Biphoton ququarts as either pure or mixed states, features and reconstruction from coincidence measurements

Biphoton polarization-frequency ququarts can be produced in processes of collinear Spontaneous Parametric Down-Conversion (SPDC) non-degenerate with respect to frequencies of photons.

“Biphoton polarization-frequency ququarts can be produced in processes of collinear Spontaneous Parametric Down-Conversion (SPDC) non-degenerate with respect to frequencies of photons [1].”

• cites [1] Y.I. Bogdanov, E.V. Moreva, G.A. Maslennikov, R.F. Galeev, S.S. Straupe, S.P. Kulik, Phys. Rev. A 73, 063810 (2006) sourceSupported

The degree of polarization of the mixed state coincides with the degree of polarization of the original two-qudit ququart (P = P(4)).

“and, hence, P = P (4) , i.e., the degree of polarization of the mixed state coincides with the degree of polarization of the original two-qudit ququart”

Pure states of qutrits (4) and ququarts (3) are characterized, correspondingly, by three and four complex parameters, {C1,B+,C4} and {C1,B+,C4,B-}, which corresponds to 6 and 8 real constants.

“Pure states of qutrits (4) and ququarts (3) are characterized, correspondingly, by three and four complex parameters, { C 1 , B + , C 4 } and { C 1 , B + , C 4 , B -} , which corresponds to 6 and 8 real constants.”

When |B-|=1, the qutrit's contribution to the ququart's wave function (3) equals zero (as C1=C4=B+=0).

“and in the second case the qutrit's contribution to the quqaurt's wave function (3) equals zero (as C 1 = C 4 = B + = 0).”

The averaged polarization density matrix ρ can be presented as a sum of products of 2×2 single-photon polarization matrices (Eq 14).

“One of them used below consists in the presentation of ρ in the form of a sum of products of 2 × 2 single-photon polarization matrices:”

In all other cases (|B-|≠1,0), there is no factorization for frequency and polarization parts in Ψ(4).

“In all other cases ( | B -| = 1 , 0) there is no factorization for frequency and polarization parts in Ψ (4)”

The only two exceptions to ρ characterizing a mixed polarization state occur when B-=0 and when |B-|=1.

“The only two exceptions occur in the cases B -= 0 and | B -| = 1.”

Both the degree of polarization of the mixed state and that of the original two-qudit ququart are determined by the Schmidt parameter of the mixed state K via Eq (20).

“and they both are determined by the Schmidt parameter of the mixed state K via Eq. (20).”

Parameters characterizing MPS are the same as in the case of pure-state ququarts, {C1,B+,C4,B-}.

“Parameters characterizing these states are the same as in the case of purestate ququarts, { C 1 , B + , C 4 , B -} .”

In a general case, the density matrix ρ characterizes a mixed polarization state.

“In a general case the density matrix ρ characterizes a mixed polarization state.”

Both terms in the ququart's wave function (3) obey the entanglement criterion (2).

“Besides, owing to the symmetry, both terms in the ququart's wave function (3) obey the entanglement criterion (2)”

Tr(σρr) ≡ Tr(σρ(4)).

“Evidently, Tr ( ⃗ σρ r ) ≡ Tr ( ⃗ σ ρ (4) )”

MPS are characterized completely by 5 independent real constants: |C1|, φ1, B+, |C4|, φ4, where φ1,4 are phases of the parameters C1,4, with the constant |B-| to be found from the normalization condition.

“Thus, we find that in this case MPS are characterized completely by 5 independent real constants: | C 1 | , ϕ 1 , B + , | C 4 | , ϕ 4 , where ϕ 1 , 4 are phases of the parameters C 1 , 4 with the constant | B -| to be found from the normalization condition.”

In some cases the entanglement of biphoton ququarts is purely frequency entanglement (e.g., if C4=B+=B-=0 and C1=1).

“In some cases this is a purely frequency entanglement (e.g., if C 4 = B + = B -= 0 and C 1 = 1)”

The ququart's wave function (3) contains both symmetric and antisymmetric wave functions of the polarization and frequency Bell states.

“Note that the ququart's wave function (3) contains both symmetric and antisymmetric wave functions of the polarization and frequency Bell states.”

The derived expressions for the Schmidt parameter and concurrence (16) and (17) depend on B- only as on |B-|.

“This is seen well from the derived expressions for the Schmidt parameter and concurrence (16) and (17), which depend on B -only as on | B -| .”

This density matrix characterizes pure states.

“This density matrix characterizes pure states.”

The biphoton wave function corresponding to the state vector of Eq. (1) can be written in a form related to the use of polarization and frequency Bell states Ψ±.

“The result can be written in different forms. The form most convenient for the further consideration is that related to the use of polarization and frequency Bell states Ψ ± :”

These conditions reduce the number of independent real constant parameters characterizing qutrits and ququarts to 4 and 6, respectively.

“These conditions reduce the amount independent real constant parameters characterizing qutrits and ququarts, correspondingly, to 4 and 6.”

The density matrix of MPS does not depend on the phase of the qutrit's wave function Ψ(3), which enters into the definition of ρ as determining one of its components.

“Also, as seen well from the definition of ρ in the form of Eqs. (11), (13), the density matrix of MPS does not depend on the phase of the qutrit's wave function Ψ (3) , which enters into the definition of ρ as determining one of its components.”

The symmetry required by Bose-Einstein statistics is clearly violated in the two-qubit wave function of Eq. (24), where the symmetric and antisymmetric Bell-state wave functions are summed on equal terms.

“This feature is clearly violated in the twoqubit wave function of Eq. (24) where the symmetric and antisymmetric Bell-state wave functions are summed on equal terms.”

In those same special cases, the relation C = √2(1 - K^-1) becomes valid again.

“In these cases the relation C = √ 2 ( 1 -K -1 ) becomes valid again”

Averaging of states of biphoton ququarts over frequencies ω1,2 gives rise to the MPS considered here and in Ref. [3].

“Averaging of states of bophoton ququarts over frequencies ω 1 , 2 gives rise to”

• Also stated in: § IV. COMPARISON WITH A TWO-QUBIT PURE-STATE MODEL OF BIPHOTON QUQUARTS · p. 4 · plain

  “MPS considered here and in Ref. [3].”

• cites [3] M.V. Fedorov, P.A. Volkov, Yu.M. Mikhailova, Phys. Rev. A 84, 032335 (2011) sourceNot checked — no full text of the source

When B-=0, the ququart is reduced to qutrit.

“In the first case the ququart is reduced to qutrit”

The antisymmetric Bell-state wave functions Ψ_pol^- and Ψ_fr^- appear only in the form of their product, which makes the total wave function Ψ(4) symmetric with respect to transposition of photon variables 1↔2.

“But the antisymmetric Bell-state wave functions Ψ pol -and Ψ fr -appear only in the form of their product, which makes the total wave function Ψ (4) symmetric with respect to the transposition of photon variables 1 ⇀ ↽ 2.”

In a general case, the expressions for K and C from the two-qubit model differ from K and C of Eqs. (16) and (17).

“In a general case, these expressions differ from K and C of Eqs. (16) and (17).”

Such terms (the 12 off-diagonal products) do not contribute to conditional probabilities analyzed in section V.

“Such terms do not contribute to conditional probabilities analyzed below in section V.”

In the special cases B-=0 or |B-|=1, when states averaged over frequencies remain pure, all discussed parameters of quantum and classical correlations coincide with each other and equal half of the von Neumann mutual information.

“Note finally that in other special cases, B -= 0 or | B -| = 1, when states averaged over frequencies remain pure, all discussed parameters of quantum and classical correlations coincide with each other and are equal to a half of the von Neumann mutual information”

In addition to the normalization and common phase conditions, MPS have one more condition: features of mixed states do not depend on the phase of the parameter B-.

“But, in addition to the normalization and common phase conditions we have now one condition more: as seen well from the structure of the density matrix ρ written in the form (12), features of mixed states do not depend of the phase of the parameter B -.”

In both cases B-=0 and |B-|=1, the ququart's wave function Ψ(4) factorizes into parts depending on polarization and frequency variables separately.

“In both cases B -= 0 and | B -| = 1 the ququart's wave function Ψ (4) factorizes for parts depending on the polarization and frequency variables separately.”

The Schmidt parameter of MPS remains related to their degree of polarization P by the same relation as in pure biphoton qutrits (Eq 20).

“On the other hand, the Schmidt parameter of MPS remains related to their degree of polarization P by the same relation as in the case of pure states of biphoton qutrits”

The conditional probability of registering a high-frequency photon polarized in direction α in channel 1 and a low-frequency photon polarized in direction α+90° in channel 2 is related to the unknown phase φ− by a formula very similar to Eq. (53).

“This conditional probability is related to the unknown phase ϕ -by a formula very similar to that of Eq. (53)”

In the case B+=0, Eq. (35) takes the form Bα+ = √2 cos α sin α (C4−C1).

“Indeed, as mentioned above, in the case B + = 0 Eq. (35) takes the form B α + = √ 2 cos α sin α ( C 4 -C 1 ).”

The Schmidt parameter of MPS, K, is related directly to the degree of polarization of ququarts.

“In particular, the Schmidt parameter of MPS, K is found to be related directly to the degree of polarization of ququarts.”

By assuming φ+=0, cos φ− can be expressed in terms of experimentally measurable relative amounts of photon counts.

“By assuming again that ϕ + = 0, we find from this equation cos ϕ -expressed in terms of the experimentally measurable relative amounts of photon counts.”

• AssumptionThe authors state this as an assumption: ϕ+ = 0.

MPS considered above occupy an intermediate position between qutrits and ququarts.

“MPS considered above occupy an intermediate position between qutrits and ququarts.”

When only polarization measurements are used (or equivalently, when the ququart's state is averaged over photon frequencies), the originally pure biphoton ququart state appears as a mixed two-qubit biphoton polarization state.

“If in experiments one uses only polarization measurements, the originally pure states of ququarts can be seen as mixed biphoton polarization states.”

• Also stated in: § I. INTRODUCTION · p. 1 · plain

  “In a general case, this gives rise to two-qubit biphoton mixed polarization states (MPS) [3].”

• cites [3] M.V. Fedorov, P.A. Volkov, Yu.M. Mikhailova, Phys. Rev. A 84, 032335 (2011) sourceNot checked — no full text of the source

In a general case, entanglement of biphoton ququarts is an inseparable mixture of polarization and frequency entanglement.

“but in a general case entanglement of biphoton ququarts is an inseparable mixture of the polarization and frequency entanglement.”

The correct results are those based on the picture of two-qudit polarization-frequency biphoton ququarts and of MPS arising after averaging over frequencies, i.e., the results determined by Eqs. (16)-(20).

“We believe that the correct results are those based on the picture of two-qudit polarizationfrequency bipohoton ququarts and of MPS arising after averaging over frequencies, i.e., the results determined by Eqs. (16)-(20).”

In experiments, biphoton ququarts can display different features depending on whether experiments involve purely polarization or polarization-frequency measurements.

“In experiments, biphoton ququarts can display different features in dependence on whether experiments involve purely polarization or (alternatively) polarization-frequency measurements.”

The symmetry of the total wave function under transposition of photon variables is an obligatory feature of two-boson pure states.

“This is the obligatory feature of two-boson pure states, which often is not taken seriously but which manifests itself, e.g., in the existence of MPS discussed below and in Ref. [3].”

• cites [3] M.V. Fedorov, P.A. Volkov, Yu.M. Mikhailova, Phys. Rev. A 84, 032335 (2011) sourceNot checked — no full text of the source

The density matrix of MPS can be reduced over polarization variables of one of the two photons to give the mixed-state reduced density matrix of Eq (15).

“The density matrix of MPS can be further reduced over polarization variables of one of two photons to give rise to the mixed-state reduced density matrix of the form [3]”

• cites [3] M.V. Fedorov, P.A. Volkov, Yu.M. Mikhailova, Phys. Rev. A 84, 032335 (2011) sourceNot checked — no full text of the source

All biphoton ququarts are entangled.

“and, hence, all biphoton ququarts are entangled.”

For MPS with C1=C4=0, the relative entropy is less than concurrence for any values of the remaining nonzero parameters |B-| and |B+|=1-|B-|.

“For MPS with C 1 = C 4 = 0 the relative entropy was found in [3] and shown to be less than concurrence at any values of the remaining nonzero parameters | B -| and | B + | = 1 - | B -| .”

• cites [3] M.V. Fedorov, P.A. Volkov, Yu.M. Mikhailova, Phys. Rev. A 84, 032335 (2011) sourceNot checked — no full text of the source

The only exceptions to S_rel < C occur at |B-|=0, 1, and 1/√2, where concurrence and relative entropy are equal.

“The only exceptions occur at | B -| = 0 , 1 and 1 / √ 2, where the concurrence and relative entropy are equal.”

• cites [3] M.V. Fedorov, P.A. Volkov, Yu.M. Mikhailova, Phys. Rev. A 84, 032335 (2011) sourceNot checked — no full text of the source

In the case of mixed states, the Schmidt parameter K is related to the amount of classical rather than quantum correlations.

“As the degree of polarization is a classical concept, we can deduce from Eq. (20) that in the case of mixed states the Schmidt parameter K is related to the amount of classical rather than quantum correlations.”

• AssumptionRests on an assumption: The degree of polarization is a classical concept.

Photons of SPDC pairs are always indistinguishable.

“Photons of SPDC pairs are always indistinguishable.”

In principle, differences between predictions of the theory of mixed states and of the two-qubit model of ququarts can be seen in experiments on measurement of the degree of polarization of biphoton polarization-frequency ququarts.

“In principle, differences between predictions of the theory of mixed states and of the two-qubit model of ququarts can be seen in experiments on measurement of the degree of polarization of biphoton polarization-frequency ququarts.”

Owing to this, the phase of Ψ(3) can be chosen, e.g., in a way making the parameter B+ real and positive.

“Owing to this, the phase of Ψ (3) can be chosen, e.g., in a way, making the parameter B + real and positive.”

In pure bipartite states, the Schmidt parameter K can be used to characterize the amounts of both quantum and classical correlations.

“this is the reason why in pure bipartite states the Schmidt parameter K can be used for characterization of the amounts of both quantum and classical correlations.”

In all other cases, the state arising after averaging over frequency variables is mixed.

“and, hence, the state, arising after averaging over frequency variables, is mixed.”

Averaging ρ(4) over frequency variables turns it into the density matrix of a mixed two-qubit polarization state.

“But being averaged over frequency variables, ρ (4) turns into the density matrix of a mixed two-qubit polarization state [3].”

• cites [3] M.V. Fedorov, P.A. Volkov, Yu.M. Mikhailova, Phys. Rev. A 84, 032335 (2011) sourceNot checked — no full text of the source

There is no way to get such MPS states in a two-qubit model.

“There is no way to get such states in a two-qubit model.”

The symmetry of two-photon wave functions is dictated by the Bose-Einstein statistics of photons.

“Its symmetry is dictated by the Bose-Einstein statistics of photons.”

In these cases, averaging a pure polarization-frequency state over frequency variables leaves the remaining polarization state pure, because Ψ(4) factorizes.

“This is the reason why in these cases averaging of a pure polarization-frequency state over frequency variables leaves the remaining polarization state pure.”

Wave functions of two photons in a pure state cannot be asymmetric with respect to the transposition of their variables.

“Wave functions of two photons in a pure state cannot be asymmetric with respect to the transposition of their variables.”

To get a correct ququart's wave function (3), in addition to symmetrization, one has to give freedom to photon frequencies ω1,2 by considering them as variables which can take one of two values each: either ω1=ωh and ω2=ωl or ω1=ωl and ω2=ωh.

“To get a correct quqaurt's wave function (3), in addition to symmetrization, one has to give freedom to photon frequencies ω 1 , 2 by considering them as variables which can take one of two values each: either ω 1 = ω h and ω 2 = ω l or ω 1 = ω l and ω 2 = ω h .”

In the two-qubit model, owing to fixed frequencies, two photons of SPDC pairs are considered as 'partially distinguishable', so the polarization biphoton wave function appears not necessarily symmetric under transposition of particle variables 1 ⇀ ↽ 2.

“Owing to this, two photons of SPDC pairs are considered as 'partially distinguishable', owing to which the polarization biphoton wave function appears to be not necessarily symmetric with respect to the transposition of particle's variables 1 ⇀ ↽ 2”

The picture of mixed two-qubit polarization states described in the paper differs significantly from the traditionally widely used model of two-qubit pure-state ququarts.

“A picture of mixed two-qubit polarization states described above differs significantly from traditionally widely used model of two-qubit pure-state ququarts.”

In the experimental scheme of Fig. 2, one has to install in front of detectors both polarizers and frequency filters.

“This means that in the experimental scheme of Fig. 2 one has to install in front of detectors both polarizers and frequency filters.”

Eq. (54) can be used for finding the phase φ− from data obtained from coincidence polarization-frequency measurements in the case of ququarts with B+=0.

“Eq. (54) can be used for finding the phase ϕ -from the data to be obtained from the coincidence polarization-frequency measurements in the case of ququarts with B + = 0.”

Additional information about parameters of MPS can be obtained from coincidence measurements with polarizers in both channels 1 and 2 turned for the same angle α with respect to the horizontal or vertical axes.

“Additional information about parameters of MPS can be obtained from coincidence measurements with polarizers in both channels 1 and 2 turned for the same angle α with respect the horizontal or vertical axes.”

The results of photon-counting measurements with polarizers turned by α are related to the corresponding conditional probabilities by the same relation as in the case α = 0 (Eq. 29).

“The results of such measurements (photon counting) are related to the corresponding conditional probabilities by the same relation as in the case α = 0 (29):”

This concludes determination of all five parameters of MPS in terms of conditional probabilities.

“This concludes determination of all five parameters of MPS in terms of conditional probabilities.”

• AssumptionThe authors state this as an assumption: B+ is taken real and positive.

Characteristics of MPS do not depend on φ−.

“Though characteristics of MPS do not depend of ϕ -, features of pure states of ququarts can be phase-sensitive.”

All biphoton polarization-frequency ququarts are entangled.

“Owing to this, all biphoton polarization-frequency ququarts are entangled and, in a general case, their entanglement is an inseparable mixture of the polarization and frequency entanglement.”

The only remaining unknown parameter of the ququart's states is the phase φ− of the parameter B−.

“The only remaining unknown parameter of the ququart's states is the phase ϕ -of the parameter B -.”

With |B-| found from Eq. (28), the described procedure finalizes the finding of all parameters of MPS of a general form.

“With | B -| found from Eq. (28), this finalizes the procedure of finding all parameters of MPS of a general form.”

Because of photon indistinguishability, the coincidence count N_V|H equals N_H|V.

“because of photon indistinguishability N V | H = N H | V”

Features of MPS can be used straightforwardly for experimental measurement of ququart's parameters if they are not known in advance.

“Features of MPS can be used straightforwardly for experimental measurement of ququart's parameters if they are not known in advance.”

The transformations (31) do not affect the frequency part of the ququart's wave function (3).

“Evidently, the transformations (31) do not affect the frequency part of the ququart's wave function (3).”

In the case B+=0, a direct measurement of the parameter k in Eq. (49) is sufficient for completing the reconstruction of the parameters characterizing MPS.

“In this case a direct measurement of the parameter k in Eq. (49) appears to be sufficient for completing the reconstruction of the parameters characterizing MPS.”

Eqs. (27) and (29) can be used for finding two real constants characterizing MPS, |C1|^2 and |C4|^2, directly from experimental measurements with polarizer in both channels 1 and 2 oriented either horizontally or vertically.

“Now Eqs. (27) and (29) can be used for finding two real constants characterizing MPS, | C 1 | 2 and | C 4 | 2 , directly from experimental measurements with polarizer in both channels 1 and 2 oriented either horizontally or vertically.”

For finding B+, |B-|^2, φ1 and φ4, other measurements are needed.

“For finding them other measurements are needed.”

Coincidence measurements sufficient for determining the phase φ− are most simple in the case of ququarts with B+=0.

“Such coincidence measurements sufficient for determining the phase ϕ -are most simple in the case of ququarts with B + = 0.”

With three values of wα|α(α) at α=-α0, 0, and α0, one can reconstruct the function wα|α(α) in a small vicinity of α=0.

“With this three values we can reconstruct the function w α | α ( α ) in a small vicinity of the point α = 0.”

When B+ is found, Eq. (28) yields |B-| = [2 w_{H|V} - |B+|^2]^{1/2}.

“When B + is found, Eq. (28) yields | B -| = [ 2 w H | V -| B + | 2 ] 1 / 2 .”

• AssumptionThe authors state this as an assumption: B+ is taken real and positive.

In both cases (a) and (b), with known |C1| and θ1, Eq. (43) can be used to find the phase φ1 of C1 as a function of B+.

“In both cases, with known | C 1 | and θ 1 , we can use Eq. (43) for finding the phase ϕ 1 of the parameter C 1 as a function of B + .”

For MPS with C4=0, horizontal-vertical coincidence measurements plus a measurement with polarizers at 135° and 45° suffice to determine the absolute values of C1, B+, and B-.

“Thus, for measuring absolute values of all three constants, | C 1 | , | B + | , and | B -| ( with C 4 = 0), it is sufficient to complete horizontal-vertical coincidence measurements by measurement of the coincidence number of photons with polarizers P 1 and P 2 turned, correspondingly, for 135 ◦ and 45 ◦ with respect to the horizontal direction.”

For relative amounts of registered photons, losses from undivided pairs and detector inefficiency do not matter.

“But for relative amounts of registered photons all these losses do not matter.”

If horizontal-vertical coincidence measurements give C1=C4=0, then one additional coincidence measurement with identically oriented polarizers at an arbitrary angle suffices to determine |B+| and |B-|.

“Thus, if a complete set of horizontal-vertical coincidence measurements gives C 1 = C 4 = 0, for finding | B + | and | B -| one has to make only one coincidence measurement more, with identically oriented polarizers P 1 and P 2 and with arbitrary chosen angle of their orientation α = 0 , π .”

When B+=0, one can choose an arbitrary phase of Ψ(3) to provide φ1=0, leaving only one phase parameter φ4 to be determined from experiments in addition to C1 and C4.

“Moreover, as B + = 0, we can choose now an arbitrary phase of Ψ (3) in a way providing ϕ 1 = 0, which leaves only one phase parameter ϕ 4 to be determined form experiments in addition to C 1 and C 4 .”

Features of pure states of ququarts can be phase-sensitive.

“Though characteristics of MPS do not depend of ϕ -, features of pure states of ququarts can be phase-sensitive.”

The described scheme of measurements is valid also in the case C1=0, C4=0.

“Note also that the described scheme of measurements is valid also in the case C 1 = 0 , C 4 = 0.”

Because undivided pairs do not participate in coincidence measurements, the amount of photons with any given polarization accessible for registration is twice less than in the original beam.

“As the undivided pairs do not participate in such measurements, the amount of photons with any given polarization accessible for registration is twice less than in the original beam.”

In picture (c) of Fig. 2, the parabola connecting points A, O, and B has a horizontal tangent at O.

“The line tangent to this parabola in the point O is horizontal.”

In a general case, the entanglement of biphoton polarization-frequency ququarts is an inseparable mixture of polarization and frequency entanglement.

“Owing to this, all biphoton polarization-frequency ququarts are entangled and, in a general case, their entanglement is an inseparable mixture of the polarization and frequency entanglement.”

In the case C4=0, the MPS correlation parameters K and C do not depend on the phase φ1.

“But it should be noted that in the case C 4 = 0 the MPS correlation parameters K (16) and C (17) do not depend on ϕ 1 .”

In simpler purely polarization (non-selective in frequencies) measurements, the same ququart states are seen as two-qubit mixed polarization states, MPS.

“If in fully polarizationfrequency coincidence measurements ququarts are seen as pure states, in simpler purely polarization (non-selective in frequencies) measurements the same states are seen as two-qubit mixed polarization states, MPS.”

The phase φ− cannot be found from any purely polarization measurements and requires combined polarization-frequency measurements.

“The phase ϕ -cannot be found from any purely polarization measurements and requires combined polarization-frequency measurements.”

In biphoton polarization-frequency ququarts, frequencies of photons are variables independent of polarizations rather than given numbers.

“consideration of frequencies of photons in biphoton polarization-frequency ququarts as variables independent of polarizations rather than as given numbers”

The derived Eqs. (40) and (41) determine directly phases φ1 and φ4 as functions of B+.

“Here expressions for the parameters | C 1 | and | C 4 | in terms of conditional probabilities are known (27), and the derived Eqs. (40) and (41) determine directly phases ϕ 1 and ϕ 4 as functions of B + .”

• AssumptionThe authors state this as an assumption: B+ is taken real and positive.

Similar measurements with polarizers deviating by a small angle α from vertical orientation can determine θ4 and then φ4 as a function of B+.

“Similar measurements and calculations can be done with polarizers deviating for a small angle α from the vertical orientation, to determine θ 4 and, then, ϕ 4 as a function of B + .”

The polarization antisymmetric Bell-state wave function is invariant with respect to the transformations (31), so Ψα expressed via (1 0)α and (0 1)α has the same form as Ψ expressed via (1 0) and (0 1) [Eq. (7)].

“Besides, as can be easily checked, the polarization antisymmetric Bell-state wave function is invariant with respect to the transformations (31), i.e., Ψ α -expressed via ( 1 0 ) α and ( 0 1 ) α has the same form as Ψ -expressed via ( 1 0 ) and ( 0 1 ) [Eq. (7)].”

By purely polarization measurements, all parameters of the 'qutrit's part' of the ququart Ψ(3) except its phase, and the parameter |B−|, can be determined.

“The analysis of the previous Section shows that by means of purely polarization measurements one can determine all parameters of the 'qutrit's part' of the ququart Ψ (3) (except its phase) and the parameter | B -| .”

In fully polarization-frequency coincidence measurements, ququarts are seen as pure states.

“If in fully polarizationfrequency coincidence measurements ququarts are seen as pure states, in simpler purely polarization (non-selective in frequencies) measurements the same states are seen as two-qubit mixed polarization states, MPS.”