Entanglement of biphoton states: qutrits and ququarts

Classified as probably not a substantial claim.

The I-concurrence is defined via the Schmidt entanglement parameter K as C_I = 2(1 - K^{-1}).

“But instead, one can use the so-called 'I-concurrence' [19], by definition, determined via the Schmidt entanglement parameter K as C I = 2(1 -K -1 ).”

• cites [19] sourceDisputed

  Section 2 (p. 4) identifies the expression in terms of Schmidt coefficients as the tangle, or squared I-concurrence, C², not I-concurrence C. The same section distinguishes C from C² by taking the square root. The displayed formulas are not decoded in our copy, so the precise expression involving K cannot be checked, but the claim's identification of the unsquared I-concurrence is contrary to the text. (LLM, conf=0.84)

  Checked by a reasoning model — may be wrong.

  Experimental — accuracy not yet measured

Classified as probably not a substantial claim.

Entanglement can only be defined for a particular decomposition of a complex system into subsystems.

“entanglement can only be defined for a particular decomposition of a complex system into subsystems”

• cites [5] sourceSupported

Probably not a substantial claim: may not pass S3 (load-bearing).

From equation (4.3), the biphoton polarization vector ξ = Tr(ρ_r σ) is given by equation (5.16).

“If we define the biphoton polarization vector as ξ = Tr(ρ r σ ) [22], where σ is the vector of Pauli matrices, from equation (4.3) we easily find”

• cites [22] Wang A-M, 2000, Chin. Phys. Lett. sourceSupported

May combine two claims.

All purely polarization Bell states are unpolarized and maximally entangled.

“In particular, all purely polarization Bell states are unpolarized and maximally entangled [23][24][25].”

• cites [23] sourceSupported
• cites [24] sourceSupported
• cites [25] sourceNot checked — no full text of the source

Probably not a substantial claim: may not pass S3 (load-bearing).

Expressions for concurrence C (5.3 and last formula of 5.4) can also be derived directly from Wootters' original definition [20].

“Note that expressions for the concurrence C (equation (5.3) and the last formula of equation (5.4)) can also be derived directly from the original Wootters' definition [20].”

• cites [20] sourceSupported

Classified as probably not a substantial claim.

According to the Schmidt decomposition (Schmidt theorem), any entangled (unfactorable) bipartite wave function can be presented as a sum of factorized terms.

“A natural extension of this definition is given by the Schmidt decomposition, or Schmidt theorem [14,15], according to which any entangled (unfactorable) bipartite wave function can be presented as a sum of factorized terms.”

• cites [14] Grobe R, 1994, J. Phys. B: At. Mol. Opt. Phys. sourcePartly supported

  The passage labelled Abstract presents an exact two-particle wavefunction as a sum of canonical terms, and the passage labelled L508 calls that superposition the Schmidt decomposition. But the paper discusses identical-particle terms—Slater determinants for fermions and orbital products for bosons—rather than stating the claim for any bipartite wavefunction. No page number is supplied for these passages. (LLM, conf=0.86)

  Experimental — accuracy not yet measured
• cites [15] sourceNot checked — no full text of the source

May combine two claims.

The trace of the squared reduced density matrix ρ_r^2 determines purity of the reduced state and coincides with the inverse Schmidt entanglement parameter K^{-1}.

“As is known [14,15], the trace of the squared reduced density matrix ρ 2 r determines purity of the reduced state coinciding with the inverse value of the Schmidt entanglement parameter K -1 .”

• cites [14] Grobe R, 1994, J. Phys. B: At. Mol. Opt. Phys. sourceSupported
• cites [15] sourceNot checked — no full text of the source

May combine two claims.

Key reasons for the differences from the cited two-qubit treatment are the symmetry of wave functions and the higher dimensionality of ququarts compared with qutrits, properly taken into account.

“As explained above, key reasons for this and other differences between this paper [31] and others are in the symmetry of wave functions and in a higher dimensionality of ququarts compared with qutrits properly taken into account in equations (7.9)-(7.12)and (7.14).”

• cites [31] sourcePartly supported

  The Introduction (p. 1) identifies ququarts as four-dimensional and qutrits as three-dimensional, and Section E (p. 4) describes a difference in their polarization transformations. The copy does not explicitly identify wave-function symmetry as a key reason for the citing paper’s differences from other treatments; that causal explanation is not established by the cited passages. (LLM, conf=0.76)

  Experimental — accuracy not yet measured

May combine two claims.

The maximal achievable values of K and C_I for ququarts are Kmax = 4 and C_I max = √3/2, in agreement with general expectations for qudits.

“As follows from equations (7.16) and (7.17), the maximal achievable values of K and C I for ququarts are K max = 4 and C I max = √ 3/2, in agreement with the general expectations for qudits K max = d and C I max = 2(1 -d -1 ) [19].”

• cites [19] sourcePartly supported

  Section 2 (p. 4) states that the squared I-concurrence reaches 2(d−1)/d for maximally entangled states, giving C_I,max = √(3/2) when d = 4. The cited paper does not state the claimed K_max = d relation or identify a measure K, so it supports only the I-concurrence part of the general expectation. (LLM, conf=0.85)

  Experimental — accuracy not yet measured

Probably not a substantial claim: may not pass S3 (load-bearing).

In the generally accepted treatment, ququarts are considered as two-qubit states.

“As mentioned in the Introduction, in the generally accepted treatment, ququarts are considered as two-qubit states.”

• cites [22] Wang A-M, 2000, Chin. Phys. Lett. sourceCould not verify

  The passage labeled Abstract (page unavailable) writes a two-qubit state using four basis states, but does not discuss ququarts or say that treating them as two-qubit states is generally accepted. Searches for “ququart” and “four-dimensional quantum system” found no passage addressing that claim. (LLM, conf=0.70)

  Experimental — accuracy not yet measured

May combine two claims.

All biphoton ququarts are entangled and non-factorable, in contrast to the earlier two-qubit model assumption that at C1C4 = C2C3 ququarts are factorable.

“This means that all biphoton ququarts are entangled and non-factorable (in contrast to this, earlier [31], in a two-qubit model, it was assumed that at C 1 C 4 = C 2 C 3 ququarts are factorable).”

• cites [31] sourceSupported

Probably not a substantial claim: may not pass S3 (load-bearing).

This procedure serves as an alternative protocol of quantum state tomography of ququarts described in the work [31].

“This procedure serves as an alternative protocol of quantum state tomography of ququarts described in the work [31].”

• cites [31] sourceSupported

Classified as probably not a substantial claim.

The orthogonality condition expression follows directly from commutation rules for the photon creation and annihilation operators.

“this expression follows directly from commutation rules for the photon creation and annihilation operators”

Classified as probably not a substantial claim.

The orthogonality condition for biphoton purely polarization state-vectors is given by equation (2.3).

“let us first write down the orthogonality condition for biphoton purely polarization state-vectors given by”

Classified as probably not a substantial claim.

Biphoton ququarts are two-qudits with single-photon Hilbert space dimensionality d = 4.

“Biphoton ququarts are shown to be two-qudits with the single-photon Hilbert space dimensionality d = 4”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

The orthogonality relation for state vectors (2.3), matrix elements of the density matrix (2.4), the coordinate-dependent wave function (2.7), and the coordinate density matrix (2.8) are symmetric with respect to particle transpositions 1↔2.

“Note that the orthogonality relation for state vectors (2.3), matrix elements of the density matrix (2.4), as well as the coordinate-dependent wave function (2.7) and the coordinate density matrix (2.8) are symmetric with respect to particle transpositions 1 2.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The corresponding family of state vectors of non-entangled qutrits is given by equation (5.13).

“With these wave functions found explicitly, we can reconstruct the corresponding family of state vectors of non-entangled qutrits,”

Classified as probably not a substantial claim.

The reduced density matrix ρ_r = Tr_2 ρ for the qutrit is given by equation (4.3).

“The next step is the reduction of the density matrix with respect to one of the photon variables, e.g. of photon 2. Mathematically, this means taking traces of all matrices with subscript 2 in equation (4.1), which gives”

May combine two claims.

For degenerate eigenvalues λ+ = λ-, the Schmidt decomposition is not unique; if two orthogonal normalized linear combinations of ψ+ and ψ- are found, the Schmidt decomposition can be rewritten in the form (6.9).

“However, it should be noted that in the case of degenerate eigenvalues of the reduced density matrices, λ + = λ -, this is not the only possible form of the Schmidt decomposition.Indeed, if one can find two linear combinations ϕ and χ of the functions ψ + and ψ -such that ϕ and χ are orthogonal and normalized, then products of these functions can be used to rewrite the Schmidt decomposition (6.2)”

Classified as probably not a substantial claim.

The existence of the two degenerate biphoton modes is displayed in the matrix elements (2.3) and (2.4).

“But their existence is displayed clearly in the matrix elements (2.3) and (2.4).”

May combine two claims.

The reduced density matrix of the HV state has degenerate eigenvalues λ+ = λ- = 1/2, and its Schmidt modes are determined by photon polarizations along 45° and 135° with respect to the horizontal axis.

“Eigenvalues of the reduced density matrix of this state are degenerate, λ + = λ -= 1 2 , and the Schmidt modes are easily found to be determined by the wave functions of photons polarized along the directions 45 • and 135 • with respect to the horizontal axis:”

Probably not a substantial claim: may not pass S3 (load-bearing).

With normalization condition (3.3), equation (5.1) reduces to the simpler form (5.2).

“With normalization condition (3.3) taken into account, equation (5.1) can be reduced to a much simpler form:”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

Substitution into Wootters' definition gives C = | -2C*_1 C*_3 + C*_2^2 | = |C*_2_- - C*_2_+ + C*_2_2 |, in agreement with equations (5.3) and (5.4).

“C = | -2C * 1 C * 3 + C * 2 2 | = |C * 2 --C * 2 + + C * 2 2 |,(5.9)”

• Also stated in: § 5. Degree of entanglement · boosted

  “in complete agreement with equations (5.3) and (5.4).”

May combine two claims.

The qutrit basis wave function HV (3.6) is maximally entangled (K = 2, C = 1) and hence inseparable.

“As mentioned above, one of the basis wave functions of qutrits, HV (3.6), is maximally entangled (K = 2, C = 1) and, hence, inseparable.”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

The same conclusions about entanglement of states like |1_H, 1_V> can be obtained from consideration in terms of coordinate-dependent basis wave functions and their superpositions.

“The same conclusions about entanglement of states like |1 H , 1 V can be obtained from consideration in terms of 'coordinate'-dependent basis wave functions and their superpositions.”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

The same basis wave functions can be written equivalently in the form of two-row columns, which is more convenient for matrix calculations.

“The same basis wave functions can be written equivalently in the form of two-row columns, which is more convenient for calculation of matrices”

Probably not a substantial claim: may not pass S3 (load-bearing).

The full density matrix of qutrit (3.11) can be presented in two forms, equations (4.1) and (4.2).

“The full density matrix of qutrit (3.11) can be presented in the following two forms:”

Probably not a substantial claim: may not pass S3 (load-bearing).

In terms of constants C± (3.13), equations (5.2) and (5.3) take the form (5.4).

“Finally, in terms of the constants C ± (3.13), equations (5.2) and (5.3) take the form”

Probably not a substantial claim: may not pass S3 (load-bearing).

The wave functions ψ± and ψ+ describe three Bell states.

“± and + are the wave functions describing three Bell states.”

May combine two claims.

The paper finds a relationship between entanglement and polarization of biphoton qutrits, and derives a series of other results.

“We also find the relationship between entanglement and polarization of biphoton qutrits, and a series of other results is derived.”

May combine two claims.

If φ2 = 1/2(φ1 + φ3), then for complex constants C1,2,3 equation (5.3) reduces to C = |2|C1||C3| - |C2|^2|, and this case is equivalent to the case of real constants C1,2,3.

“Hence, if ϕ 2 = 1 2 (ϕ 1 + ϕ 3 ), in the case of complex constants C 1,2,3 , equation (5.3) is reduced to”

• Also stated in: § 5. Degree of entanglement · hedged

  “i.e. this case appears to be equivalent to the case of real constants C 1,2,3 .”

May combine two claims.

For qutrits with real coefficients, the subsystem entropy S_r real is minimal and equals zero at C+ = ±1/√2 and is maximal and equals 1 at C+ = 0 and C+ = ±1.

“S r real is minimal and equals zero at C + = ±1/ √ 2 and S r real is maximal and equals 1 at C + = 0 and C + = ±1.”

Classified as probably not a substantial claim.

The transformation (4.5) is provided by the matrix U (4.6).

“Evidently, the transformation”

• Also stated in: § 4. Density matrices · plain

  “is provided by the matrix”

May combine two claims.

Qutrits are maximally entangled when C=1 and K=2; for wave functions with real constants C1,2,3, C± this occurs in two cases: C+ = ±1 and C+ = 0.

“Qutrits are maximally entangled when C = 1 and K = 2, and for wave functions with real constants C 1,2,3 , C ± this occurs in two cases: C + = ±1 and C + = 0.”

Probably not a substantial claim: may not pass S3 (load-bearing).

In general form, the qutrit's wave function corresponding to the state vector (3.1) is given by ψ = C1 ψ_HH + C2 ψ_HV + C3 ψ_VV.

“In a general form, the qutrit's wave function corresponding to the state vector (3.1) is given by”

Classified as probably not a substantial claim.

The four modes correspond to single-photon Hilbert-space dimensionality d = 4, in contrast to d = 2 and two modes for qutrits and the traditional two-qubit model of ququarts.

“Four modes correspond to the dimensionality of the single-photon Hilbert space d = 4, in contrast to d = 2 and two modes in the cases of qutrits and of the traditional two-qubit model of ququarts.”

Classified as probably not a substantial claim.

Under the assumption that the two photons in each non-degenerate SPDC pair definitely have different frequencies, products of creation operators with coinciding frequencies are excluded, leaving only four two-photon basis state-vectors.

“But if we assume that two photons in each non-degenerate SPDC pair definitely have different frequencies, the products of two creation operators corresponding to coinciding frequencies have to be excluded to give finally only four two-photon basis state-vectors:”

• AssumptionThe authors state this as an assumption: The two photons in each non-degenerate SPDC pair definitely have different frequencies.

May combine two claims.

For the state φ, the eigenvalues are λ1 = λ2 = (1/2) cos^2 φ and λ3 = λ4 = (1/2) sin^2 φ, yielding S_r(φ) = 1 - 2(cos^2 φ log2|cos φ| + sin^2 φ log2|sin φ|).

“Evidently, its eigenvalues are λ 1 = λ 2 = 1 2 cos 2 φ and λ 3 = λ 4 = 1 2 sin 2 φ, which yields S r (φ) = 1 -2(cos 2 φ log 2 | cos φ| + sin 2 φ log 2 | sin φ|).”

Classified as probably not a substantial claim.

Polarization and frequency together make four combined polarization-frequency modes per photon: Hh, Hl, Vh, and Vl.

“Together, they make four states that can be occupied by each photon, or four combined polarization-frequency modes, Hh, Hl, Vh and Vl (instead of two polarization modes H and V in the case of degenerate photons).”

May combine two claims.

Biphoton ququarts are two-qudit rather than two-qubit states, with one-photon Hilbert space dimension d = 4 and two-photon Hilbert space dimension D = d^2 = 16, in contrast to qutrits with d = 2 and D = 4.

“Thus, biphoton ququarts are shown to be two-qudit rather than two-qubit states, with the dimensionality of the one-photon Hilbert space d = 4 and the dimensionality of the two-photon Hilbert space D = d 2 = 16 (in contrast with d = 2 and D = d 2 = 4 in the case of qutrits).”

May combine two claims.

The obtained 4×4 coherence matrix coincides with that of the two-qubit ququart model, although the reduced density matrices of the two-qubit and two-qudit systems do not coincide.

“Moreover, the thus obtained 4 × 4 coherence matrix coincides with that of the two-qubit ququart's model, although the reduced density matrices of the two-qubit and two-qudit do not coincide.”

May combine two claims.

A simple symmetrization of wave function (7.2) is insufficient and reduces it to the qutrit wave function (3.11), with C2 + C3 playing the role of C2 and with ququart peculiarities lost.

“A simple symmetrization of wave function (7.2) is insufficient and, actually, it does not help much, because it reduces wave function (7.2) to the qutrit's wave function (3.11), with C 2 + C 3 playing the role of C 2 in equation (3.11) and with all peculiarities of ququarts completely lost.”

May combine two claims.

The ququart 16×16 density matrix contains many zeros and can be reduced by unitary transformations to a 4×4 coherence matrix plus 12 zero lines and columns.

“Note, however, that as the ququart 16 × 16 density matrix contains many zeros, by a series of unitary transformation it can be reduced to the 4 × 4 coherence matrix plus 12 zero lines and columns (similar to what was done for qutrits in equations (4.6) and (4.7)).”

May combine two claims.

The single-photon states from which biphoton ququarts are constructed are not qubits and form a Hilbert space of dimensionality d = 4, rather than d = 2 as for qutrits.

“Single-photon states from which the biphoton ququarts are constructed are not qubits and they form the Hilbert space of dimensionality d = 4, rather than d = 2 occurring in the case of qutrits.”

Classified as probably not a substantial claim. May combine two claims.

One restriction is the assumption that the frequencies of photons are always different; this excludes same-frequency states and gives six symmetric wave functions.

“One of these restrictions was mentioned above: we assume that the frequencies of photons are always different.This excludes states |h, σ ; h, σ and |l, σ ; l, σ , which give rise to six symmetric wave functions.”

• AssumptionThe authors state this as an assumption: The two photons in each non-degenerate SPDC pair definitely have different frequencies.

Probably not a substantial claim: may not pass S3 (load-bearing).

For the HV state, the Schmidt decomposition (6.9) takes the form of the original definition of the wave function HV (3.6).

“and the Schmidt decomposition (6.9) takes the form of the original definition of the wave function HV (3.6).”

Classified as probably not a substantial claim.

The dimensionality of the biphoton Hilbert space is D = d^2 = 16.

“The dimensionality of the biphoton Hilbert space is D = d 2 = 16.”

May combine two claims.

For the state φ = cos φ HH + sin φ VV, equations (7.16) and (7.17) give K(φ) = 4/(1 + cos^2 2φ) and C_I(φ) = 1 + (1/2) sin^2 2φ.

“For this state, equations (7.16) and (7.17) give K (φ) = 4/(1 + cos 2 2φ) and C I (φ) = 1 + 1 2 sin 2 2φ.”

May combine two claims.

At φ = 0, when the entropy S_r(φ) is minimal, two eigenvalues become zero, two Schmidt-mode products disappear, and the degree of entanglement is minimal.

“However, at φ = 0, when the entropy S r (φ) is minimal, two of these eigenvalues become zero, λ 3 = λ 4 = 0, and two Schmidt-mode products in the Schmidt decomposition disappear.Hence, in this case the degree of entanglement is minimal.”

May combine two claims.

Equations (A.2) and (A.8) are identical, proving that coincidence measurements after the beam splitter can be used to obtain information about unconditional single-particle probabilities before the beam splitter.

“Coincidence of equations (A.2) and (A.8) proves that, indeed, coincidence measurements after the BS can be used for obtaining information about such features of the beam before the BS as characterizing unconditional (single-particle) probabilities for photons to have horizontal or vertical polarizations.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Relative numbers of coincidence counts coincide with conditional probabilities w(c)_sigma|sigma for a photon '1' to have polarization sigma under the condition that another photon '2' has polarization sigma.

“Relative numbers of coincidence counts coincide with conditional probabilities w (c) σ | σ for a photon '1' to have polarization σ under the condition that another photon '2' has polarization σ .”

Probably not a substantial claim: may not pass S3 (load-bearing).

Experimental measurement of conditional probabilities is straightforward because they are expressed explicitly in terms of coincidence detector counts.

“Equations (A.4) and (A.5) show that, in principle, experimental measurement of conditional probabilities is straightforward because they are expressed explicitly in terms of the coincidence detector counts N d (c) σ | σ .”

May combine two claims.

For real C_i, the inverse Schmidt entanglement parameter and concurrence can be expressed in terms of single-particle probabilities as K^-1 = 1/2 {1 + (w(s)_H - w(s)_V)^2 + (w(s)_H(45) - w(s)_V(45))^2} and C = 1 - (w(s)_H - w(s)_V)^2 - (w(s)_H(45) - w(s)_V(45))^2.

“Altogether this gives K -1 = 1 2 {1 + (w (s) H -w (s) V ) 2 + (w (s) H (45) -w (s) V (45) ) 2 } (A.27) and C = 1 -(w (s) H -w (s) V ) 2 -(w (s) H (45) -w (s) V (45) ) 2 .(A.28)”

May combine two claims.

Single-particle probabilities are related to qutrit parameters by w(s)_H = |C1|^2 + 1/2 |C2|^2 and w(s)_V = |C3|^2 + 1/2 |C2|^2.

“w (s) H = |C 1 | 2 + 1 2 |C 2 | 2 , w (s) V = |C 3 | 2 + 1 2 |C 2 | 2 . (A.10)”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

In experiments, the numbers of generated biphoton pairs are not directly observable.

“Although these equations are appropriate, unfortunately in experiments the numbers of generated pairs are not directly observable.”

May combine two claims.

The beam splitter splits between channels I and II only photons of half of all incoming biphoton pairs; the other half remain unsplit and are equally distributed between channels I and II.

“As follows from equation (A.1) and from the described features of the BS, the latter splits between channels I and II only photons of half of all incoming biphoton pairs, whereas the other half remain unsplit, and these unsplit pairs are equally distributed between channels I and II.”

May combine two claims.

In ququarts, all basis biphoton states are of type a†_i a†_j |0 with i = j, so the distribution of photons of split pairs in channels I and II corresponds to the type shown in figure A.2(b).

“Because in ququarts all the basis biphoton states are of type a † i a † j |0 with i = j (where i and j numerate above indicated modes), the distribution of photons of split pairs in channels I and II corresponds to the type shown in figure A.2(b).”

Classified as probably not a substantial claim. May combine two claims.

The final expression for |C1 C4 - C2 C3|^2 is given by equation (B.11), where all terms on the right-hand side are expressible via conditional probabilities either in the original x y basis or in the basis turned by 45 degrees.

“|C 1 C 4 -C 2 C 3 | 2 = 2 |C 1 | 2 |C 4 | 2 + |C 2 | 2 |C 3 | 2 -|C 1 (45 • )| 2 + |C 4 (45 • )| 2-1 2 2 , (B.11) where all terms on the right-hand side are expressible via conditional probabilities either in the original (x y) basis or in the basis turned by 45 • .”

Probably not a substantial claim: may not pass S3 (load-bearing).

Finding the phases of C_i requires additional measurements, e.g. with polarizers installed at 45 degrees to the x- and y-axes.

“In addition to absolute values of C i , we have to find their phases, ϕ 1,2,3 , and this requires additional measurements, e.g. with polarizers installed at 45 • to the xand y-axes, as shown in figure A.1(b).”

May combine two claims.

Although direct measurement of single-particle probabilities after the beam splitter is problematic due to unsplit pairs, probabilities w(s)_H and w(s)_V can be found from coincidence measurements.

“In other words, equations (A.8) show that although, owing to unsplit pairs, direct measurement of single-particle probabilities after the BS is problematic, nevertheless, probabilities w (s) H and w (s) V can be found from the results of coincidence measurements.”

May combine two claims.

The absolute values of all three coefficients C1, C2, C3 in the qutrit wave function are related to conditional probabilities by w(c)_H|H = |C1|^2, w(c)_V|V = |C3|^2, and w(c)_V|H = w(c)_H|V = 1/2 |C2|^2.

“w (c) H | H = |C 1 | 2 , w (c) V | V = |C 3 | 2 , w (c) V | H = w (c) H | V = 1 2 |C 2 | 2 . (A.9)”

May combine two claims.

The number of H- or V-polarized photons arising from split HH or VV pairs equals 1/2 N_tot_HH or 1/2 N_tot_VV, whereas the number of the same photons arising from split HV pairs equals 1/4 N_tot_HV.

“As seen from figure A.2, the number of H-or V-polarized photons arising from split HH or VV pairs equals 1 2 N tot HH or 1 2 N tot VV (figure A.2(a)), whereas the number of the same photons arising from split HV pairs is equal to 1 4 N tot HV (figure A.2(b)).”

May combine two claims.

The absolute values of ququart parameters C_i are related to conditional probabilities by |C1|^2 = 2w(c)_Hh|Hl = 2w(c)_Hl|Hh, |C2|^2 = 2w(c)_Hh|Vl = 2w(c)_Vl|Hh, |C3|^2 = 2w(c)_Hl|Vh = 2w(c)_Vh|Hl, and |C4|^2 = 2w(c)_Vh|Vl = 2w(c)_Vl|Vh.

“|C 1 | 2 = 2w (c) Hh | Hl = 2w (c) Hl | Hh , |C 2 | 2 = 2w (c) Hh | Vl = 2w (c) Vl | Hh , |C 3 | 2 = 2w (c) Hl | Vh = 2w (c) Vh | Hl , |C 4 | 2 = 2w (c) Vh | Vl = 2w (c) Vl | Vh . (B.”

May combine two claims.

Relations between C_i(45 degrees) and C_i for ququarts are C1(45) = 1/2(C1+C2+C3+C4), C2(45) = 1/2(-C1+C2-C3+C4), C3(45) = 1/2(-C1-C2+C3+C4), and C4(45) = 1/2(C1-C2-C3+C4).

“C 1 (45 • ) = 1 2 (C 1 + C 2 + C 3 + C 4 ) , C 2 (45 • ) = 1 2 (-C 1 + C 2 -C 3 + C 4 ) , C 3 (45 • ) = 1 2 (-C 1 -C 2 + C 3 + C 4 ) , C 4 (45 • ) = 1 2 (C 1 -C 2 -C 3 + C 4 ) . (B.4)”

Probably not a substantial claim: may not pass S3 (load-bearing).

The sum of all conditional probabilities w(c)_sigma|sigma equals unity.

“Evidently, the sum of all conditional probabilities (A.4) equals unity:”

Probably not a substantial claim: may not pass S3 (load-bearing).

If all parameters C_i are known in advance to be real, the procedure of finding the degree of entanglement from experimental data significantly simplifies.

“Note that if it is known in advance that all parameters C i are real, the procedure of finding the degree of entanglement from experimental data significantly simplifies.”

May combine two claims.

The numbers of detector coincidence counts N_d(c)_sigma|sigma obtainable with sigma- and sigma-polarizers in channels I and II are N_d(c)_H|H = eta/2 N_tot_HH, N_d(c)_V|V = eta/2 N_tot_VV, and N_d(c)_H|V = N_d(c)_V|H = eta/4 N_tot_HV.

“N d (c) H | H = η 2 N tot HH , N d (c) V | V = η 2 N tot VV , N d (c) H | V = N d (c) V | H = η 4 N tot HV , (A.3)”

May combine two claims.

The same relations between detector counts, conditional probabilities, and expansion coefficients can be written in the x y basis turned by 45 degrees, so all sets of coincidence measurements in two bases can determine all absolute values of ququart parameters in both bases.

“The same relations (B.1)-(B.3)between numbers of detector counts, conditional probabilities and expansion coefficients of the ququart wave function can also be written in the x y basis turned by 45 • with respect to the horizontal-vertical basis.Thus, all sets of coincidence measurements in two bases can be used to determine all absolute values of ququart parameters in both bases, |C i | 2 and |C”

Classified as probably not a substantial claim.

The definition of configurational entanglement is absolutely unambiguous in the case of bipartite states with distinguishable particles when all bipartite modes are non-degenerate.

“This definition is absolutely unambiguous in the case of bipartite states with distinguishable particles when all bipartite modes are non-degenerate”

Probably not a substantial claim: may not pass S2 (checkable), S4 (based).

Many widespread common opinions and evaluations in the field of entanglement of biphoton qutrits and ququarts need corrections.

“many widespread common opinions and evaluations in this field that need corrections are shown below.”

Classified as probably not a substantial claim. May combine two claims.

Entangled systems differ from separable ones in their behavior under local operations, but the definition of which operations are local relies on a particular choice of subsystems.

“Entangled systems differ from separable ones in their behavior under local operations, but the mere definition of which operations we should consider as local relies on a particular choice of subsystems.”

May combine two claims.

Separation in particles' variables provides conditions for describing intrinsic quantum correlations in systems of indistinguishable particles, which coexists with other types of entanglement and is accessible for experimental measurements.

“such separation provides conditions for describing intrinsic quantum correlations in systems of indistinguishable particles, which coexists with other types of entanglement and is accessible for experimental measurements.”

Probably not a substantial claim: may not pass S2 (checkable).

It is rather difficult to draw conclusions about the numbers of modes by looking at only the state vector or density matrix in the symbolical operator form.

“it is rather difficult to draw conclusions about the numbers of modes by looking at only the state vector or density matrix in the symbolical operator form”

Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing).

The question about the number of modes and configurations is not so clear.

“But the question about the number of modes and configurations is not so clear.”

Probably not a substantial claim: may not pass S2 (checkable).

It is up to us to choose a division of a quantum system into subsystems and to induce a corresponding tensor product structure, giving rise to possible entanglement.

“It is up to us to choose such a division and to induce a corresponding tensor product structure, giving rise to possible entanglement.”

• AssumptionRests on an assumption: The Hilbert space of a quantum system has no a priori division into subsystems.

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

For two distinguishable particles, the choice of subsystem decomposition is obvious.

“Of course, sometimes such a choice is obvious, that is, for example, the case of two distinguishable particles.”

Classified as probably not a substantial claim.

There is still no single unified and widely accepted approach to the problem of entanglement quantifiers for systems of indistinguishable particles.

“there is still no single unified and widely accepted approach to the problem.”

• cites [1][2][3][4][5][6][7][8][9][10]Not matched

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

Probably not a substantial claim: may not pass S2 (checkable).

The separation between symmetry and configurational entanglement is rather conventional.

“On the other hand, as shown above, separation for symmetry and configurational entanglement is rather conventional.”

Probably not a substantial claim: may not pass S3 (load-bearing).

As qutrit (3.1) is a two-photon state, its polarization wave function depends on two variables.

“As qutrit (3.1) is a two-photon state, its polarization wave function depends on two variables.”

Classified as probably not a substantial claim.

In all cases, the key point is the correct calculation of matrix elements as shown in equation (2.3).

“In all cases, the key point is in the correct calculation of matrix elements as shown in equation ( 2.3).”

May combine two claims.

The degrees of polarization and entanglement anticorrelate: maximally entangled qutrits are non-polarized, and maximally polarized states are non-entangled.

“Thus, the degrees of polarization and entanglement anticorrelate with each other: the maximally entangled qutrits are non-polarized, and maximally polarized states are non-entangled.”

May combine two claims.

The suggested interpretation of degenerate biphoton modes is valid for any states of two photons in different single-photon modes, for general-form biphoton qutrits and ququarts, and for any biphoton states of higher dimensionality.

“The suggested interpretation is valid not only for the state |1 H , 1 V but also for any other states of two photons in different single-photon modes, as well as for the general-form biphoton qutrits and ququarts, and for any biphoton states of higher dimensionality.”

May combine two claims.

Because the total phase of the state vector (3.1) or wave function does not affect any measurable characteristics of qutrits, one of the phases φ1,2,3 (or a linear combination of phases) can be taken equal to zero, so the general form of the qutrit's state vector (3.1) is characterized by four independent constants.

“Actually, as the total phase of the state vector (3.1) or wave function (see below) does not affect any measurable characteristics of qutrits, one of the phases ϕ 1,2,3 , or a linear combination of phases, can be taken equal to zero, and hence the general form of the qutrit's state vector (3.1) is characterized by four independent constants (e.g.|C 1 |, |C 3 |, ϕ 1 and ϕ 3 with ϕ 2 = 0).”

• AssumptionRests on an assumption: The total phase of the state vector or wave function does not affect any measurable characteristics of qutrits.

Classified as probably not a substantial claim.

Symmetry entanglement is entanglement that exists only in systems of indistinguishable particles and arises exclusively owing to the symmetry of wave functions or matrix elements dictated by Bose-Einstein or Fermi-Dirac statistics.

“In this sense, it is possible to speak about symmetry entanglement as entanglement existing only in systems of indistinguishable particles and arising exclusively owing to the symmetry of wave functions or matrix elements dictated by the Bose-Einstein or Fermi-Dirac statistics.”

May combine two claims.

In the case C = 1, λ+ = λ- = 1/2, the two Schmidt-decomposition products have equal weights, and the wave function is non-separable.

“In the case C = 1, when entanglement is maximal, λ + = λ -= 1 2 , i.e. two products in the Schmidt decomposition (6.2) are presented with equal weights and, clearly, the wave function is non-separable.”

May combine two claims.

Although K, concurrence C, and subsystem entropy S_r characterize entanglement in different metrics, their behavior is very similar, confirming the conclusions about separability and non-separability of qutrits and their wave functions.

“Therefore, although Schmidt entanglement parameter K , concurrence C and subsystem entropy S r characterize the degree of entanglement in different metrics, their behavior is very similar, which confirms all conclusions made above about conditions of separability and non-separability of qutrits and their wave functions.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

A general rule for obtaining multipartite wave functions from state vectors is known in quantum-field theory, and for bosons the corresponding formula has the form given in equation (3.4).

“A general rule for obtaining multipartite wave functions from state vectors is known pretty well in quantum-field theory, and for bosons the corresponding formula has the form [16] (in slightly modified notation)”

• cites [16] Schweber S S, 1961, An Introduction to Relativistic Quantum Field Theory sourceNot checked — no full text of the source

Classified as probably not a substantial claim. May combine two claims.

In a slightly modified interpretation, symmetry entanglement can be included in generalized configurational entanglement, and then the main mathematical requirement is simply a correct calculation of matrix elements of the type (2.3).

“In a slightly modified interpretation, the symmetry entanglement can be included in the generalized configurational entanglement and then, mathematically, the main requirement consists simply in a correct calculation of matrix elements of the type (2.3).”

Probably not a substantial claim: may not pass S3 (load-bearing).

The single-configuration description corresponds precisely to the approximation of leaving only one term in expressions like equation (2.3).

“whereas the single-configuration description corresponds precisely to this approximation: only one term left in expressions like that of equation ( 2.3)”

May combine two claims.

The missing fourth antisymmetric Bell state in expansion (3.12) is related to symmetry restrictions for two-boson states, and the existence of biphoton qutrits as superpositions of only three basis wave functions occurs exclusively owing to symmetry restrictions eliminating the fourth antisymmetric basis Bell state.

“This obligatory zero coefficient in front of -or the missing fourth antisymmetric Bell state in expansion (3.12) is related to restrictions imposed by the symmetry requirements for twoboson states.Thus, even the existence of biphoton qutrits as superpositions of only three basis wave functions occurs exclusively owing to symmetry restrictions eliminating the fourth (antisymmetric) basis Bell state”

May combine two claims.

Equation (7.25) takes the form Kqqrt = 2Kqtr, so in terms of the Schmidt entanglement parameter, post-selection after the beam splitter doubles the degree of entanglement of the original qutrit.

“Hence, equation (7.25) takes the simplest form: K qqrt = 2 K qtr , i.e. in terms of the Schmidt entanglement parameter, post-selection after the BS doubles the degree of entanglement of the original qutrit.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

The exclusion of same-frequency states does not have fundamental reasons and is related rather to the most often met experimental conditions.

“Their exclusion does not have fundamental reasons and is related rather to the most often met experimental conditions.”

May combine two claims.

All ququarts are entangled and unseparable, in contrast to earlier predictions of the two-qubit model.

“One of the qualitative consequences of these results is that all ququarts are entangled and unseparable, in contrast to earlier predictions of the two-qubit model.”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

Non-degeneracy of photons can be provided by the different directions of photons with equal frequencies in the non-collinear SPDC process.

“Non-degeneracy of photons can be provided by the different direction of photons with equal frequencies in the non-collinear SPDC process.”

Probably not a substantial claim: may not pass S4 (based).

In a nonselective beam splitter, each photon has equal probabilities of transmitting or being reflected.

“In a nonselective BS each photon has equal probabilities of transmitting or being reflected.”

May combine two claims.

A set of coincidence photon-counting measurements after a non-selective beam splitter, using polarizers in the horizontal-vertical basis and in the basis turned 45 degrees, is sufficient for determining the degree of entanglement and for full reconstruction of all parameters of arbitrary biphoton qutrits.

“As shown below, this set of measurements is sufficient for determining the degree of entanglement as well as for a full reconstruction of all parameters of arbitrary biphoton qutrits.”

May combine two claims.

Biphoton ququarts are produced by biphoton beams with non-degenerate photons (either in frequencies or in directions of propagation) and therefore have more degrees of freedom than qutrits, e.g. polarization and frequency.

“As ququarts are produced by biphoton beams with non-degenerate photons (either in frequencies or in directions of propagation), they have more degrees of freedom than qutrits, e.g.polarization and frequency.”

May combine two claims.

If SPDC photons in pairs have two different frequencies ωh and ωl, one cannot say which photon gets the higher or lower frequency, so frequency becomes a second photon variable additional to polarization.

“Indeed, if for example SPDC photons in pairs have two different frequencies ω h (high) and ω l (low), ω h > ω l , one cannot say for sure which of the two photons gets a higher or lower frequency.Hence, the photon frequency ω becomes a second photon variable, additional to polarization and taking two values, ω h (high) and ω l .”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

Polarization-frequency and polarization-angular ququarts are equivalent.

“As mentioned earlier, polarization-frequency and polarization-angular ququarts are equivalent.”

May combine two claims.

Consequently, the maximal achievable Schmidt entanglement parameter for ququarts is Kmax = 4, compared with Kmax = 2 for qutrits.

“Consequently, a two times higher dimensionality of the double-qubit Hilbert space (compared with qubits) doubles the maximal achievable degree of entanglement of ququarts if it is evaluated by the Schmidt entanglement parameter, K max = 4 (to be compared with K max = 2 in the case of qutrits).”

May combine two claims.

The remaining two terms in the Schmidt decomposition indicate that at φ = 0 the state φ is entangled and non-factorable.

“However, the remaining two terms in the Schmidt decomposition indicate that in the case φ = 0 the state φ is entangled and non-factorable.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Biphoton ququarts are also two-qudit states.

“Then, apart from qutrits being two-qubit states, the biphoton ququarts are also two-qudit states.”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

The nature of entanglement of different basis states is different.

“It may be interesting to note that the nature of entanglement of different basis states is different.”

Classified as probably not a substantial claim.

Because the single-photon Hilbert-space dimensionality is d = 4, these states are four-qudits rather than qubits.

“Because the dimensionality of the single-photon Hilbert space is d = 4, these states are d = four-qudits, rather than qubits.”

May combine two claims.

In ququarts all 12 excluded basis wave functions either are missing in all possible superpositions or can be present symbolically only with obligatory zero coefficients.

“Thus, in ququarts all 12 excluded basis wave functions either are missing in all possible superpositions or can be made present symbolically but with obligatory zero coefficients, like the only antisymmetric wave function in the qutrit wave function”

• AssumptionThe authors state this as an assumption: The two photons in each non-degenerate SPDC pair definitely have different frequencies.
• AssumptionRests on an assumption: All biphoton wave functions must be symmetric under exchange of the two photons; antisymmetric biphoton wave functions cannot be realized with photons.

Probably not a substantial claim: may not pass S3 (load-bearing).

These expressions properly take into account the higher dimensionality of biphoton ququarts compared with qutrits and with the traditional simplified two-qubit version.

“Also, these expressions properly take into account a higher dimensionality of the biphoton ququarts compared with qutrits and compared with the traditional simplified version of biphoton ququarts based on equations (7.1)-( 7.2).”

May combine two claims.

For ququarts, photon frequencies are variables rather than given numbers, and frequency entanglement must be taken into account together with polarization entanglement.

“In such cases the frequencies of photons are variables rather than simply some given numbers, and frequency entanglement has to be taken into account together with polarization entanglement.”

Probably not a substantial claim: may not pass S4 (based). May combine two claims.

The Schmidt entanglement parameter, concurrence, and subsystem von Neumann entropy are good for pure bipartite states; they characterize entanglement in different metrics and their predictions must be compatible with each other.

“All of them are good for pure bipartite states; they characterize the degree of entanglement in different metrics but, of course, their predictions must be compatible with each other.”

May combine two claims.

Biphoton qutrits and ququarts are two-boson formations whose wave functions are symmetric under permutation of photon variables, and this symmetry is important for correct evaluation of the degree of entanglement.

“As both qutrits and ququarts are two-boson formations, their wave functions are obliged to be symmetric with respect to permutations of photon variables, and this condition is important for correct evaluation of the degree of entanglement.”

Probably not a substantial claim: may not pass S4 (based).

The common phase of the biphoton wave function does not affect any measurements.

“As is well known and mentioned above, the common phase of the biphoton wave function does not affect any measurements.”

Probably not a substantial claim: may not pass S3 (load-bearing).

For ququarts, in addition to polarizers, frequency filters must be installed in front of the detectors to count photons in given frequency-polarization modes Hh, Hl, Vh, and Vl.

“Differently from qutrits, in addition to polarizers one has to install frequency filters in front of the detectors to count the numbers of photons in the given frequency-polarization Hh, Hl, Vh and V l.”

May combine two claims.

The described procedure provides a full reconstruction of the ququart wave function and can be used for determining the degree of entanglement.

“Thus the procedure described above provides a full reconstruction of the ququart wave function (qqrt) (7.14) and, in particular, can be used for determining the degree of entanglement.”

Probably not a substantial claim: may not pass S3 (load-bearing).

One possible way of measuring directly the parameters of ququarts, and in particular their degree of entanglement, is similar to that described for qutrits: split the original biphoton beam into two channels by a non-selective beam splitter and carry out a full set of coincidence measurements in the x y and x y bases.

“One possible way of measuring directly the parameters of ququarts, and in particular their degree of entanglement, is similar to that described above for qutrits and illustrated by figure A.2: the original biphoton beam has to be split into two channels by a non-selective BS and a full set of coincidence measurements has to be carried out in the x y and x y bases of figure A.2(b).”

May combine two claims.

The described procedure permits complete reconstruction of all qutrit parameters and determination of the degree of entanglement from experimental coincidence measurements.

“This procedure permits us to reconstruct completely all qutrit parameters and to find the degree of entanglement (A.22) from experimental coincidence measurements.”