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Optimising quantum data hiding

Francesco Anna Mele, Ludovico Lami

TL;DR

This work resolves an open problem in quantum data hiding by constructing explicit bipartite, separable, orthogonal data hiding states that are nearly indistinguishable under LOCC. The authors start from a pair of separable, PPT states $oldsymbol{\sigma_0^{(d)}}, oldsymbol{\sigma_1^{(d)}}$ and apply a parity-encoding technique across $k$ copies to obtain $oldsymbol{ ho_0^{(k,d)}}, oldsymbol{ ho_1^{(k,d)}}$, which remain separable and orthogonal while achieving LOCC indistinguishability that decays as $oldsymbol{2oldsymbol{oldsymbol{}}_d^k}$. By bounding the LOCC norm via the PPT norm and reducing the problem to a tractable linear program, the authors derive an explicit bound on the local dimension: $oldsymbol{D_oldsymbol{ u} obreak obreak obreak le 40igl(2/oldsymbol{ u}igr)^{10}}$ for any $oldsymbol{ u} obreak obreak (0,1)$. The result demonstrates nonlocality without entanglement at maximal extent and introduces novel numerical-analytic techniques (Sanov-type large deviations, Tikhonov regularisation) with potential wider applicability to convex optimisation in quantum information.

Abstract

Quantum data hiding is the existence of pairs of bipartite quantum states that are (almost) perfectly distinguishable with global measurements, yet close to indistinguishable when only measurements implementable with local operations and classical communication are allowed. Remarkably, data hiding states can also be chosen to be separable, meaning that secrets can be hidden using no entanglement that are almost irretrievable without entanglement -- this is sometimes called `nonlocality without entanglement'. Essentially two families of data hiding states were known prior to this work: Werner states and random states. Hiding Werner states can be made either separable or globally perfectly orthogonal, but not both -- separability comes at the price of orthogonality being only approximate. Random states can hide many more bits, but they are typically entangled and again only approximately orthogonal. In this paper, we present an explicit construction of novel group-symmetric data hiding states that are simultaneously separable, perfectly orthogonal, and even invariant under partial transpose, thus exhibiting the phenomenon of nonlocality without entanglement to the utmost extent. Our analysis leverages novel applications of numerical analysis tools to study convex optimisation problems in quantum information theory, potentially offering technical insights that extend beyond this work.

Optimising quantum data hiding

TL;DR

This work resolves an open problem in quantum data hiding by constructing explicit bipartite, separable, orthogonal data hiding states that are nearly indistinguishable under LOCC. The authors start from a pair of separable, PPT states and apply a parity-encoding technique across copies to obtain , which remain separable and orthogonal while achieving LOCC indistinguishability that decays as . By bounding the LOCC norm via the PPT norm and reducing the problem to a tractable linear program, the authors derive an explicit bound on the local dimension: for any . The result demonstrates nonlocality without entanglement at maximal extent and introduces novel numerical-analytic techniques (Sanov-type large deviations, Tikhonov regularisation) with potential wider applicability to convex optimisation in quantum information.

Abstract

Quantum data hiding is the existence of pairs of bipartite quantum states that are (almost) perfectly distinguishable with global measurements, yet close to indistinguishable when only measurements implementable with local operations and classical communication are allowed. Remarkably, data hiding states can also be chosen to be separable, meaning that secrets can be hidden using no entanglement that are almost irretrievable without entanglement -- this is sometimes called `nonlocality without entanglement'. Essentially two families of data hiding states were known prior to this work: Werner states and random states. Hiding Werner states can be made either separable or globally perfectly orthogonal, but not both -- separability comes at the price of orthogonality being only approximate. Random states can hide many more bits, but they are typically entangled and again only approximately orthogonal. In this paper, we present an explicit construction of novel group-symmetric data hiding states that are simultaneously separable, perfectly orthogonal, and even invariant under partial transpose, thus exhibiting the phenomenon of nonlocality without entanglement to the utmost extent. Our analysis leverages novel applications of numerical analysis tools to study convex optimisation problems in quantum information theory, potentially offering technical insights that extend beyond this work.

Paper Structure

This paper contains 12 sections, 13 theorems, 91 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Construction of separable, orthogonal quantum data hiding states
  4. Two special states
  5. Bounding the LOCC norm via the PPT norm
  6. Bounding the PPT norm
  7. Concluding the proof
  8. Conclusions
  9. Acknowledgements
  10. $\pazocal{G}$-twirling
  11. LOCC distinguishability of the two special states
  12. Invariance of the two special states under partial transposition

Key Result

Theorem 2

For every $\varepsilon\in(0,1)$ there exist bipartite states $\rho_1,\rho_2$ on $\mathds{C}^{D}\otimes\mathds{C}^{D}$ that are both separable and orthogonal, and satisfy with local dimension bounded as $D \le 40\bigl(\tfrac{2}{\varepsilon}\bigr)^{10}$.

Figures (1)

  • Figure 1: Behaviour of $\mu_d$ for $2 \le d \le 1000$. The function is monotonically decreasing in the parameter $d$, with $\mu_2 = 1$, $\mu_3 \approx 0.993$, and asymptotic value $\lim_{d \to \infty} \mu_d = \sqrt{3/8} \approx 0.612$. Crucially, it satisfies $\mu_d<1$ for all $d\ge3$.

Theorems & Definitions (28)

  • Definition 1: ($\varepsilon$-quantum data hiding states)
  • Theorem 2: (Existence of separable, orthogonal quantum data hiding states)
  • Remark 3: (Required dimension for quantum data hiding)
  • Conjecture 4
  • Conjecture 5: (Construction of orthogonal, separable quantum data hiding states)
  • Definition 6: (Two special states)
  • Lemma 7
  • Lemma 8
  • proof : Proof of Lemma \ref{['lemma_sepsep']}
  • Proposition 9: (Bounds on the LOCC norm between the two special states)
  • ...and 18 more