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REVIEW 4 major objections 4 minor 76 references

Teleportation with non-maximally entangled states and underlying unitary algebras of certain bipartite systems

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A determinant tells which qubit pairs are entangled

desk verdict The qutrit entanglement test is built on a false eigenvalue/singular-value identification and fails on a simple product state; the qubit parts are standard, the teleportation protocol is unfinished, and the paper does not warrant peer review. read the letter →

arxiv 2505.21084 v1 pith:7J3UFRUB submitted 2025-05-27 quant-ph

Prabal Dasgupta , Debashis Gangopadhyay This is my paper
classification quant-ph MSC 81P4581R05
keywords bipartiteentanglementqubitqutritSchmidtdecompositiondeterminantcriterionteleportationnon-maximallyentangledstatesSU(2)andSU(3)algebras
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that entanglement of pure bipartite qubit and qutrit states can be read directly from a coefficient matrix built out of the wavefunction amplitudes. For a two-qubit state, the determinant of the 2x2 matrix $A$ is zero exactly for product states and nonzero exactly for entangled states. For two-qutrit states, the paper claims that $\det P = 0$ together with $\operatorname{Tr} P = \pm 1$ characterizes unentangled states, with any departure implying entanglement. The same matrix language is used to construct a teleportation protocol with non-maximally entangled qubit channels, where the determinant acts as an extra security-relevant parameter, and to reveal underlying SU(2) and SU(3) algebras. A sympathetic reader would care because these rules would turn entanglement detection into a quick algebraic test and would extend standard teleportation to imperfectly entangled resources.

What carries the argument

The central object is the coefficient matrix, a $2\times 2$ matrix $A$ or $3\times 3$ matrix $P$ whose entries are the amplitudes of the computational basis states of the bipartite wavefunction. For qubits the load-bearing quantity is $\det A$; for qutrits it is the pair $(\det P,\operatorname{Tr} P)$. The teleportation protocol uses the non-maximally entangled channel together with $\det A$ as an additional parameter that the receiver must know. The algebra structures are obtained by identifying the Bell-state matrices with scaled Pauli matrices and the qutrit matrices with scaled Gell-Mann matrices; antiunitary imaginary matrices are converted into unitary ones by multiplication by $\sqrt{2}e^{-i\pi/2}$, which the paper connects with time reversal.

What would settle it

Take the non-normal coefficient matrix $$P=\begin{pmatrix}1/2 & 1/\sqrt{2} & 0\\0 & 1/2 & 0\\0 & 0 & 0\end{pmatrix}.$$ It satisfies $\det P=0$ and $\operatorname{Tr} P=1$, so the paper's rule labels the state unentangled. But $\rho_A=P P^\dagger$ has two nonzero eigenvalues, $(1\pm \sqrt{3}/2)/2$, so the Schmidt rank is two and the state is entangled.

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Extended reading notes

Core claim

For a bipartite qubit state $|\psi\rangle=\sum a_{ij}|ij\rangle$, define the coefficient matrix $A=(a_{ij})$. The paper's central claim is that $\det A=0$ exactly when the state factorizes and $\det A\neq 0$ exactly when it is entangled, with $0\le |\det A|\le 1/2$ where the upper bound is reached by the maximally entangled Bell states. For qutrits, the analogous matrix $P=(a_{ij})$ is claimed to give a two-number test: if $\det P=0$ and $\operatorname{Tr} P=\pm 1$ are satisfied together, the Schmidt decomposition has one term and the state is unentangled, while any departure from these simultaneous conditions means the state is entangled. The paper also reports that the amplitude matrices for Bell states and entangled qutrit bases can be rescaled into generators of SU(2) and SU(3) algebras, and that teleportation of an arbitrary qubit works through the non-maximally entangled channel $a_{00}|00\rangle+a_{11}|11\rangle$ with fidelity bounded below by $4|\det A|^2$.

Load-bearing premise

The qutrit part of the paper assumes that the Schmidt coefficients can be obtained by diagonalizing the coefficient matrix $P$ itself; this holds only when $P$ is a normal matrix, whereas for a general two-qutrit state the eigenvalues of $P P^\dagger$, not of $P$, are the squared Schmidt coefficients.

Editorial extensions

If this is right

  • If the qubit rule is correct, one number, $|\det A|$, monitors how entangled a pure two-qubit state is, from $0$ for product states up to $1/2$ for Bell states.
  • The teleportation protocol would let a general qubit be sent through a non-maximally entangled channel, with Bob requiring $\det A$ as well as Alice's measurement basis, and with fidelity between $4|\det A|^2$ and $1$.
  • If the qutrit rule is correct, the same determinant-and-trace style of test would detect entanglement in three-level bipartite systems using just two matrix invariants.
  • The amplitude matrices of the entangled bases close into SU(2) for qubits and SU(3) for qutrits under rescalings, suggesting a structural link between these entanglement criteria and time-reversal operations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The qutrit criterion is only as strong as the identification of the eigenvalues of $P$ with the Schmidt coefficients; for a non-normal $P$ the Schmidt coefficients are the singular values of $P$, so a safe version of the test needs an extra rank-one check beyond $\det P$ and $\operatorname{Tr} P$.
  • The extra parameter $\det A$ in the teleportation protocol could double as a shared secret: an eavesdropper who knows the measurement basis but not the channel amplitudes could not separate $\alpha$ from $\beta$ without also knowing $\det A$.
  • A numerical scan over random two-qutrit states with non-normal coefficient matrices would quickly reveal how often the proposed det-and-trace rule misclassifies entangled states.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The manuscript claims new determinant-based rules for detecting entanglement of pure bipartite qubit and qutrit states, a teleportation protocol using non-maximally entangled states with enhanced cryptographic security, underlying SU(2) and SU(3) algebras for the coefficient matrices, and closed-form entanglement entropies expressed through determinants and traces. Section 2 presents a qubit criterion relating Det A to the eigenvalues of the reduced density matrix. Section 6 extends this to qutrits via a coefficient matrix P and asserts that Det P = 0 and Tr P = ±1 characterize product states. Section 3 applies the qubit formalism to teleportation, and Sections 4 and 6 identify the coefficient matrices after redefinitions with the Pauli and Gell-Mann generators. Section 7 gives entropy formulas.

Significance. The qubit determinant relation in Section 2 is correct and follows directly from the Schmidt decomposition; it is a simple but valid observation. The qutrit criterion, however, is the central new claim and is false, as shown by a separable product state that violates the stated condition. The teleportation fidelity and the claimed security enhancement are not properly defined or analyzed. The algebraic structures in Sections 4 and 6 are the result of rescaling and phase rotations of the standard generators and do not constitute new physics. Since the load-bearing qutrit assertion is wrong and the other claims are either elementary or unsupported, the manuscript is not suitable for publication in its present form.

major comments (4)
  1. [6 (after Eq. (27))] The identification of the eigenvalues of ρA = PP† as |μ0|², |μ1|², |μ2|², where μi are the eigenvalues of P, is valid only for normal P. In general, the eigenvalues of ρA are the squared singular values of P. This invalidates the qutrit entanglement criterion and the cubic equation for μk. A direct counterexample is the product state |ψ⟩ = |0⟩_A ⊗ (|0⟩+|1⟩)_B/√2, for which P = [[1/√2,1/√2,0],[0,0,0],[0,0,0]]. This state is separable, yet Det P = 0 and Tr P = 1/√2 ≠ ±1, so the paper's rule labels it entangled. The criterion is therefore not a valid entanglement test.
  2. [6 (Eq. (33))] Because the μi used in Sχ are not the Schmidt coefficients, the qutrit entanglement entropy formula is not the von Neumann entropy of the reduced state. The roots of the cubic written in Section 6 do not, in general, coincide with the eigenvalues of ρA. Consequently, Eq. (33) and the associated Figure 11 do not describe entanglement entropy, and the interpretation of the 'yellow plane' and 'green surface' is unsupported.
  3. [3 (Eqs. (15)–(18))] The teleportation fidelity F = (a00α + a11β)² is computed for the single branch in which Alice obtains |00⟩, without applying the usual unitary correction conditioned on the measurement outcome. It is therefore not the average teleportation fidelity of a complete protocol. The claimed 'enhanced cryptographic security' from the parameter Det A is not backed by any security definition, attack model, or comparison; the fact that Bob can infer α and β from the measurement statistics is a property of the state parameterization, not a cryptographic advantage.
  4. [4 and 6 (Eqs. (19)–(22) and (29)–(30))] The 'underlying SU(2)/SU(3) algebras' are constructed by rescaling and phase-multiplying the coefficient matrices of the Bell and qutrit basis states. After these redefinitions the matrices become exactly the Pauli and Gell-Mann generators, so the commutation relations in Eqs. (22) and (30) reproduce the standard algebras by construction. The paper does not show that these algebras act as symmetries of any physical Hamiltonian or that they have dynamical significance; the connection to time reversal in Section 8(e) is speculative and not derived.
minor comments (4)
  1. [General] The manuscript references Figures 2–11 in the text, but the figures are not included in the submitted text; the experimental proposal in Section 5 relies on these missing figures.
  2. [References and Abstract] There are numerous typographical errors in the reference list, e.g., 'Wooters', 'Deutsh', 'PhysRevLett.81.3018', and inconsistent numbering (a), (2), (c) in the abstract; these should be corrected.
  3. [4 (Eqs. (21) and (25))] In Eqs. (21) and (25), the anticommutator expressions are unclear and appear dimensionally inconsistent; for the Pauli matrices one expects {Ai, Aj} = δij, so the stated formula with a sign factor depending on i,j needs to be checked.
  4. [2 (Eq. (3))] In Section 2, the normalization condition in Eq. (3) is written as a00²+...=1; this is only true for real coefficients, so the restriction to real aij should be stated before Eq. (4) and the complex case should be addressed or explicitly excluded.

Circularity Check

2 steps flagged · score 6.0 of 10

Qutrit entanglement criterion reduces to an assumed identification of P's eigenvalues with Schmidt coefficients; SU(2)/SU(3) algebras are imposed by rescaling to Pauli/Gell-Mann matrices.

  1. self definitional [Section 6, after Eq. (27)]
    "P is a 3x3 matrix and it will have three eigenvalues µ0, µ1, µ2. Then the eigenvalues of ρA = P P† are |µ0|2, |µ1|2 and |µ2|2."

    For a general coefficient matrix P, the eigenvalues of ρA = P P† are the squared singular values of P, not the squared eigenvalues |µ_i|^2; the stated equality holds only when P is normal. By imposing this equality, the paper defines the Schmidt coefficients to be the eigenvalues of P, so the subsequent qutrit rule (DetP = 0 and TrP = ±1 iff unentangled, otherwise entangled) is not a derived prediction. It is a direct restatement of the assumed spectral identification. A rank-one product state with P = [[1/√2,1/√2,0],[0,0,0],[0,0,0]] has DetP = 0 and TrP = 1/√2, yet the rule classifies it as entangled, confirming that the criterion is forced by the assumption, not by the Schmidt structure.

  2. renaming known result [Section 6, after Table 2, Eq. (30)]
    "Redefining P matrices in table (2) as P ′0 = √3P0; P ′1 = √2P1 ; P ′2 = √2e(−iπ/2)P2 ; P ′3 = √2P3 ; P ′4 = √2P4 ; P ′5 = √2e(−iπ/2)P5 ; P ′6 = √2P6 ; P ′7 = √2e(−iπ/2)P7 ; P ′8 = √2P8. We have [P ′i, P′j] = 2i P k fijk P ′k (30) which is a SU(3) algebra where fijk’s are the usual structure constants."

    Table 2 already defines the Pα matrices as the Gell-Mann matrices divided by √2 (with i factors for the antisymmetric ones, e.g. P2 = iλ2/√2). The redefinitions multiply those matrices by √2 or by √2 e^(-iπ/2) = -i√2, which converts iλk/√2 into λk. Hence the P′i are exactly the Gell-Mann matrices by construction, and Eq. (30) is simply the standard su(3) Lie algebra. The 'underlying SU(3) algebra' is therefore a renaming of the known Gell-Mann commutation relations, not a structurally new result derived from bipartite entanglement.

full rationale

The paper is self-contained and correct for the qubit determinant criterion: for a real 2x2 amplitude matrix A, ρA = AA†, and DetA = 0 exactly when the coefficient matrix has rank one, i.e. the state is a product state. That part is an honest derivation. The circularity is concentrated in the qutrit section. There, the central step 'the eigenvalues of ρA = P P† are |µ0|^2, |µ1|^2 and |µ2|^2' is not a theorem; it conflates eigenvalues of P with singular values of P. Once that identification is assumed, the paper's qutrit entanglement rule and the qutrit entanglement entropy formula Eq. (33) follow by construction, because µ_i are simultaneously declared to be eigenvalues of P and Schmidt amplitudes. The rule is therefore not an independent prediction; it is an artifact of the assumed spectral relation. A simple product-state counterexample (DetP = 0, TrP = 1/√2, but state unentangled) shows that the 'otherwise entangled' branch is false. The SU(2) and SU(3) algebra claims are also imposed by deliberately rescaling the Bell-basis and qutrit-basis coefficient matrices into the Pauli and Gell-Mann matrices, which is a renaming of known results rather than a new algebraic derivation. Because the central qutrit claim reduces to its own assumption, but the qubit part and teleportation protocol retain independent content, the overall circularity score is 6 rather than higher.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

No free parameters are fitted to data. The central qutrit claim relies on an ad hoc and incorrect identification of eigenvalues of P with Schmidt coefficients. No new physical entities are postulated.

assumptions (2)
  • ad hoc to paper The eigenvalues of ρA = P P† are the squared eigenvalues of P
    Invoked without proof in Section 6; false for non-normal P, so the qutrit derivation and criterion collapse.
  • standard math Standard assumptions of pure bipartite quantum states and Schmidt decomposition
    Used throughout the paper; these are accepted background.

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Pith. "Pith review of Teleportation with non-maximally entangled states and underlying unitary algebras of certain bipartite systems." pith.science (2026). https://pith.science/paper/7J3UFRUB

@misc{pith2026250521084,
  author       = {Pith},
  title        = {Pith review of: Teleportation with non-maximally entangled states and underlying unitary algebras of certain bipartite systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7J3UFRUB}},
  note         = {Machine review of arXiv:2505.21084}
}
abstract

New convenient thumbrules are obtained to test entanglement of wavefunctions for bipartite qubit and qutrit systems. All results are analytic. The new results are: (a) For bipartite qubit systems there exists a matrix $A$ for which $\det A = 0$ implies unentanglement while $\det A \ne 0$ implies entanglement. There is an underlying SU(2) algebra. (2) Teleportation for a general qubit state is possible by using non-maximally entangled bipartite qubit states. This protocol has an additional parameter, viz., $\det A$, which enhances the cryptographic security of the teleportation. (c) For qutrits there is a matrix $P$ for which $\det P = 0$ simultaneously with ${\rm tr}P=\pm 1$ imply unentanglement. Any departure from these conditions implies entanglement. There exists an underlying SU(3) algebra. (d) Physical interpretation of the underlying algebras are given and plausible experimental scenarios are proposed for the SU(2) case in the context of two entangled electrons. (e) The entanglement entropy in both cases, viz., for qubits and qutrits respectively, are expressed in terms of the determinants and trace of the matrices mentioned above.

Figures

Figures reproduced from arXiv: 2505.21084 by the authors.

Figure 1
Figure 1. Teleportation Circuit Let the information qubit be |ψ⟩I = α|0⟩ + β|1⟩ ,(α 2 + β 2 = 1). To send the information qubit via a quantum teleportation channel we use the non-maximally entangled state |γ⟩ = a00|00⟩ + a11|11⟩ , (a 2 00 + a 2 11 = 1). This is a non-maximally entangled state as the probability amplitudes a00 ̸= 1/ √ 2 , a11 ̸= 1/ √ 2 and DetA = a00.a11 ̸= 1/2 (from equation 10). Consider the teleportation pr… view at source ↗
Figure 2
Figure 2. Block Diagram for experiment on |ϕ0⟩ [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. The Trajectory of |0⟩A in S.G like region 7 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: The Trajectory of |0⟩B in S.G like region [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: The Trajectory of |1⟩A in S.G like region [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: The Trajectory of |1⟩B in S.G like region 8 [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: Block Diagram for experiment on |ϕ1⟩ [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: Trajectory of |01⟩ in S.G like region 9 [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]
Figure 9
Figure 9. Figure 9: Trajectory of |10⟩ in S.G like region 6 The underlying SU(3) algebra for bipartite qutrit systems11 A natural generalisation of two state systems to a three state system is the qutrit. We now study entanglement in a bipartite qutrit system75−77.The most general qutrit …
Figure 10
Figure 10. Figure 10: S vs |DetA| 2 for bipartite qubit system 12 [PITH_FULL_IMAGE:figures/full_fig_p012_10.png]
Figure 11
Figure 11. Figure 11: Variation of Entanglement Entropy with TrP and ( [PITH_FULL_IMAGE:figures/full_fig_p013_11.png]

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