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Entanglement cost of quantum depolarization

Kun Fang

arXiv:2610.00187v1quant-phcs.ITmath.IT

TL;DR

Exact entanglement cost had remained unresolved even for qubit isotropic states, where collective preparation might save asymptotic resources. The paper proves exact results for qubits and derives near-tight bounds for qudit isotropic states, extending these findings to depolarizing-channel simulation.

  • Problem

    Even the entanglement cost of qubit isotropic states remained unresolved, with the central difficulty being whether collective preparation of many copies could save asymptotic resources.

    Source: §1 · p. 2

  • Method

    Tuned qubit semigroups turn log-Sobolev entropy estimates into supporting lines of Wootters’ function, while the full depolarizing 2 →3 norm supplies a qudit lower bound.

    Source: §1 · p. 2

  • Results

    At most 2% of log d separates the qudit isotropic-state lower bound from entanglement of formation, while the cost is exact for every two-qubit Bell-diagonal state.

    Source: §1 · p. 2, §5 · p. 13

  • Takeaways & Limitations

    Parallel and adaptive sequential simulation costs for depolarizing channels equal the entanglement cost of their isotropic Choi states, giving exact qubit costs and narrow qudit intervals.

    Source: §5 · p. 13

    The tuned semigroup family is not covered by the cited generalized-depolarizing results or King’s theorem, and the general estimate is insufficient for supporting-line matching.

    Source: §3.1 · p. 6

Abstract

from arXiv · show

Entanglement cost is the asymptotic rate of Bell pairs required to prepare a quantum state. Its regularized definition has made exact evaluation difficult, even for isotropic states (i.e., depolarized maximally entangled states). In this work, we determine the entanglement cost of every qubit isotropic state and, more generally, every two-qubit Bell-diagonal state. Our proof uses a family of suitably tuned qubit semigroups to transform a general log-Sobolev entropy bound into supporting lines of Wootters' function. A recent exact tensorization theorem of Dong et al. [arXiv:2606.17729] extends these bounds to arbitrarily many copies, yielding a general lower bound on entanglement cost. Matching this bound with the entanglement of formation also determines the exact cost for a broader class of two-qubit states. For qudit isotropic states, we derive another general lower bound from the full depolarizing $2\to 3$ norm. This bound substantially improves on the PPT-relative entropy of entanglement and nearly matches the entanglement of formation, leaving a gap of at most $2\%$ of $\log d$ that vanishes as $d$ grows. Finally, the exact qubit results and qudit bounds extend to the entanglement cost of preparing quantum depolarizing channels under both parallel and sequential strategies.

1 Introduction

Regularization makes exact entanglement costs difficult to determine, and even qubit isotropic states had remained unresolved. The paper resolves the qubit cases, bounds the qudit isotropic cost, and extends the results to depolarizing-channel simulation.

1 Introduction

  • Entanglement cost is a regularized quantity, so an exact one-copy entanglement-of-formation formula does not generally determine the operational cost [HHT01].The unresolved obstacle for qubit isotropic states was excluding savings from collective preparation.Source: §1 · pp. 1–2
  • Every two-qubit Bell-diagonal state, including every qubit isotropic state, has an exact entanglement cost; the paper also gives a general lower bound for two-qubit states.The bound is exact for the class where normalized negativity equals concurrence.Source: §1 · p. 2
  • For qudit isotropic states, the cubic-norm bound improves substantially on the PPT-relative-entropy benchmark and leaves a gap of at most 2% of log d from entanglement of formation.The gap vanishes as dimension grows.Source: §1 · p. 2
  • The state results yield exact entanglement costs for qubit depolarizing channels under parallel and adaptive sequential simulation, and tight estimates for qudit channels.The reduction identifies both simulation costs with the cost of the isotropic normalized Choi state [Wil18, Theorem 1 and Section IV].Source: §1 · p. 2

2 Preliminaries

The preliminaries set the paper’s state, entanglement, and quantum Markov semigroup conventions. They define the cost and entangled fraction and establish notation used in the proofs.

2 Preliminaries

  • The conventions use base-two logarithms except where natural logarithms are specified, and define von Neumann entropy, binary entropy, and relative entropy in nats.The Schatten norms and finite-dimensional operator notation are also specified.Source: §2 · pp. 2–3
  • The normalized entangled fraction maximizes overlap with maximally entangled rank-one projectors in the relevant dimension.The set of these projectors is denoted ME.Source: §2 · p. 2
  • Entanglement cost is the minimum asymptotic Bell-pair rate for preparing copies by local operations and classical communication with vanishing error; entanglement of formation is also introduced.The cost is related to regularized entanglement of formation [HHT01].Source: §2 · p. 3
  • A quantum Markov semigroup is a continuous family of completely positive unital maps with identity initial map and the semigroup composition law.The text also defines invariant states, primitivity, and σ-reversibility [DGO+26, Definitions 2.7 and 2.10].Source: §2 · p. 3
  • Positive off-diagonal scaling means that the semigroup preserves the diagonal algebra and scales off-diagonal matrix units by nonnegative factors.The property is specified in a fixed eigenbasis of the invariant state [DGO+26, Definition 2.11].Source: §2 · p. 3

3 Entanglement cost for qubit states

The section proves a general entangled-fraction lower bound for two-qubit entanglement cost and identifies when it is exact. The exact class includes all Bell-diagonal states and qubit isotropic states.

3.1 A general lower bound for qubit states

  • For every two-qubit state, entanglement cost is bounded below by W evaluated at the normalized entangled fraction.The bound is exact when normalized negativity equals concurrence.Source: §3 · p. 3, §3.1 · p. 4
  • A tuned qubit semigroup turns a log-Sobolev entropy estimate into supporting lines of W, and exact tensorization extends the bound to arbitrary numbers of pairs.Averaging over pure-state decompositions and taking the regularized limit proves the general bound.Source: §3.1 · pp. 4, 6
  • The tuned generator has off-diagonal rate η_c = (1 + c)/2 < 1 and is not Hilbert–Schmidt self-adjoint, so cited generalized-depolarizing estimates do not suffice for supporting-line matching.The general estimate gives only η_c a_c < a_c.Source: §3.1 · p. 6

3.2 Tightness of the entangled-fraction bound

  • For entangled two-qubit states, equality of normalized negativity and concurrence characterizes the exact-cost class; one equivalent condition is a maximally entangled vector spanning the negative eigenspace of the partial transpose.On this class the entangled-fraction lower bound coincides with entanglement of formation.Source: §3.2 · pp. 6–7
  • Every two-qubit Bell-diagonal state and every qubit isotropic state has entanglement cost equal to its entanglement of formation.The proof uses that normalized negativity equals concurrence for these states.Source: §3.2 · p. 7

4 Entanglement cost for qudit states

For dimensions d ≥3, the paper derives a general entangled-fraction lower bound on entanglement cost and evaluates it for isotropic states. The resulting interval substantially improves the PPT-relative-entropy benchmark, nearly matches entanglement of formation, and has an absolute gap that vanishes uniformly over the noise range as d grows.

4.2 The depolarizing cubic norm and its tensorization

  • The depolarizing 2 →3 norm yields the affine entropy bounds, and tensorization preserves the exact norm constant across products, including when the individual maps are not contractive.The proof adapts Dong et al. [DGO+26] while retaining the full 2 →3 norm constants.Source: §4 · p. 7, §4.2 · pp. 10–11

4.1 A general lower bound for qudit states

  • For every d × d state with d ≥3, Theorem 8 gives a cubic-norm lower bound on entanglement cost based on normalized entangled fraction.The proof bounds pure-state entropy by affine functions of local Bell overlaps, averages over preparation ensembles, and optimizes over r.Source: §4 · p. 7, §4.1 · p. 8

4.3 Explicit evaluation of the cubic-norm bound

  • Optimizing the affine entropy bound over r gives three branches: r = 1, an interior optimizer, and the contractive endpoint r = r0,d.The intermediate regime retains the full norm and yields a stronger nonlinear branch than the contractive endpoint.Source: §4.3 · p. 11

4.4 Isotropic states: comparison and tightness

  • The paper pairs its general lower bound with the known entanglement of formation to certify an interval for isotropic-state entanglement cost.The interval width bounds the possible advantage of collective preparation over independent preparation.Source: §4.4 · p. 12
  • For isotropic states, the cubic-norm bound substantially improves the PPT benchmark in examples d = 3, 10, and 60; its gap from entanglement of formation stays below 2% of log d for 3 ≤ d ≤ 104.The maximum normalized width in the scan is approximately 1.989% at d = 8. The absolute gap vanishes uniformly over the noise parameter as d grows.Source: §4.4 · p. 12, Figure 1 · p. 13

5 Conclusion

The paper determines exact entanglement costs for qubit isotropic and two-qubit Bell-diagonal states, while giving narrowing bounds for qudit isotropic states and depolarizing-channel costs.

5 Conclusion

  • The exact entanglement cost is determined for every qubit isotropic state and every two-qubit Bell-diagonal state.The exact qudit isotropic-state cost remains open.Source: §5 · p. 13
  • For qudit isotropic states, the cubic-norm bound substantially improves the PPT-relative-entropy benchmark and nearly matches entanglement of formation.The remaining interval vanishes in absolute width with increasing dimension, uniformly over the noise parameter.Source: §5 · p. 13
  • Depolarizing-channel results give exact simulation costs for qubits and the same narrow cost intervals for qudits under parallel and adaptive sequential strategies.Both channel costs equal the entanglement cost of the corresponding isotropic Choi state [Wil18, Theorem 1 and Section IV].Source: §5 · p. 13

A Calculations for the tuned qubit semigroup

This appendix supplies calculations for the qubit semigroup constructed in the proof of Lemma 3, deriving a matching identity and verifying its hypotheses and sharp log-Sobolev constant.

A Calculations for the tuned qubit semigroup

A.1 Dirichlet form and algebraic matching

The calculations match coefficients so the reference-state term and Dirichlet form yield the supporting-line expression, including its endpoint case.

A.1 Dirichlet form and algebraic matching

  • For n pairs, grouping the other row and column indices into α and β expresses M as a 2 × 2 block matrix at each site.The product weights cancel on the other sites, and summing the site contributions proves Eq. (28).Source: §A.1 · p. 15
  • Each summand acts on one qubit and as identity on the other qubits; applying the one-site reference formula to the blocks proves Eq. (29) without assuming M factorizes.Source: §A.1 · p. 15
  • The rates are chosen so the reference-state term minus the Dirichlet form divided by a_c depends only on ∥M∥_2^2 and |M_00 + M_11|^2.The coefficient constraints equate the |M_00|^2 and |M_11|^2 terms and relate their difference from the |M_01|^2 coefficient to Re(M_00M_11).Source: §A.1 · p. 15
  • The resulting identities give the supporting-line expression in Eq. (30), which remains valid at c = 1 with a_1 = 1/2 and η_1 = 1.The slope is fixed by √(µ_cν_c)/a_c = (ln 2)W′(c), while the contact state determines the intercept.Source: §A.1 · p. 15

A.2 Semigroup properties and the sharp log-Sobolev constant

The semigroup is completely positive, unital, and primitive; its sharp log-Sobolev constant is established by reduction to the diagonal two-point algebra and continuity at the endpoint.

A.2 Semigroup properties and the sharp log-Sobolev constant

  • The generator L_c has a Lindblad representation, and Ψ_c,t = e^−tL_c is completely positive and unital.It is σ_c-reversible, preserves the diagonal algebra, and scales off-diagonal matrix units by η_c.Source: §A.2 · p. 16
  • The diagonal algebra is invariant, and on it L_c is the classical two-point generator with probabilities (µ_c, ν_c) and a sharp 2-log-Sobolev constant.The entropy difference depends only on the eigenvalues of M, so it can be evaluated at u = 1, where M and X are diagonal.Source: §A.2 · p. 16
  • The classical inequality gives α_2(L_c) ≥ a_c, while restricting to diagonal X gives the reverse inequality and hence equality.Source: §A.2 · p. 16
  • At c = 1, continuity extends the displayed expressions with a_1 = 1/2, also the sharp constant on the diagonal algebra.Source: §A.2 · p. 16

B Uniform large-dimension estimate

The appendix analytically estimates the interval width studied numerically in Figure 2. The bound is conservative in finite dimensions but proves that the width vanishes in absolute ebits uniformly over the noise range.

B Uniform large-dimension estimate

  • The analytical estimate proves that interval width vanishes in absolute ebits, uniformly over the entire noise range, despite being conservative in finite dimensions.It estimates the maximum interval width studied numerically in Figure 2.Source: B Uniform large-dimension estimate · p. 16
  • The estimate compares isotropic entanglement of formation U_d with the cubic-norm lower bound B_d, using a convexity-based chord and a clipped affine lower bound.The difference between these bounds increases to the clipping point and then decreases.Source: B Uniform large-dimension estimate · p. 17
  • For p at or above the separability threshold, both endpoints of the certified interval are zero.Source: B Uniform large-dimension estimate · p. 17
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