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Computable entanglement cost under positive partial transpose operations

Ludovico Lami, Francesco Anna Mele, Bartosz Regula

TL;DR

This work addresses the long-standing problem of computing the asymptotic zero-error PPT entanglement cost for arbitrary finite-dimensional quantum states, a task complicated by regularisation over many copies. The authors construct two SDP-based hierarchies, the \chi-p and \kappa-q sequences, that bound the cost from below and above and prove that their limits converge to the true cost with exponential speed in the hierarchy level p (uniformly over states) and without requiring a closed-form single-letter formula. They prove additivity for the \chi-hierarchy, show counterexamples to additivity for E_{\kappa}, and derive a tight, efficiently computable upper and lower bound that collapses to E^{\mathrm{exact}}_{c,\mathrm{PPT}} in the limit, enabling a polynomial-time algorithm to approximate the cost to any prescribed additive accuracy. This yields the first efficiently computable, operationally meaningful asymptotic entanglement measure applicable to all states, not just those with zero bi-negativity, and provides a practical tool to bound LOCC limits via PPT transformations. The results have broad implications for entanglement theory and quantum information tasks where exact LOCC capabilities are hard to characterize, offering a concrete, scalable route to quantify entanglement resources.

Abstract

Quantum information theory is plagued by the problem of regularisations, which require the evaluation of formidable asymptotic quantities. This makes it computationally intractable to gain a precise quantitative understanding of the ultimate efficiency of key operational tasks such as entanglement manipulation. Here we consider the problem of computing the asymptotic entanglement cost of preparing noisy quantum states under quantum operations with positive partial transpose (PPT). By means of an analytical example, a previously claimed solution to this problem is shown to be incorrect. Building on a previous characterisation of the PPT entanglement cost in terms of a regularised formula, we construct instead a hierarchy of semi-definite programs that bypasses the issue of regularisation altogether, and converges to the true asymptotic value of the entanglement cost. Our main result establishes that this convergence happens exponentially fast, thus yielding an efficient algorithm that approximates the cost up to an additive error $\varepsilon$ in time $\mathrm{poly}(D,\,\log(1/\varepsilon))$, where $D$ is the underlying Hilbert space dimension. To our knowledge, this is the first time that an asymptotic entanglement measure is shown to be efficiently computable despite no closed-form formula being available.

Computable entanglement cost under positive partial transpose operations

TL;DR

This work addresses the long-standing problem of computing the asymptotic zero-error PPT entanglement cost for arbitrary finite-dimensional quantum states, a task complicated by regularisation over many copies. The authors construct two SDP-based hierarchies, the \chi-p and \kappa-q sequences, that bound the cost from below and above and prove that their limits converge to the true cost with exponential speed in the hierarchy level p (uniformly over states) and without requiring a closed-form single-letter formula. They prove additivity for the \chi-hierarchy, show counterexamples to additivity for E_{\kappa}, and derive a tight, efficiently computable upper and lower bound that collapses to E^{\mathrm{exact}}_{c,\mathrm{PPT}} in the limit, enabling a polynomial-time algorithm to approximate the cost to any prescribed additive accuracy. This yields the first efficiently computable, operationally meaningful asymptotic entanglement measure applicable to all states, not just those with zero bi-negativity, and provides a practical tool to bound LOCC limits via PPT transformations. The results have broad implications for entanglement theory and quantum information tasks where exact LOCC capabilities are hard to characterize, offering a concrete, scalable route to quantify entanglement resources.

Abstract

Quantum information theory is plagued by the problem of regularisations, which require the evaluation of formidable asymptotic quantities. This makes it computationally intractable to gain a precise quantitative understanding of the ultimate efficiency of key operational tasks such as entanglement manipulation. Here we consider the problem of computing the asymptotic entanglement cost of preparing noisy quantum states under quantum operations with positive partial transpose (PPT). By means of an analytical example, a previously claimed solution to this problem is shown to be incorrect. Building on a previous characterisation of the PPT entanglement cost in terms of a regularised formula, we construct instead a hierarchy of semi-definite programs that bypasses the issue of regularisation altogether, and converges to the true asymptotic value of the entanglement cost. Our main result establishes that this convergence happens exponentially fast, thus yielding an efficient algorithm that approximates the cost up to an additive error in time , where is the underlying Hilbert space dimension. To our knowledge, this is the first time that an asymptotic entanglement measure is shown to be efficiently computable despite no closed-form formula being available.
Paper Structure (23 sections, 30 theorems, 171 equations)

This paper contains 23 sections, 30 theorems, 171 equations.

Table of Contents

  1. End matter
  2. Introduction
  3. Quantum states and channels
  4. Positive partial transpose
  5. Zero-error PPT entanglement cost
  6. Violations of additivity of E_kappa
  7. New entanglement monotones: E_chi,p and E_kappa,q
  8. Rationale of our construction
  9. Relation between the chi- and kappa-hierarchies
  10. and as PPT monotones
  11. Dual expressions
  12. Additivity properties
  13. Continuity
  14. Main results
  15. Enter the regularisation
  16. ...and 8 more sections

Key Result

Theorem 1

For all bipartite states $\rho=\rho_{AB}$ on a system of minimal local dimension $d\coloneqq \min\left\{|A|,|B|\right\}\geq 2$, and all positive integers $p\in \mathds{N}^+$, it holds that entailing that

Theorems & Definitions (70)

  • Theorem 1: (Exact expression of the cost)
  • Corollary 2: (Simple formula for the cost of qubit-qudit states)
  • Theorem 3: (Time to compute the cost)
  • Lemma 4
  • Proposition 5
  • proof : Proof sketch of Theorem \ref{['convergence_thm']}
  • proof : Proof sketch of Theorem \ref{['efficient_algorithm_thm']}
  • Lemma S6: (Variational characterisation of the trace norm)
  • proof
  • Definition S7: (Punch card states)
  • ...and 60 more