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Limiting one-way distillable secret key via privacy testing of extendible states

Vishal Singh, Karol Horodecki, Aby Philip, Mark M. Wilde

TL;DR

The paper derives a tight, computable bound on the maximum probability that a $k$-extendible state passes a privacy test, and shows this bound equals the maximum fidelity with the maximally entangled state, providing a fundamental limit for private tasks under $k$-extendibility. This leads to efficient semidefinite-programming bounds on the one-shot and $n$-shot, one-way distillable key and the one-shot and $n$-shot forward-assisted private capacity of channels, with single-letter bounds in the i.i.d. setting. The authors introduce a family of $k$-unextendible divergences (hypothesis-testing, sandwiched Rényi, and geometric Rényi) that serve as resource monotones and yield practical, tight bounds that outperform previous efficiently computable bounds in key examples like isotropic and erasure channels. The work unifies private-key and private-channel capacities under the $k$-unextendibility framework and demonstrates the operational relevance of these divergences through numerical results and explicit bounds. The results hold broad implications for secure quantum communication and provide tools for evaluating private capacities with tractable optimization.

Abstract

The notions of privacy tests and $k$-extendible states have both been instrumental in quantum information theory, particularly in understanding the limits of secure communication. In this paper, we determine the maximum probability with which an arbitrary $k$-extendible state can pass a privacy test, and we prove that it is equal to the maximum fidelity between an arbitrary $k$-extendible state and the standard maximally entangled state. Our findings, coupled with the resource theory of $k$-unextendibility, lead to an efficiently computable upper bound on the one-shot, one-way distillable key of a bipartite state, and we prove that it is equal to the best-known efficiently computable upper bound on the one-shot, one-way distillable entanglement. We also establish efficiently computable upper bounds on the one-shot, forward-assisted private capacity of channels. Extending our formalism to the independent and identically distributed setting, we obtain single-letter efficiently computable bounds on the $n$-shot, one-way distillable key of a state and the $n$-shot, forward-assisted private capacity of a channel. For some key examples of interest, our bounds are significantly tighter than other known efficiently computable bounds.

Limiting one-way distillable secret key via privacy testing of extendible states

TL;DR

The paper derives a tight, computable bound on the maximum probability that a -extendible state passes a privacy test, and shows this bound equals the maximum fidelity with the maximally entangled state, providing a fundamental limit for private tasks under -extendibility. This leads to efficient semidefinite-programming bounds on the one-shot and -shot, one-way distillable key and the one-shot and -shot forward-assisted private capacity of channels, with single-letter bounds in the i.i.d. setting. The authors introduce a family of -unextendible divergences (hypothesis-testing, sandwiched Rényi, and geometric Rényi) that serve as resource monotones and yield practical, tight bounds that outperform previous efficiently computable bounds in key examples like isotropic and erasure channels. The work unifies private-key and private-channel capacities under the -unextendibility framework and demonstrates the operational relevance of these divergences through numerical results and explicit bounds. The results hold broad implications for secure quantum communication and provide tools for evaluating private capacities with tractable optimization.

Abstract

The notions of privacy tests and -extendible states have both been instrumental in quantum information theory, particularly in understanding the limits of secure communication. In this paper, we determine the maximum probability with which an arbitrary -extendible state can pass a privacy test, and we prove that it is equal to the maximum fidelity between an arbitrary -extendible state and the standard maximally entangled state. Our findings, coupled with the resource theory of -unextendibility, lead to an efficiently computable upper bound on the one-shot, one-way distillable key of a bipartite state, and we prove that it is equal to the best-known efficiently computable upper bound on the one-shot, one-way distillable entanglement. We also establish efficiently computable upper bounds on the one-shot, forward-assisted private capacity of channels. Extending our formalism to the independent and identically distributed setting, we obtain single-letter efficiently computable bounds on the -shot, one-way distillable key of a state and the -shot, forward-assisted private capacity of a channel. For some key examples of interest, our bounds are significantly tighter than other known efficiently computable bounds.

Paper Structure

This paper contains 32 sections, 11 theorems, 150 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Secret key distillation from states
  3. Private communication over channels
  4. Summary of results
  5. Background
  6. Quantum states and channels
  7. Secret keys and private states
  8. k-Extendibility
  9. k-Unextendible hypothesis testing divergence
  10. k-Unextendible sandwiched Rényi divergence
  11. Privacy test for k-extendible states
  12. Limits on one-way secret-key distillation from states
  13. Upper bounds on the one-shot, one-way distillable key of a state
  14. Upper bounds on the n-shot, one-way distillable key of a state
  15. Limits on forward-assisted private capacity of channels
  16. ...and 17 more sections

Key Result

Proposition 1

A bipartite state is $k$-extendible if and only if it can be written as a convex combination of $k$-pure extendible states.

Figures (4)

  • Figure 1: Upper bound on the number of secret bits that can be distilled from a single copy of an isotropic state with $\varepsilon = 0.05$. The bound from Theorem \ref{['theo:dd_1shot_key_bnd_k_ext']} is compared against the bound from SW25 for different values of the parameter $F$ of the isotropic state (see \ref{['eq:iso_st_defn']} for reference).
  • Figure 2: Upper bounds on the $n$-shot, one-way distillable key rate of a two-dimensional isotropic state with $F = 0.95$ and $\varepsilon = 10^{-5}$. The bounds are computed for different values of $k$ using Theorem \ref{['theo:dd_1shot_key_bnd_k_ext']}, and they are compared against the hypothesis-testing relative entropy of entanglement bound. The bounds from Theorem \ref{['theo:dd_1shot_key_bnd_k_ext']} can only be computed for a finite number of copies of the state, say $n$, since $E^{\varepsilon}_k\!\left(\rho^{\otimes n}\right)$ must be less than $\log_2 k$ for the bound to hold. This restriction manifests itself in the plot as the curves corresponding to $k=2$ and $k=3$ end abruptly.
  • Figure 3: Lower bound on the minimum number of copies of a two-dimensional isotropic state needed to distill a single secret bit, with error tolerance $\varepsilon$, using a one-way LOCC protocol. The lower bound on the minimum number of copies is computed for different values of the parameter $F$ (see \ref{['eq:iso_st_defn']}) and three different values of the error tolerance $\varepsilon$. When $F=1$, the isotropic state is a maximally entangled state, and only a single copy of the state would suffice to distill a secret bit with any $\varepsilon \in [0,1]$. However, the isotropic state becomes increasingly noisy as $F$ decreases, which means that a larger number of copies are needed to distill a single secret bit with the desired error tolerance.
  • Figure 4: (a) Upper bound on the $n$-shot, forward-assisted private capacity of an erasure channel using Theorem \ref{['theo:priv_cap_ub_hypo_test']} and SW25. The erasure probability is set equal to 0.3 and the error tolerance is set equal to $10^{-5}$. The upper bound from SW25 holds for only 70 channel uses for this choice of parameters. However, the bound from Theorem \ref{['theo:priv_cap_ub_hypo_test']} holds for 104 channel uses with this choice of parameters. (b) Lower bound on the minimum number of uses of an erasure channel needed to securely transmit a single bit over the channels, assisted by local operations and forward public communication.

Theorems & Definitions (16)

  • Definition 1: $k$-pure extendible state
  • Proposition 1
  • Theorem 1
  • Definition 2
  • Theorem 2
  • Corollary 1
  • Definition 3
  • Remark 1
  • Lemma 1
  • Theorem 3
  • ...and 6 more