument shows that, of course, no single-particle fringes can be observed.”

Quantum interference may occur entirely due to indistinguishability of particles.

“Quantum interferencefoot_6 may occur also entirely due to indistinguishability of particles.”

Hong-Ou-Mandel interference provides a powerful tool to estimate the degree of indistinguishability of two separately emitted photons.

“The Hong-Ou-Mandel interference provides a powerful tool to estimate the degree of indistinguishability of two separately emitted photons.”

Shih and Alley and Ou and Mandel were the first to achieve effects characteristic for a maximally entangled polarization state.

“Shih and Alley (1988), and Ou and Mandel (1988a) were the first to achieve this.”

• cites Shih and Alley (1988) sourceNot checked — no full text of the source
• cites Ou and Mandel (1988a)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

A coincident detection at the two outputs c and d can occur only due to the entangled |ψ−cd component of the total state.

“A coincident detection at the two outputs, c and d, can occur only due the entangled |ψ - cd component of the total state.”

The minus sign in |Ψ− leads to a cancellation of the photon bunching amplitudes, that is to a fermionic-like behavior.

“The minus sign in |Ψ -leads to a cancellation of the photon bunching amplitudes, that is to a”

Such interferometers play a crucial role in creation, manipulation and projection of various multiphoton entangled states.

“Such interferometers play a crucial role in creation, manipulation and projection of various multiphoton entangled states.”

A deterministic CNOT gate would make Bell-state measurement possible.

“A deterministic CNOT gate 19 would make such a the Bellstate measurement possible.”

The experimental results confirmed that the quantum interference can be revived.

“The experimental results confirmed that the quantum interference can be indeed revived.”

• cites Pittman et al. (1996) sourceNot checked — no full text of the source

One can construct a GHZ-state analyzer with which one can identify two out of the 2^N maximally entangled GHZ states.

“One can construct a GHZ-state analyzer (Pan and Zeilinger, 1998), with which one can identify two out of the 2 N maximally entangled GHZ states.”

• cites Pan and Zeilinger, 1998 sourceNot checked — no full text of the source

CNOT gates are difficult to realize with linear optics and single photons.

“However, CNOT gates are difficult to realize with linear optics and single photons (see section VII).”

A modified Bell-state analyzer introduced by Pan and Zeilinger can directly be generalized to the N-particle case.

“A modified version of a Bell-state analyzer, which can directly be generalized to the N -particle case was introduced by Pan and Zeilinger (1998).”

• cites Pan and Zeilinger (1998) sourceNot checked — no full text of the source

This makes Hong-Ou-Mandel interferometers very stable.

“This makes the Hong-Ou-Mandel interferometers very stable.”

By exploiting quantum interference effects due to the bosonic nature of photons, photonic Bell-state and GHZ-state analyzers can be realized in a probabilistic way.

“Nevertheless, by exploiting quantum interference effects due to the bosonic nature of photons discussed in the above section, photonic Bell-state and GHZ-state analyzers can be realized in a probabilistic way.”

Later experiments evolved into observations of a Hong-Ou-Mandel dip for photons originating from two sources that were progressively more independent of each other.

“Later experiments of this kind evolved into observations of a Hong-Ou-Mandel dip for photons originating from two sources, which were progressively more and more independent of each other.”

This kind of post-selected entanglement was used to violate Bell's inequality in experiments of Shih and Alley and Ou and Mandel.

“This kind of post-selected entanglement was used to violate Bell's inequalityfoot_8 , both in experiments of Shih and Alley (1988), and Ou and Mandel (1988a).”

• cites Shih and Alley (1988) sourceNot checked — no full text of the source
• cites Ou and Mandel (1988a)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

By taking advantage of hyper-entanglement, one can implement a complete Bell-state analysis.

“Finally, let us mention that by taking the advantage of the properties of hyper-entanglement one can implement a complete Bell-state analysis (Kwiat and Weinfurter, 1998;Walborn et al., 2003b).”

• cites Kwiat and Weinfurter, 1998 sourceNot checked — no full text of the source
• cites Walborn et al., 2003bNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The origin of Hong-Ou-Mandel interference is due to the indistinguishability of two-photon amplitudes describing the various alternatives leading to a coincidence count.

“However, it is important to note that essentially, the origin of this interference is due to the indistinguishability of two-photon amplitudes describing the various alternatives leading to a coincidence count.”

Bell state measurement schemes can be generalized to the N-particle case.

“Bell state measurement schemes can be generalized to the N -particle case.”

The GHZ analyzer setup with three photons entering modes a, b and c and polarization beam splitters transmitting H and reflecting V can filter out two GHZ states |Φ±0 = (|HHH ± |VVV) from the eight.

“Consider now the setup of Fig. 7 and suppose that three photons enter the GHZ analyzer by modes a, b and c.The polarization beam splitters transmit H and reflect V polarizations, thus a coincidence detection at the three outputs can only originate from either the case that all photons are transmitted (this corresponds to the input state |H |H |H ) or all reflected (|V |V |V ).The two cases are ful”

Kim et al. (2006) proposed and implemented a polarization Sagnac interferometric configuration with bidirectional pumping of a type-II phase-matched periodically poled KTiOPO4 crystal.

“Kim et al. (2006) proposed and implemented a polarization Sagnac interferometric configuration with bidirectional pumping of a type-II phase-matched periodically poled KTiOPO 4 (PPKTP) crystal.”

• cites Kim et al. (2006) sourceNot checked — no full text of the source

After compensation, interference occurs pairwise between processes in which the photon pair is created at distances ±x from the middle of the crystal.

“After the compensation, interference occurs pairwise between processes in which the photon pair is created at distances ±x from the middle of the crystal.”

If d is larger than the coherence width, the terms can become partially labeled by their spatial location.

“Similarly, if d is larger than the coherence width, the terms can become partially labeled by their spatial location.”

The compensation method also provides reasonable compensation for the transverse walk-off effect.

“The method also provides reasonable compensation for the transverse walk-off effect.”

Early Bell-test experiments used entangled photons from atomic cascades.

“The early Bell-test experiments used entangled photons from atomic cascades, see Clauser and Shimony (1978).”

• cites Clauser and Shimony (1978) sourceNot checked — no full text of the source

Because the pulses are separated on the order of a few nanoseconds, much shorter than the timescale of any phase drifts in the fiber, the drifts do not affect the quality of entanglement.

“Note that because the pulses are only separated on the order of a few nanoseconds, and this is much shorter than the timescale of any phase drifts in the fiber, the drifts do not affect the quality of entanglement.”

It is possible to produce entangled states involving large numbers of photons, approaching the macroscopic domain, via twin-beam multi-photon entanglement.

“It is also possible to produce entangled states involving large numbers of photons, approaching the macroscopic domain.Such entanglement is related to experiments on twin beams (Smithey et al., 1992) and could be called a twin-beam multi-photon entanglement.”

• cites Smithey et al., 1992 sourceNot checked — no full text of the source

Polarization-path entanglement can be generated by a double pass of a pump laser through a BBO crystal.

“Polarization-path entanglement: Polarization-path entanglement can be generated by a double pass of a pump laser through a BBO crystal (Chen et al., 2003).”

• cites Chen et al., 2003 sourceNot checked — no full text of the source

Due to phase-matching, idler and signal photons of given frequencies are correlated in emission directions.

“Due to the phase-matching relation, idler and signal photons of given frequencies are correlated in emission directions.”

An advantage of time-bin entanglement is that it is insensitive to polarization fluctuations.

“An advantage of the time-bin entanglement is that it is insensitive to polarization fluctuations.”

Using reference laser pulses to actively lock the phase, time-bin entanglement can be robustly distributed over long distances in optical fibers.

“Using reference laser pulses to actively lock the phase, it can be robustly distributed over long distances in optical fibers.”

A more robust distribution of hyper-entanglement is possible with photon pairs entangled both in time (time-bin) and in polarization.

“Polarization-time entanglement: A more robust distribution of hyper-entanglement is possible with photon pairs which are entangled both in time (i.e., time-bin entanglement) and in polarization (Genovese and Novero, 2002;Schuck et al., 2006;Chen et al., 2006d).”

• cites Genovese and Novero, 2002 sourceNot checked — no full text of the source
• cites Schuck et al., 2006 sourceNot checked — no full text of the source
• cites Chen et al., 2006dNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The visibility of four-particle fringes can be impaired by the statistical properties of the emission process.

“The influence of photon statistics.-Thevisibility of the four-particle fringes in the set-up of Fig. 14 can be impaired by the statistical properties of the emission process.”

To ensure that the source of photons is unknowable, the photons should be detected behind a narrow band filtering system whose inverse bandwidth (coherence time) clearly exceeds the pulse duration τ.

“To ensure that the source of photons is unknowable, the (initially spectrally broadband) photons should be detected behind a narrow band filtering system (to be called later simply a filter) whose inverse of the bandwidth (coherence time) clearly exceeds the pulse duration τ (e.g., by an order of magnitude).”

Depending on phase shifts, detection at c(+) and d(+) acts like a Bell-state measurement, projecting the two photons into state 1/√2(|c|d + e^{i(φc+φd)}|c'|d').

“Note that depending on the phase shifts the detection at, e.g., c(+) and d(+), acts like a Bell-state measurement, projecting the two photons into the state 1 √ 2 |c |d + e i(φc+φ d ) |c |d .”

The original motivation to observe entanglement of more than two particles stems from the observation by GHZ that three-particle entanglement leads to a dramatic conflict between local realism and EPR's ideas with predictions of quantum mechanics.

“The original motivation to observe entanglement of more than two particles, with measurements on the particles performed at spatially separated stations, stems from the observation by GHZ that three-particle entanglement leads to a dramatic conflict between local realism and EPR's ideas with predictions of quantum mechanics (GHZ, 1989;Greenberger et al., 1990;Mermin, 1990a), see section V.B.1.”

• cites GHZ, 1989Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Greenberger et al., 1990 sourceNot checked — no full text of the source
• cites Mermin, 1990aNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The basic principle of stimulated entanglement creation was first experimentally demonstrated in the few-photon regime.

“The basic principle of 22 It is interesting to note that this simplified Hamiltonian H sim SPDC and the state generated thereby [Eq.( 42)] are closely related to squeezing and continuous-variable entanglement, see Braunstein and van Loock (2005) for a review.a stimulated entanglement creation was first experimentally demonstrated in the few-photon regime (Lamas-Linares et al., 2001).”

• cites Lamas-Linares et al., 2001 sourceNot checked — no full text of the source

Twin-beam multi-photon entanglement can be generated via standard SPDC with a strong pump pulse.

“The twin-beam multi-photon entanglement can be generated via standard SPDC, but with a strong pump pulse.”

By going to higher values of τ, bipartite entangled states consisting of large numbers of photons are generated.

“By going to higher values, bipartite entangled states constituting of large numbers of photons are generated.”

To have high visibility, the ratio of the probability of each pulse to produce a single down converted pair to the probability of not producing anything must be less than 14%.

“Thus, to have high visibility the ratio of the probability of each pulse to produce a single down converted pair to the probability of not producing anything must be less than 14%.”

Yurke and Stoler (1992a, 1992b) showed that in theory multiparticle entanglement effects should be in principle observable for particles originating from independent sources.

“Yurke andStoler (1992a, 1992b) showed that in theory multiparticle entanglement effects should be in principle observable for particles originating from independent sources.”

• cites Yurke and Stoler (1992a, 1992b)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Simultaneous firings of four detectors can exhibit interference effects provided the two contributing processes are indistinguishable.

“Simultaneous firings of the four detectors can exhibit interference effects provided the two contributing processes are indistinguishable: detection of the particles from source A in d(+) and a(+), and detection of the particles from B in c(+) and b(+); detection of the particles from source A in c(+) and a(+), and detection of the particles form B in d(+) and b(+).”

In the four-particle interference setup, if it cannot be determined which source produced the particle activating detectors d(+) and c(+), then four-particle interference effects may occur.

“If it cannot be determined, even in principle, which source produced the particle which activated the detectors, say d(+) and c(+), then four-particle interference effects may occur.”

The state of the four particles after passage of photons 2 and 3 via PBS1, provided the photons are indistinguishable, reads as Eq. (49).

“The state of the four particles, after the passage of 2 and 3 via PBS1, and provided the photons are indistinguishable”

• AssumptionThe authors state this as an assumption: Sources A and B simultaneously emit a photon pair each.
• AssumptionThe authors state this as an assumption: Pairs in beams x, y (1, 3, and 2, 4) are in identical polarization states 1/√2(|Hx,Hy> + |Vx,Vy>).
• AssumptionThe authors state this as an assumption: The photons are indistinguishable.

These processes erase the possibility of distinguishing the photons from different pairs by their arrival time or spatial modes.

“These processes erase the possibility of distinguishing the photons from different pairs by their arrival time or spatial modes”

Five-photon entanglement was demonstrated by interfering a four-photon GHZ state with a source of pseudo-single photons from an attenuated laser light.

“Later on five-photon entanglement was demonstrated by interfering a four-photon GHZ state (Pan et al., 2001a) with a source of pseudo-single photons from an attenuated laser light (Zhao et al. 2004).”

• cites Pan et al., 2001aNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Zhao et al. 2004 sourceNot checked — no full text of the source

The scheme in Fig. 15 also allows one to generate unconditional three-particle GHZ states via a method based on entangled entanglement.

“The scheme in Fig. 15 also allows one to generate unconditional three-particle GHZ statesfoot_12 via a method based on the notion of entangled entanglement (Krenn and Zeilinger, 1996).”

• cites Krenn and Zeilinger, 1996 sourceNot checked — no full text of the source

Four-fold coincidences can reveal four-particle GHZ correlations.

“Therefore, four-fold coincidences can reveal four-particle GHZ correlations.”

The number of GHZ-like entangled qubits climbed up to ten by entangling both the polarization and momentum degrees of freedom of five photons.

“The number of GHZ-likeentangled qubits climbed up to ten by entangling both the polarization and momentum degrees of freedom of five photons (Gao et al. 2010a).”

• cites Gao et al. 2010aNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The dominance of the two terms is a necessary but not yet sufficient condition for demonstrating a GHZ entanglement, as there could be just a statistical mixture of the two states.

“The dominance of the two terms is a necessary but not yet sufficient condition for demonstrating a GHZ entanglement, as there could, in principle, be just a statistical mixture of the two states.”

The resulting counts originate from a superposition between |H1H2V3> and |V1V2H3>, that is, the three-photon GHZ state.

“Thus the resulting counts originate from a superposition between |H 1 H 2 V 3 and |V 1 V 2 H 3 , that is, the three-photon GHZ state:”

Only the superposition |H1,H2,H3,H4> + |V1,V2,V3,V4>, which is a GHZ state, leads to four-fold coincidence.

“Only the superposition |H 1 , H 2 , H 3 , H 4 +|V 1 , V 2 , V 3 , V 4 , which is a GHZ state, leads to four-fold coincidence.”

The generic idea of observing photonic GHZ entanglement, later put into practice, was given by Zeilinger et al. (1997).

“The generic idea of observing photonic GHZ entanglement, later put into practice, was given by Zeilinger et al. (1997)”

• cites Zeilinger et al. (1997) sourceNot checked — no full text of the source

With further improvements in high-power pump source and photon collection efficiency, GHZ entanglement among six photons and eight photons was also observed.

“With further improvements in high-power pump source and photon collection efficiency, GHZ entanglement among six photons (Lu et al. (2007)) and eight photons (Yao et al. (2011)) were also observed.”

• cites Lu et al. (2007)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Yao et al. (2011)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Engineering phase-matching conditions in photonic crystal fibers allowed creation of photon pairs at 597 nm and 860 nm in an intrinsically factorable state of frequencies.

“Engineering of the phase matching conditions in the fibers allowed creation of photon pairs at 597 nm and 860 nm in an intrinsically factorable state of frequencies.”

• cites Grice, U'Ren, and Walmsley (2001 2009) sourceNot checked — no full text of the source
• cites Fulconis et al. 2007 sourceNot checked — no full text of the source

Such a photon-pair source has almost no spectral correlations.

“Thus there were almost no spectral correlations.”

A GHZ-type entanglement among three spatially distributed photons, using the above methods of entanglement construction, was first observed by Bouwmeester et al. (1999a).

“A GHZ-type entanglement among three spatially distributed photons, using the above methods of entanglement construction, was first observed by Bouwmeester et”

• cites Bouwmeester et al. (1999a)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

By further developing the ideas of Yurke and Stoler (1992a), Żukowski (2000) showed that the above procedure indeed permits a valid GHZ test.

“By further developing the ideas of Yurke and Stoler (1992a), Żukowski (2000) showed that the above procedure indeed permits a valid GHZ test.”

• cites Yurke and Stoler (1992a)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Żukowski (2000) sourceNot checked — no full text of the source

Femtosecond pulse pumped SPDC shows relatively poor quantum interference visibilities.

“Unfortunately, femtosecond pulse pumped SPDC shows relatively poor quantum interference visibilities (Keller and Rubin, 1997).”

• cites Keller and Rubin, 1997 sourceNot checked — no full text of the source

All the above states can be observed also by a single, tunable setup.

“Finally, all the above states can be observed also by a single, tunable setup.”

• cites Wieczorek et al., 2008 sourceNot checked — no full text of the source
• cites Lanyon et al., 2009bNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The symmetric Dicke state can be obtained by using collinear type-II SPDC and splitting the four (six) photons into four (six) output modes.

“Another important multipartite entangled state, the symmetric Dicke state, can be obtained by using collinear type-II SPDC and splitting the four (six) photons into four (six) output modes (Kiesel et al. 2007;Wieczorek et al. 2009a;Prevedel et al. 2009).”

• cites Kiesel et al. 2007 sourceNot checked — no full text of the source
• cites Wieczorek et al. 2009aNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Prevedel et al. 2009 sourceNot checked — no full text of the source

Detection of genuine multiparticle entanglement with appropriate witnesses usually requires an experimental effort that increases only polynomially with the number of qubits.

“Detection of a genuine multiparticle entanglement with appropriate witnesses usually requires an experimental effort that increases only polynomially with the number of qubits.”

This enables a decoherence-free communication of a qubit encoded in four-photon states.

“This enables a decoherence free communication of a qubit encoded in four-photon states (Bourennane et al. 2004b).”

• cites Bourennane et al. 2004bNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The GHZ theorem was the initial motivation for experimental efforts to produce entanglement of more than two particles.

“This result was the initial motivation for experimental efforts to produce entanglement of more than two particles.”

Violations of Bell's inequality also occur for a wide range of mixed states.

“This is also the case for a wide range of mixed states (Horodecki, Horodecki and Horodecki, 1995).”

• cites Horodecki, Horodecki and Horodecki, 1995 sourceNot checked — no full text of the source

The high symmetry of the Dicke state makes it an ideal resource to synthesize a number of different multipartite entangled states by projection.

“The high symmetry of this state makes it an ideal resource to synthesize a number of different multipartite entangled states by projection as shown above for the GHZ-states (Wieczorek et al. 2009b) but also for entanglement enhanced sub shot-noise measurements.”

• cites Wieczorek et al. 2009bNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Quantum information processing is able to perform tasks impossible with classical methods.

“That is why quantum information processing is able to perform tasks impossible with the classical methods.”

Because any classical information processing or communication has a local realistic model, theorems and experiments that reveal phenomena impossible to describe by such formalism indicate the existence of a different method of processing and transferring information.

“Note that, as any classical information processing, or communication, has a local realistic model, the theorems and experiments that reveal phenomena impossible to describe by such formalism, clearly indicate existence of a different method of processing and transferring information.”

Bell (1964) proved that there is a deep conflict between quantum mechanics and local realistic theories.

“Bell (1964) proved, that despite the hopes of Einstein at al., there is a deep conflict between quantum mechanics and local realistic theories.”

• cites Bell (1964) sourceNot checked — no full text of the source

For the spin-1/2 singlet, S_Bell can acquire the maximal value 2√2 allowed by quantum mechanics, known as the Cirel'son bound.

“For example the predictions for the spin-1/2 singlet give correlations for which S Bell can acquire the maximal value 2 √ 2 allowed by quantum mechanics, known as the Cirel'son 37 bound (Cirel'son, 1980;Landau, 1987).”

• cites Cirel'son, 1980 sourceNot checked — no full text of the source
• cites Landau, 1987 sourceNot checked — no full text of the source

Some quantum processes involving entangled states violate the Bell inequality.

“Some quantum processes involving entangled states violate the inequality.”

The conflict between quantum mechanics and local realistic theories leads to drastically different predictions concerning two-particle interference phenomena.

“but one which straightforwardly leads to drastically different predictions concerning two-particle interference phenomena.”

After beam-splitting of these two modes into four modes and conditioning on detection of one photon in each of the four modes, the observed state can be written as a superposition of a four-photon GHZ state and a product of two EPR pairs.

“After beam-splitting of these two modes into four modes and again conditioning on detection of one photon in each of the four modes, the observed state can be written as a superposition of a four-photon GHZ state and a product of two EPR pairs.”

The method of entanglement witnesses allows detection of entanglement in a given state.

“Themethod of entanglement witnesses (Horodecki et al., 1996;Terhal, 2000;Lewenstein et al., 2000;Bruß et al., 2002) allows to detect the entanglement in a given state.”

• cites Horodecki et al., 1996Not checked — no full text of the source
• cites Terhal, 2000 sourceNot checked — no full text of the source
• cites Lewenstein et al., 2000 sourceNot checked — no full text of the source

Predictions for any pure, nonfactorisable two-system state lead to violations of Bell's inequality.

“In fact, predictions for any pure, nonfactorisable (i.e., not necessarily maximally entangled) two-system state lead to violations (Gisin and Peres, 1992).”

• cites Gisin and Peres, 1992 sourceNot checked — no full text of the source

Equation (56) indicates that one can predict with certainty the result of measuring the observable pertaining to one of the particles by choosing to measure suitable observables for the other two.

“Equation ( 56) indicates that we can predict with certainty the result of measuring the observable pertaining to one of the particles (say, C) by choosing to measure suitable observables for the other two.”

Although the settings in Aspect's experiments were quickly varying, they were derived from independent function generators with certain correlation times and thus not fully random as assumed in the derivation of Bell inequalities.

“Although the settings were quickly varying, they were derived from independent function generators with certain correlation times and thus not fully random as it is assumed in the derivation of Bell inequalities.”

If φa + φb + φc = π/2 + kπ (where k is an integer), one has perfect correlations and perfect predictability of the common measurement result.

“Therefore, if φ a + φ b + φ c = π/2 + kπ (where k is an integer), one has perfect correlations and perfect predictability of the common measurement result.”

One can find many ad hoc local realistic models that violate the fair sampling assumption.

“One can find many ad hoc local realistic models that violate it, see e.g.Cabello and Santos (1996).”

• cites Cabello and Santos (1996)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• AssumptionThe authors state this as an assumption: Fair sampling: whether a detector registers a particle or not is statistically independent of all other processes in the experiment.

This is in full contradiction with the quantum mechanical prediction A_n(π/2)B_n(π/2)C_n(π/2) = -1.

“This, however, is in a full contradiction with the quantum mechanical prediction obtained from (55) which reads:”

Independent randomness was experimentally realized by Weihs et al. (1998).

“This independent randomness was experimentally realized by Weihs et al. (1998).”

• cites Weihs et al. (1998) sourceNot checked — no full text of the source

For φa = π/2, φb = 0, and φc = 0, whatever the results of local measurements for the n-th triple represented by the GHZ state, they satisfy A_n(π/2)B_n(0)C_n(0) = 1.

“For instance, for φ a = π/2, φ b = 0 and φ c = 0, whatever may be the results of local measurements of the observables for, say, the particles belonging to the n-th triple represented by the GHZ quantum state, they have to satisfy”

In a local realistic model, from Eqs. (56)-(58) one can deduce A_n(π/2)B_n(π/2)C_n(π/2) = 1.

“Now, in a local realistic model, from these results we can deduce a further correlation by simply multiplying Eqs.(56-58).Since X n (0) 2 = +1, regardless of whether X n (0) = +1 or -1, we obtain”

The noise resistance of violations of such inequalities by N-qubit GHZ states grows exponentially with N.

“The noise resistance 42 of violations of such inequalities by N -qubit GHZ states is growing exponentially with N .”

The reasoning leading to inequality (62) is trivially generalizable to an arbitrary number of parties.

“The reasoning is trivially generalizable to an arbitrary number of parties.”

The detection loophole was first closed in an ion-trap experiment by Rowe et al. (2001) utilizing the nearly perfect detection efficiency of fluorescence detection of single ions.

“It was first closed in an ion-trap experiment by Rowe et al. (2001) utilizing the nearly perfect detection efficiency of fluorescence detection of single ions.”

• cites Rowe et al. (2001) sourceNot checked — no full text of the source

The later experiment and the subsequent ones were done with moving reference frames.

“The later experiment and the subsequent ones (Stefanov et al., 2002;2003) were done with moving reference frames.”

• cites Stefanov et al., 2002 sourceNot checked — no full text of the source
• cites Stefanov et al., 2003 sourceNot checked — no full text of the source

Inequality (62) is the necessary and sufficient condition for the correlation functions E(A_k, B_l, C_m) involved in it to have a local realistic model.

“which is the necessary and sufficient condition for correlation functions involved in it, E(A k , B l , C m ), to have a local realistic model [for proofs see Werner and Wolf (2001), Żukowski and Brukner (2002)].”

• cites Werner and Wolf (2001) sourceNot checked — no full text of the source
• cites Żukowski and Brukner (2002) sourceNot checked — no full text of the source

Instead of purely statistical reasoning for deriving Bell's inequality, one can follow the spirit of the EPR paper and first define elements of reality based on specific perfect correlations of the entangled state, then show contradicting predictions between local realistic theories and quantum mechanics.

“More remarkably, instead of purely statistical reasoning for deriving Bell's inequality, one can fully follow the spirit of the EPR paper, and first try to define “elements of reality” based on specific perfect correlations of the entangled state.”

Bell experiments were also performed using fiber-based entangled photon sources, from which two photons were distributed over a distance of more than 10 km apart.

“The Bell experiments were also performed using fiber-based entangled photon sources, from which two photons were distributed over a distance of more than 10 km apart, see, e.g., (Tittel et al., 1998;Zbinden et al., 2001).”

• cites Tittel et al., 1998 sourceNot checked — no full text of the source
• cites Zbinden et al., 2001 sourceNot checked — no full text of the source

Hence, in a local realistic model the values X_n(φ) for the angles specified in the equality are EPR's elements of reality.

“Hence, in a local realistic model the values X n (φ) for the angles specified in the equality are EPR's elements of reality.”

• AssumptionThe authors state this as an assumption: Locality assumption forces X_n(φ) to depend solely on the local phase.

The efficiency loophole emerges due to low collection and detection efficiency of the particles.

“The efficiency loophole emerges due to low collection and detection efficiency of the particles.”

Non-classicality of GHZ correlations can be significant even for many particles.

“Thus non-classicality of the GHZ correlations can be significant even for many particles (Mermin, 1990b;Klyshko, 1993;Roy and Singh, 1991;Żukowski 1993;Gisin and Bechmann-Pasquinucci, 1998).”

• cites Mermin, 1990bNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Klyshko, 1993Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Roy and Singh, 1991 sourceNot checked — no full text of the source

The falsification of Bell's inequalities by those experiments was up to certain loopholes due to experimental imperfections, which still allow local realistic models for the measurement results.

“However, this falsification was up to certain loopholes, which are due to experimental imperfections, and still allow to build local realistic models for the measurement results obtained in the experiments.”

This corresponds to a violation of the local realistic threshold of 2 by 30 standard deviations.

“This corresponds to a violation of the local realistic threshold of 2 by 30 standard deviations.”

Eberhard (1993) lowered this threshold to 67% by employing Clauser and Horne (1974) inequalities, albeit for non-maximally entangled states only.

“Eberhard (1993) managed to lower this threshold to 67% by employing, effectively, the Clauser and Horne (1974) inequalities, albeit for non-maximally entangled states, only.”

• cites Eberhard (1993) sourceNot checked — no full text of the source
• cites Clauser and Horne (1974) sourceNot checked — no full text of the source

Violation of Bell's inequality was also demonstrated using Josephson phase qubits with deterministic entangled-state preparation and single-shot readout.

“Violation of Bell's inequality was also demonstrated using Josephson phase qubits with deterministic entangled-state preparation and single-shot readout (Ansmann, 2009).”

• cites Ansmann, 2009Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

If there are N > 2 maximally entangled quantum systems (qubits), and measurements are performed in N mutually spatially separated regions by N independent observers, the correlations violate bounds imposed by local realism much more strongly than in the two-particle case.

“If there are N > 2 maximally entangled quantum systems (qubits), and if measurements on them are performed in N mutually spatially separated regions by N independent observers, the correlations in such an experiment violate bounds imposed by local realism much stronger than in the two-particle case.”

For an efficiency lower than about 83%, one cannot derive a generalized CHSH-type inequality that is violated by quantum predictions.

“For an efficiency lower than about 83% one can show that one cannot derive a (generalized) CHSH-type inequality that is violated by quantum predictions.”

• cites Garg and Mermin, (1987) sourceNot checked — no full text of the source

Thus the EPR elements of reality program breaks down, because it leads to a 1 = -1 contradiction.

“Thus the EPR elements of reality program breaks down, because it leads to a 1 = -1 contradiction.”

In Hardy's thought experiment the two particles are nonmaximally entangled.

“A crucial distinguishing feature in Hardy's thought experiment is that the two particles are nonmaximally entangled.”

• cites Hardy 1993 sourceNot checked — no full text of the source

The GHZ reasoning can be reduced to a two-party case while retaining its all-versus-nothing feature.

“Interestingly, the GHZ reasoning can be reduced to a two-party (thus two space-like separated regions) case while its all-versus-nothing feature is still retained.”

Hardy's thought experiment provides an alternative way to demonstrate a violation of local realism without inequalities for two spin-half particles or photon polarizations.

“Hardy's thought experiment (Hardy 1993) provides an alternative way to demonstrate a violation of local realism without inequalities for two spin-half particles, or equivalently for polarizations of two photons.”

• cites Hardy 1993 sourceNot checked — no full text of the source

The three-photon GHZ state (63) satisfies the eigenequations (64).

“The GHZ state (63) satisfies the following eigenequations”

An all-versus-nothing violation of local realism can be shown for two-particle four-dimensional entangled systems.

“Later, after a considerable debate (Cabello, 2001a(Cabello, , 2001b(Cabello, , 2003;;Lvovsky, 2002;Marinatto, 2003;Chen et al., 2003), it was shown that an all versus nothing violation of local realism can be shown for two particle four-dimensional entangled systems.”

• cites Cabello, 2001aNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Lvovsky, 2002 sourceNot checked — no full text of the source
• cites Chen et al., 2003 sourceNot checked — no full text of the source

Leggett (2003) discussed a broad class of nonlocal hidden-variable theories that provide an explanation for all existing Bell-type experiments.

“In a novel approach, Leggett (2003) discussed a broad class of nonlocal hidden-variable theories, which, based on a very plausible type of realism, provide an explanation for all existing Bell-type experiments.”

• cites Leggett (2003) sourceNot checked — no full text of the source

The first laboratory test of the GHZ-type paradox was done by Pan et al. (2000).

“The first laboratory test of the GHZ-type paradox was done by Pan et al. (2000).”

• cites Pan et al. (2000) sourceNot checked — no full text of the source

A GHZ-type argument involving non-contextuality can be applied in a two-particle scheme by encoding three two-state quantum systems in distinct degrees of freedom of two photons.

“One option is to encode three two-state quantum systems in distinct degrees of freedom of only two photons.Thereby a GHZ-type argument, now also necessarily involving non-contextuality 44 , can be applied in this two-particle scheme (Zukowski 1991, Michler et al. 2000a).”

• cites Zukowski 1991 sourceNot checked — no full text of the source
• cites Michler et al. 2000aNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

If two indistinguishable photons enter a shared 50-50 beamsplitter from different input ports, they exit randomly via just one port.

“Instead of annihilation, they relied on the Hong-Ou-Mandel bunching effect: if two indistinguishable photons enter the shared beamsplitter, such that each enters by a different input port, they exit randomly via just one port (see section III.C).”

With the settings of Eq. (64) one can obtain a GHZ paradox falsifying elements of reality.

“With these settings one can get a GHZ paradox falsifying the elements of reality of the form described in the previous subsection.”

Violation of local realism implies that either locality, realism, or both cannot provide a foundational basis for quantum theory, provided the freedom assumption holds.

“Violation of local realism implies that either locality, or realism, or both, cannot provide a foundational basis for quantum theory (provided the freedom assumption of randomness and independence of setting choices holds).”

• AssumptionThe authors state this as an assumption: The freedom assumption of randomness and independence of setting choices holds.

In Hardy's ideal situation, quantum mechanics predicts that approximately 9% of photon pairs give coincidence counts absolutely not allowed by local realism.

“In such a case, in the ideal situation, for a specific set of measurements quantum mechanics predicts that approximately 9% of the pairs of photons must give measurement results in a form of coincidence counts absolutely not allowed by local realism.”

• cites Hardy 1993 sourceNot checked — no full text of the source

A third two-observer GHZ-like protocol uses two-photon hyper-entanglement, in which nine variables for each party can be regarded as simultaneous EPR elements of reality.

“A third protocol of the two-observer GHZ-like theorem uses a two-photon hyper-entanglement (Chen et al., 2003).Due to the specific properties of the hyperentanglement, nine variables for each party can be regarded as simultaneous EPR elements of reality.”

• cites Chen et al., 2003 sourceNot checked — no full text of the source

Quantum theory violates inequality (68).

“Quantum theory violates inequality (68).”

The Leggett-type inequality (68) follows from identity (67) plus assumptions (L1)-(L3).

“This, plus the above assumptions, implies a Leggett-type inequality (for details of the derivation see Groblacher et al., 2007) of the following form:”