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REVIEW 3 major objections 5 minor 48 references

Explicit Formulas for Estimating Trace of Reduced Density Matrix Powers via Single-Circuit Measurement Probabilities

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A single controlled-SWAP circuit can estimate every power trace of a reduced density matrix from one set of measurement probabilities.

desk verdict Useful single-circuit power-trace formulas with a real proof gap in the general case and misstated Tsallis/q-concurrence applications; worth referee time. read the letter →

arxiv 2507.17117 v1 pith:ML5UEFA7 submitted 2025-07-23 quant-ph

Rui-Qi Zhang , Xiao-Qi Liu , Jing Wang , Ming Li , Shu-Qian Shen , Shao-Ming Fei This is my paper · ORCID
classification quant-ph
keywords reduceddensitymatrixpowertracescontrolledSWAPtestquantumentanglementmeasurescircuitestimationHoeffdinginequalityKrausoperatorsclassicalsymmetric-polynomialiteration
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 a single quantum circuit, a controlled-SWAP test acting on $n$ identical copies of a state $\rho$, produces enough data to estimate the traces $\operatorname{tr}(\rho_A^2),\ldots,\operatorname{tr}(\rho_A^n)$ of the reduced density matrix on any chosen subsystem $A$. The central result is an explicit formula expressing each trace as an affine function of measurement probabilities on the $n-1$ control qubits, so that one run of one circuit replaces a family of purpose-built circuits. If the formula is right, the same data feed entanglement quantifiers and nonlinear functionals like entropy and trace exponentials, with a rigorous $O(\epsilon^{-2}\log(n/\delta))$ sample bound. A hybrid version using classical symmetric-polynomial iteration reduces the number of copies to the rank of $\rho_A$ when that rank is known.

What carries the argument

The load-bearing object is the controlled-SWAP test on $n$ copies with $n-1$ control qubits, whose effect is encoded by Kraus operators $K_z=2^{-(n-1)}\prod_{k=1}^{n-1}[I+(-1)^{z_k}S_k]$, where $S_k$ swaps copies $k$ and $k+1$. The argument runs on two mechanisms: trace lemmas that evaluate products of SWAP operators against identical copies, and a parity filter that, when probabilities are summed over outcomes with $c^z_{1,\ldots,k-1}$ even, selects the cyclic term $\operatorname{tr}(\rho^k)$ while cancelling the remaining 'waste' terms $S_{\mathrm{waste}}$ collected in the expansion. This cancellation is what lets one set of $2^{n-1}$ outcome probabilities yield $n-1$ separate trace estimates.

What would settle it

Pick a specific $n\ge 5$ and a state with a known reduced spectrum, such as a random mixed state on two qubits; expand $K_z^\dagger K_z$ exactly through all $2^{2n-2}$ terms, compute $2\sum_{c^z_{1,\ldots,k-1}\ \mathrm{even}} p(z)-1$ for every $k$, and compare with $\operatorname{tr}(\rho_A^k)$ obtained by exact diagonalization. Any mismatch for some $k$ or some state disproves Theorem 3.

Watch

Extended reading notes

Core claim

Theorem 3 states that for any $N$-partite state $\rho$ and any $k=2,\dots,n$, the power trace of the reduced state on the first subsystem obeys $\operatorname{tr}(\rho_{A_1}^k)=2\sum_{c^z_{1,2,\ldots,k-1}\ \mathrm{even}} p(z)-1$, where $p(z)$ is the probability of reading the bitstring $z$ on the $n-1$ control qubits and the sum runs over outcomes in which the first $k-1$ control bits have an even number of $1$s. The paper derives this from the Kraus-operator expansion of the circuit and uses it to claim simultaneous estimation of all these traces from a single set of measurement statistics. A technical point of the derivation is that neighbouring SWAP operators do not commute, so $K_z^\dagger K_z\neq K_z$; the paper shows that earlier work missed the resulting cross terms and supplies lemmas that account for them.

Load-bearing premise

The proof of Theorem 3 assumes that all leftover 'waste' terms in the Kraus expansion cancel when the measurement probabilities are summed over outcomes whose first $k-1$ control bits contain an even number of $1$s, for every $n$ and every state; the paper verifies this cancellation only for $n=3$ and $n=4$ and for larger $n$ asserts it 'by a similar discussion'.

Editorial extensions

If this is right

  • One fixed circuit with $n$ copies replaces up to $n-1$ purpose-built circuits for estimating $\operatorname{tr}(\rho_A^2),\dots,\operatorname{tr}(\rho_A^n)$.
  • The same measurement data feed directly into entanglement quantifiers such as concurrence, q-concurrence, Tsallis-q entanglement, and informationally complete entanglement measures.
  • Nonlinear functionals like $\operatorname{tr}(e^{\beta\rho})$ and von Neumann entropy can be approximated by truncating expansions in the estimated power traces.
  • The hybrid algorithm needs only $r$ copies when the rank $r$ of $\rho_A$ is known, and the sample bound $O(\epsilon^{-2}\log(n/\delta))$ holds simultaneously for all $k$.

Reading between the lines

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

  • If the waste-term cancellation holds generally, the same probability data could also estimate products of traces such as $\operatorname{tr}(\rho^2)\operatorname{tr}(\rho^3)$, which appear as distinct parity classes in the expansion; the paper does not use this.
  • A direct numerical test on random mixed states with $n\ge 5$ copies would be a stronger check of Theorem 3 than the symmetric GHZ and W simulations reported here.
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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

3 major / 5 minor

Summary. The paper proposes estimating tr(rho_A^k) for k=2..n using a single controlled-SWAP circuit with n copies of the state. The main result, Theorem 3, claims that tr(rho_A^k) = 2 * sum_{z: c^z_{1..k-1} even} p(z) - 1, and Corollary 1 says one set of measurement data suffices for all powers. The paper derives the n=3 and n=4 cases explicitly, develops a purely quantum algorithm and a hybrid Newton-Girard iteration algorithm, gives a Hoeffding-based sample complexity O(1/epsilon^2 log(n/delta)), applies the formulas to nonlinear functionals and entanglement measures, and reports Qiskit simulations for GHZ and W states.

Significance. If Theorem 3 holds, the single-circuit simultaneous estimation of all power traces is a practical advance over multi-circuit approaches, and the identification that [25]'s probability representation neglects the non-commutativity of neighboring SWAP operators is a useful correction. The n=3,4 derivations are explicit and internally consistent, the formulas are derived from first principles with no fitted parameters, and the Qiskit simulations match the theoretical probabilities for GHZ and W states. However, the general proof of Theorem 3 rests on an unproved cancellation of the remainder S_waste, so the central claim is currently conditional on a nontrivial combinatorial identity that should be established rigorously.

major comments (3)
  1. [Appendix B, Theorem 3 proof (Eqs. S4-S6)] The proof of Theorem 3 is incomplete at its load-bearing step. After expanding p(z) into cyclic terms and a remainder S_waste, the text asserts that the contribution of S_waste vanishes after summing over z with c^z_{1,...,k-1} even, because each remaining term has a parity structure 'other than' the cyclic ones. This cancellation is verified only for k=2 and k=n; for all intermediate k the proof says 'by a similar discussion,' and the full expansion of K_z^dagger K_z, which has 2^{2n-2} terms, is explicitly not given. Since S_waste contains cross terms such as (tr rho_A^2)^2, the explicit formula tr(rho_A^k) = 2 sum p(z) - 1 depends entirely on this unproved identity. Please supply a general combinatorial or representation-theoretic proof of the cancellation, or give a complete expansion for general n.
  2. [Section VI (Numerical simulations)] The numerical validation uses only the GHZ and W states, whose reduced density matrices are highly symmetric rank-2 states. This tests a very special slice of the parameter space and cannot substitute for a proof of the S_waste cancellation for generic rho_A or for intermediate k. Adding simulations on random pure states with generic spectra, and on random mixed states if the mixed-state claim is retained, would materially strengthen the evidence and could reveal whether the formula holds beyond the symmetric cases.
  3. [Section V.B, Definition 4 and following display] The probabilistic representation of q-concurrence is inconsistent with the definition. From Definition 4, C_q = 1 - tr(rho_A^q), and Theorem 3 gives tr(rho_A^q) = 2 sum_{even} p(z) - 1, so C_q = 2 - 2 sum_{even} p(z). The paper instead writes C_q = 2 sum p(z), which would imply tr(rho_A^q) = 1 - 2 sum p(z), contradicting Theorem 3. Please correct the displayed formula.
minor comments (5)
  1. [Remark after Theorem 3] The remark extending the result to mixed states via convex decomposition and linearity of the trace is not correct as stated: p(z) = tr(K_z^dagger K_z rho^{otimes n}) is a degree-n function of rho, not an affine function of rho. The Kraus-operator proof itself already applies to arbitrary rho through the spectral decomposition used in the lemmas, so please delete or rephrase this remark.
  2. [Section V.A, von Neumann entropy] The Taylor expansion of tr(rho ln rho) around the identity requires ||rho - I|| < 1, which holds only for full-rank rho. Please state this condition explicitly or use a series valid on the relevant part of the spectrum.
  3. [Section IV, Table I and Section VI, Table II] The table references are inconsistent: Section IV introduces a comparison table as Table I, while the error table in Section VI is also called Table I in the text but labeled Table II in the caption. The sentence introducing the comparison table is also incomplete. Please renumber and clean up the references.
  4. [Appendix B, Eq. (S4)] There are several typos in the proof of Theorem 3, including the definition of sigma_n (the text says permutations of 1,...,n-2 and 1,...,n-1 while sigma_n appears in the notation), and garbled expressions in the proof of Lemma 4 such as '|iim+1>' and 'Sis'. These should be corrected in a careful copyedit.
  5. [Section III, hybrid algorithm] The improved Newton-Girard algorithm is stated to require r quantum copies and assumes prior knowledge of r = rank(rho_A). This assumption should be stated explicitly in the algorithm description in Section III, not only in the conclusions.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation found; the general-n proof has an unproved S_waste cancellation step, which is a correctness gap rather than a circularity.

full rationale

The central formula tr(ρ^k_{A1}) = 2 Σ_{c^z_{1,...,k-1} even} p(z) − 1 is derived from the Kraus expansion of the controlled-SWAP channel and from trace identities for products of SWAP operators (Lemmas 1-4). The probabilities p(z) are not fitted to the target traces, and the numerical validation uses analytically known GHZ and W values as external benchmarks, so the result is not equivalent to its inputs by construction. The author-overlapping citations, such as [25] and [27], are used for context, for circuit inspiration, or as the target of correction; they do not carry the proof of Theorem 3. The genuine weakness is the S_waste cancellation asserted in the proof of Theorem 3. In the main text and again in Appendix B, the paper states that expanding K_z†K_z fully is "impractical and unnecessarily complex" and denotes the remaining terms by S_waste, then asserts that these terms cancel over the even-parity sums without exhibiting the general expansion or proving the cancellation for general n; only n = 3 and n = 4 are worked out, and intermediate k is dismissed with "By a similar discussion." This is an omitted proof and a correctness risk, not a circular reduction: S_waste is not defined as the cancellation conclusion, and the asserted cancellation is a separate combinatorial-mathematical claim about the expansion rather than a restatement of the theorem's input. Accordingly, the circularity score is low.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters are fitted to data. The central formula is derived from the Kraus operator model of the circuit; the only postulates are standard quantum mechanics and the circuit definition. The entropy application adds a norm condition that restricts its validity.

assumptions (3)
  • standard math Any multipartite density matrix admits an expansion as a linear combination of pure separable states (Lemma 5).
    Used in the proofs of Theorems 4 and 5 to analyze the evolution of mixed states through the circuit; it follows from the Bloch representation and spectral decomposition.
  • domain assumption The controlled-SWAP channel on n copies has Kraus operators K_z = 2^{-(n-1)} ∏_{k=1}^{n-1} [I + (-1)^{z_k} S_k], so p(z) = tr(K_z^† K_z ρ^{⊗n}).
    This is the quantum mechanical model of the circuit in Figure 1; the entire derivation starts from it.
  • domain assumption The Taylor expansion tr(ρ ln ρ) = tr[ρ Σ_{n=1}∞ (-1)^{n+1}/n (ρ-I)^n] requires ||ρ-I|| < 1 for some matrix norm.
    Stated in Section V.A; it excludes pure and near-pure states, limiting the von Neumann entropy estimation application.

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Pith. "Pith review of Explicit Formulas for Estimating Trace of Reduced Density Matrix Powers via Single-Circuit Measurement Probabilities." pith.science (2026). https://pith.science/paper/ML5UEFA7

@misc{pith2026250717117,
  author       = {Pith},
  title        = {Pith review of: Explicit Formulas for Estimating Trace of Reduced Density Matrix Powers via Single-Circuit Measurement Probabilities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ML5UEFA7}},
  note         = {Machine review of arXiv:2507.17117}
}
abstract

In the fields of quantum mechanics and quantum information science, the traces of reduced density matrix powers play a crucial role in the study of quantum systems and have numerous important applications. In this paper, we propose a universal framework to simultaneously estimate the traces of the $2$nd to the $n$th powers of a reduced density matrix using a single quantum circuit with $n$ copies of the quantum state. Specifically, our approach leverages the controlled SWAP test and establishes explicit formulas connecting measurement probabilities to these traces. We further develop two algorithms: a purely quantum method and a hybrid quantum-classical approach combining Newton-Girard iteration. Rigorous analysis via Hoeffding inequality demonstrates the method's efficiency, requiring only $M=O\left(\frac{1}{\epsilon^2}\log(\frac{n}{\delta})\right)$ measurements to achieve precision $\epsilon$ with confidence $1-\delta$. Additionally, we explore various applications including the estimation of nonlinear functions and the representation of entanglement measures. Numerical simulations are conducted for two maximally entangled states, the GHZ state and the W state, to validate the proposed method.

Figures

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