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arXiv:1206.1307

Non-Additivity of the Entanglement of Purification (Beyond Reasonable Doubt)

Jianxin Chen, Andreas Winter

Published Jun 6, 2012• 9 citations
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Abstract

We demonstrate the convexity of the difference between the regularized entanglement of purification and the entropy, as a function of the state. This is proved by means of a new asymptotic protocol to prepare a state from pre-shared entanglement and by local operations only. We go on to employ this convexity property in an investigation of the additivity of the (single-copy) entanglement of purification: using numerical results for two-qubit Werner states we find strong evidence that the entanglement of purification is different from its regularization, hence that entanglement of purification is not additive.

DOI
10.48550/arxiv.1206.1307

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In order to perform faithfully quantum communication protocols such as quantum teleportation, superdense coding, quantum key distribution, and quantum secret sharing, we need the entanglement purification process to make a maximal entanglement out of many copies of the partially entangled pairs by using local operations and classical communication. In this dissertation,\nwe analyze the efficiency of several entanglement purification protocols for qubit states and qudit states, and present a new efficient entanglement purification protocol. Furthermore, we show that our method can be applied to the existing best protocol\nso as to improve the efficiency.

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We demonstrate the convexity of the difference between the regularized entanglement of purification and the entropy, as a function of the state. This is proved by means of a new asymptotic protocol to prepare a state from pre-shared entanglement and by local operations only. We go on to employ this convexity property in an investigation of the additivity of the (single-copy) entanglement of purification: using numerical results for two-qubit Werner states we find strong evidence that the entanglement of purification is different from its regularization, hence that entanglement of purification is not additive.
The entanglement of purification EPE_PEP​, introduced by Terhal, Horodecki, Leung, and DiVincenzo in 2002, measures the total correlations of a bipartite state by the entanglement needed to purify it; whether it is additive on tensor powers has been open since its introduction. In 2012 Chen and Winter argued "beyond reasonable doubt", from numerical evidence, that the two-qubit Werner state at singlet fraction f=1/200f = 1/200f=1/200 is a counterexample, and explicitly left a completely rigorous proof to future work. To our knowledge we give the first rigorous proof: for this state WWW, EP∞(W)≤0.9663…<97/100<EP(W)E_P^\infty(W) \le 0.9663\ldots < 97/100 < E_P(W)EP∞​(W)≤0.9663…<97/100<EP​(W), hence the entanglement of purification is strictly nonadditive at some finite tensor power. The lower bound is a finite certificate: analytic reductions send the Schmidt spectrum of every purification in the complete complex Stiefel domain to a point satisfying explicit necessary inequalities in a three-dimensional box, and an exact-rational subdivision of that box has 25,383 leaves, each of which either excludes nonphysical points or proves the entropy bound. The upper bound combines an exact symmetric-support endpoint, an explicit feasible isometry, and a convexity theorem for the regularized quantity, closed by two directed rational logarithm certificates. Every computer-assisted certificate reduces to integer comparisons; a dependency-free program verifies all of them from the supplementary artifact.
We prove that the von Neumann entanglement of purification is nonadditive. For two-qubit Werner states W(f)W(f)W(f) with singlet weight 1/64≤f≤1/41/64\le f\le1/41/64≤f≤1/4, we show that canonical purification is globally optimal: its entanglement, also known as reflected entropy, gives the exact entanglement of purification. In contrast, the regularized entanglement of purification lies strictly below the reflected entropy for 0<f<f∗0<f<f_\ast0<f<f∗​, where f∗≈0.017f_\ast\approx0.017f∗​≈0.017. Together, these results establish nonadditivity throughout 1/64≤f<f∗1/64\le f<f_\ast1/64≤f<f∗​. Our canonical-optimality proof generalizes the proof in [1], which established Rényi nonadditivity for 0≤α<10\leα<10≤α<1 using classical two-qubit states. We thus resolve the additivity question at α=1α=1α=1. Remarkably, nonadditivity is again witnessed by separable states and no entanglement in the underlying states is required.
Entanglement of purification is a fundamental measure of total correlations whose additivity remains unresolved. We study its additivity for classical states on two qubits at different Rényi orders. For every α∈[0,1)α\in[0,1)α∈[0,1), we prove nonadditivity within this family, witnessed by two copies of a single state. We first solve the one-copy optimization exactly for the entire family at every Rényi order. We then restrict the two-copy optimization to a natural finite set of purifications and exhibit one whose entropy is strictly below the product value. In contrast, for α∈[2,∞]α\in[2,\infty]α∈[2,∞] we prove additivity under tensor products within this family. The interval α∈[1,2)α\in[1,2)α∈[1,2), including the von Neumann case α=1α=1α=1, remains open, and we conjecture additivity there throughout the same family.
In order to perform faithfully quantum communication protocols such as quantum teleportation, superdense coding, quantum key distribution, and quantum secret sharing, we need the entanglement purification process to make a maximal entanglement out of many copies of the partially entangled pairs by using local operations and classical communication. In this dissertation,\nwe analyze the efficiency of several entanglement purification protocols for qubit states and qudit states, and present a new efficient entanglement purification protocol. Furthermore, we show that our method can be applied to the existing best protocol\nso as to improve the efficiency.