Linear optics and quantum maps

Physical linear optical devices exist (dichroic elements) that may generate nonphysical two-qubit quantum maps.

“Surprisingly, we shall find that physical linear optical devices exist ͑dichroic elements͒ that may generate nonphysical two-qubit quantum maps ͓40͔.”

• cites [40] sourcePartly supported

  The experimental-arrangement passage (p. 2) describes physical glass elements that transmit one polarization more readily than the other, supporting the existence of polarization-dependent optical devices. The paper does not say that these elements generate nonphysical two-qubit quantum maps; that is the citing sentence’s additional claim. (LLM, conf=0.84)

  Experimental — accuracy not yet measured

A dichroic optical element necessarily performs a kind of post-selective measurement [16].

“Actually, it has been already noticed that a dichroic optical element necessarily performs a kind of post-selective measurement ͓16͔.”

• cites [16] sourceSupported

It is possible to generate two-qubit states represented by points upon and below the Werner curve in the linear entropy-tangle plane by operating on a single qubit (local operations) belonging to a pair initially prepared in the entangled singlet state.

“In the previous subsection we have shown that it is possible to generate two-qubit states represented by points upon and below the Werner curve in the linear entropy-tangle plane, by operating on a single qubit ͑local operations͒ belonging to a pair initially prepared in the entangled singlet state.”

The real-valued Mueller matrices M_A and M_B associated to M_A and M_B can be written as given in Eq. (41), using Eq. (5) with d_A = 0 = d_B and M_00 = 1.

“The real-valued Mueller matrices M A and M B associated to M A and M B , respectively, can be written as M A = ͫ 1 0 T a A ͬ , M B = ͫ 1 0 T b B ͬ , ͑41͒ where Eq. ͑5͒ with d A = 0 = d B and M 00 = 1 has been used”

• AssumptionThe authors state this as an assumption: Both single-qubit maps E_A and E_B are trace preserving.

A numerical simulation generating 10^4 states from Eq. (32) shows that a local dichroic scatterer may generate sub-Werner two-qubit states, located below the Werner curve in the linear entropy-tangle plane.

“The numerical simulation shows that a local dichroic scatterer may generate sub-Werner two-qubit states, that is, states located below the Werner curve in the linear entropy-tangle plane.”

• AssumptionRests on an assumption: The total Mueller matrices of the birefringent and dichroic scattering devices can be written as M = M_Z M_Δ, where M_Δ is a polarization-isotropic depolarizer.

For a dichroic scatterer, the output state is proportional to T_H I W T_H† I (Eq. 32).

“The analysis for the case of a dichroic scatterer can be done in the same manner leading to the result E ϰ ˜E = T H I W T H † I, ͑32͒”

• AssumptionRests on an assumption: The total Mueller matrices of the birefringent and dichroic scattering devices can be written as M = M_Z M_Δ, where M_Δ is a polarization-isotropic depolarizer.

If the input state is completely mixed, ρ_in = X_0/2, then the unnormalized output is ρ̃_out = 1/2 Σ_{α=0}^3 p_α X_α, where p_0 = M00 and (p1,p2,p3)=p is the polarizance vector.

“In addition, if in represents a completely mixed state, that is, if in = X 0 / 2, then from Eq. ͑9͒ it follows that ˜out = 1 2 ͚ =0 3 p X ,͑11͒”

Using Eq. (26) to calculate M_Δ(p) and using it in Eq. (20) gives ρ_E = p ρ_s + (1-p)/4 I_4 ≡ ρ_W.

“Thus, by using Eq.͑26͒ we can calculate M ⌬ ͑p͒ and use it in Eq. ͑20͒ to obtain E = p s + 1 -p 4 I 4 ϵ W ,͑29͒”

• AssumptionThe authors state this as an assumption: In the phenomenological model of Example 1, the second optical device T_B is absent (T_B = I), so the resulting quantum map is local.

The map E_A (for 0 ≤ p ≤ 2/3) is realized by the optical network shown in Fig. 7.

“Then, the map E A ͑for 0 Յ p Յ 2/3͒, is realized by the optical network shown in Fig. 7”

• AssumptionThe authors state this as an assumption: Both single-qubit maps E_A and E_B are trace preserving.
• AssumptionThe authors state this as an assumption: The two qubits are initially prepared in the singlet state.

For d0 ≠ d1, Eq. (34) is in contradiction with the locality constraint expressed by Eq. (12) which requires E_B = X0/2 (Eq. 35).

“This result is in contradiction, for d 0 d 1 , with the locality constraint expressed by Eq. ͑12͒ which requires E B = X 0 2 . ͑35͒”

Tr(ρ_E^2) = Tr(MM^T)/4.

“Tr͑ E 2 ͒ = Tr͑MM T ͒/4, ͑21͒”

In our case, coincidence measurements post-select only those photons that have not been absorbed by the dichroic elements present in the setup.

“In our case coincidence measurements post-select only those photons that have not been absorbed by the dichroic elements present in the setup.”

The qualitative agreement between the simulation and the experimental findings shown in Fig. 3 of Ref. [21] is evident.

“The qualitative agreement between the result of this simulation and the experimental findings shown in Fig. 3 of Ref. ͓21͔ is evident.”

• cites [21] sourceNot checked — no full text of the source

Eq. (78), together with Eq. (38), proves the equivalence between the quantum map E_A and the linear optical setup shown in Fig. 5.

“Equation ͑78͒, together with Eq. ͑38͒, proves the equivalence between the quantum map E A and the linear optical setup shown in Fig. 5.”

• AssumptionThe authors state this as an assumption: Both single-qubit maps E_A and E_B are trace preserving.
• AssumptionThe authors state this as an assumption: The two qubits are initially prepared in the singlet state.

The map E_B (for 0 ≤ p ≤ 2/3) is realized by the optical network shown in Fig. 8.

“Finally, the map E B ͑for 0 Յ p Յ 2/3͒, is realized by the optical network shown in Fig. 8”

• AssumptionThe authors state this as an assumption: Both single-qubit maps E_A and E_B are trace preserving.
• AssumptionThe authors state this as an assumption: The two qubits are initially prepared in the singlet state.

A local polarization-isotropic scatterer acting upon the two-qubit singlet state generates Werner states.

“Thus, we have found that a local polarization-isotropic scatterer acting upon the two-qubit singlet state, generates Werner states.”

The equality property in Eq. (20) is true not only for the singlet but for all four Bell states.

“Note that this pleasant property is true not only or the singlet but for all four Bell states ͓1͔, as well.”

• cites [1] sourceNot checked — no full text of the source

M represents a trace-preserving map only if M00 = 1 and d^T = (M01, M02, M03) = (0,0,0), namely, only if M describes the action of a nondichroic optical instrument.

“The equation above shows that M represents a trace-preserving map only if M 00 = 1 and d T = ͑M 01 , M 02 , M 03 ͒ = ͑0,0,0͒, namely, only if M describes the action of a nondichroic optical instrument.”

Both single-qubit maps E_A and E_B can be physically implemented as linear optical networks where polarization and spatial modes of photons are suitably coupled, without acting upon the pump beam.

“In this subsection, we show that both single-qubit maps E A and E B can be physically implemented as linear optical networks ͓6͔ where polarization and spatial modes of photons are suitably coupled, without acting upon the pump beam.”

• cites [6] sourceNot checked — no full text of the source

• AssumptionThe authors state this as an assumption: Both single-qubit maps E_A and E_B are trace preserving.
• AssumptionThe authors state this as an assumption: The two qubits are initially prepared in the singlet state.

By allowing bilocal operations performed by two separate optical devices T_A and T_B, it is possible to achieve MEMS states without using dichroic devices.

“we show that by allowing bilocal operations performed by two separate optical devices T A and T B located as in Fig. 1, it is possible to achieve MEMS states without using dichroic devices.”

• AssumptionThe authors state this as an assumption: Both single-qubit maps E_A and E_B are trace preserving.
• AssumptionThe authors state this as an assumption: The two qubits are initially prepared in the singlet state.

The map E_B (for 2/3 < p ≤ 1) can be physically implemented in the optical network shown in Fig. 6.

“In a similar manner, we can physically implement E B ͑for 2 / 3 Ͻ p Յ 1͒, in the optical network shown in Fig. 6”

• AssumptionThe authors state this as an assumption: Both single-qubit maps E_A and E_B are trace preserving.
• AssumptionThe authors state this as an assumption: The two qubits are initially prepared in the singlet state.

The locality constraint can be written as E_B = B.

“= ͉Tr͉ A ͑ E ͒, the locality constraint can be written as E B = B .”

The map represented by M (or M) is unital only if p = 0.

“Equation ͑11͒ shows that in this case Tr ˜out = M 00 , and out = ˜out / M 00 X 0 /2 if p 0, that is, the map represented by M ͑or M͒ is unital only if p =0.”

Both maps E_A and E_B must depend on the same parameter p in order to generate proper MEMS states.

“both maps E A and E B must depend on the same parameter p in order to generate proper MEMS states.”

• AssumptionThe authors state this as an assumption: Both single-qubit maps E_A and E_B are trace preserving.
• AssumptionThe authors state this as an assumption: The two qubits are initially prepared in the singlet state.

In the Jones and Stokes-Mueller calculi, the description of classical polarization of light is formally identical to the quantum description of pure and mixed states of two-level systems, respectively.

“In these calculi, the description of classical polarization of light is formally identical to the quantum description of pure and mixed states of two-level systems, respectively ͓22͔.”

• cites [22] sourceNot checked — no full text of the source

Such decomposition is not unique; for example, M = M_Δ M_D M_B is another valid decomposition.

“Such decomposition is not unique, for example, M = M ⌬ M D M B is another valid decomposition ͓43͔.”

• cites [43] sourceNot checked — no full text of the source

The need to use the proportionality symbol in Eq. (32) rather than equality is a consequence of the Hermitean character of the Jones matrix T_H that generates a non-trace-preserving map.

“This is a consequence of the Hermitean character of the Jones matrix T H that generates a non-trace-preserving map.”

Birefringent scatterers always produce generalized Werner states of the form ρ_GW = V ρ_W V†, where ρ_W denotes ordinary Werner states and V represents an arbitrary unitary operation.

“͑a͒ Birefringent scatterers always produce generalized Werner states of the form GW = V I W V † I, where W denotes ordinary Werner states ͓42͔ and V represents an arbitrary unitary operation.”

• cites [42] sourceNot checked — no full text of the source

Since λ_α ≥ 0, Eq. (7) is formally identical to the Kraus form of a completely positive one-qubit quantum map E.

“Since Ն 0, Eq. ͑7͒ is formally identical to the Kraus form ͓1͔ of a completely positive ͑CP͒ one-qubit quantum map E.”

• cites [1] sourceNot checked — no full text of the source

Eq. (35) is the reduced density matrix that could be reconstructed by Bob alone via local measurements before he and Alice had compared their experimental results and selected coincidence counts.

“Conversely, E B = X 0 /2 in Eq. ͑35͒, is the reduced density matrix that could be reconstructed by Bob alone via local measurements before he and Alice had compared their own experimental results and had selected from the raw data the coincidence counts.”

When a single photon encoding a polarization qubit passes through an optical device classically described by the Mueller matrix M = Σ T T†, its state is transformed according to ρ_in → ρ_out ∝ Σ_{α=0}^3 T_α ρ_in T_α†.

“when a single photon encoding a polarization qubit ͑represented by the 2 ϫ 2 density matrix in ͒, passes through an optical device classically described by the Mueller matrix M = ͚ T T ‫ء‬ , its state will be transformed according to in → out ϰ ͚ =0 3 T in T † ,͑8͒”

To each bilocal quantum map E_A ⊗ E_B can be associated a pair of classical Mueller matrices M_A and M_B such that (E_R)_{αβ} = Σ (M_A M_B)_{αβ,} (ρ_s^R)_.

“From the previous analysis ͓see Eqs.͑16͒-͑18͔͒ we know that to each bilocal quantum map E A E B can be associated a pair of classical Mueller matrices M A and M B such that ͑ E R ͒ ␣␤ = ͚ , ͑M A M B ͒ ␣␤, ͑ s R ͒ .͑40͒”

• AssumptionThe authors state this as an assumption: Both single-qubit maps E_A and E_B are trace preserving.
• AssumptionThe authors state this as an assumption: The two qubits are initially prepared in the singlet state.

The practical use of dichroic devices does not represent a severe limitation for SPDC setups.

“the practical use of dichroic devices does not represent a severe limitation for such setups.”

• AssumptionRests on an assumption: In any SPDC setup, the initial singlet state is actually a post-selected state (to cut off the otherwise overwhelming vacuum contribution).

Any linear optical element, either deterministic or stochastic, is equivalent to a composite device made of at most four nondepolarizing elements in parallel.

“An important theorem in classical polarization optics states that any linear optical element ͑either deterministic or stochastic͒ is equivalent to a composite device made of at most four nondepolarizing elements in parallel ͓32͔.”

• cites [32] sourceNot checked — no full text of the source

The Hermitean positive semidefinite matrix H associated to M is formally identical to the dynamical (or Choi) matrix describing a one-qubit quantum process.

“In view of the claimed connection between classical polarization optics and onequbit quantum mechanics, it is worth noting that H is formally identical to the dynamical ͑or Choi͒ matrix, describing a one-qubit quantum process ͓35͔.”

• cites [35]Not matched

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

In Ref. [21], using a setup similar to Fig. 1, the authors experimentally generated entangled two-qubit mixed states that lie upon and below the Werner curve in the linear entropy-tangle plane.

“In Ref. ͓21͔, by using a setup similar to the one shown in Fig. 1, we have experimentally generated entangled twoqubit mixed states that lie upon and below the Werner curve in the linear entropy-tangle plane ͓41͔.”

• cites [21] sourceNot checked — no full text of the source
• cites [41] sourceNot checked — no full text of the source

Equation (18) relates classical quantities (the two Mueller matrices M_A and M_B) with quantum ones (the input and output density matrices ρ^R and ρ_E^R), thus realizing the sought connection between classical polarization optics and two-qubit quantum maps.

“Equation ͑18͒ relates classical quantities ͑the two Mueller matrices M A and M B ͒ with quantum ones ͑the input and output density matrices R and E R , respectively͒, thus realizing the sought connection between classical polarization optics and two-qubit quantum maps.”

Dichroic scatterers generate sub-Werner states, that is states that lie below the Werner curve in the linear entropy-tangle plane.

“͑b͒ Dichroic scatterers generate sub-Werner states, that is states that lie below the Werner curve in the linear entropy-tangle plane.”

Only fulfilling Eqs. (52) and (53), together with E_Re = MEMS and E_Im = 0, will ensure the achievement of true MEMS states.

“Note that only fulfilling Eqs.͑52͒ and ͑53͒, together with E Re = MEMS and E Im = 0, will ensure the achievement of true MEMS states.”

• AssumptionThe authors state this as an assumption: Both single-qubit maps E_A and E_B are trace preserving.
• AssumptionThe authors state this as an assumption: The two qubits are initially prepared in the singlet state.

For partially polarized light, the coherency matrix J has the same properties as a density matrix representing a two-level quantum system in a mixed state.

“In this case the projection property is lost and J has the same properties of a density matrix representing a two-level quantum system in a mixed state.”

The two-qubit quantum map Eq. (20) is trace preserving only if the device is not dichroic, namely, only if d^T = (M01, M02, M03) = (0,0,0).

“Finally, Eq. ͑22͒, together with Eqs.͑5͒ and ͑12͒, tells us that the two-qubit quantum map Eq. ͑20͒ is trace preserving only if the device is not dichroic, namely, only if d T = ͑M 01 , M 02 , M 03 ͒ = ͑0,0,0͒.”

The degree of mixedness of the quantum state ρ_E is in a one-to-one correspondence with the classical depolarizing power of the device represented by M.

“Equation ͑21͒ shows that the degree of mixedness of the quantum state E is in a one-to-one correspondence with the classical depolarizing power ͓25͔ of the device represented by M.”

• cites [25] sourceNot checked — no full text of the source