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Quantum Information and Computation

Locally restricted measurements on a multipartite quantum system: data hiding is generic

Guillaume Aubrun, Cécilia Lancien

Published Apr 1, 2015• 10 citations
Publisher

Abstract

We study the distinguishability norms associated to families of locally restricted POVMs on multipartite systems. These norms (introduced by Matthews, Wehner and Winter) quantify how quantum measurements, subject to locality constraints, perform in the task of discriminating two multipartite quantum states. We mainly address the following question regarding the behaviour of these distinguishability norms in the highdimensional regime: On a bipartite space, what are the relative strengths of standard classes of locally restricted measurements? We show that the class of PPT measurements typically performs almost as well as the class of all measurements whereas restricting to local measurements and classical communication, or even just to separable measurements, implies a substantial loss. We also provide examples of state pairs which can be perfectly distinguished by local measurements if (one-way) classical communication is allowed between the parties, but very poorly without it. Finally, we study how many POVMs are needed to distinguish almost perfectly any pair of states on C^d, showing that the answer is exp(Θ(d^2 )).

DOI
10.26421/qic15.5-6-9

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We study the distinguishability norms associated to families of locally restricted POVMs on multipartite systems. These norms (introduced by Matthews, Wehner and Winter) quantify how quantum measurements, subject to locality constraints, perform in the task of discriminating two multipartite quantum states. We mainly address the following question regarding the behaviour of these distinguishability norms in the highdimensional regime: On a bipartite space, what are the relative strengths of standard classes of locally restricted measurements? We show that the class of PPT measurements typically performs almost as well as the class of all measurements whereas restricting to local measurements and classical communication, or even just to separable measurements, implies a substantial loss. We also provide examples of state pairs which can be perfectly distinguished by local measurements if (one-way) classical communication is allowed between the parties, but very poorly without it. Finally, we study how many POVMs are needed to distinguish almost perfectly any pair of states on C^d, showing that the answer is exp(Θ(d^2 )).
We study the distinguishability associated to either a single POVM or a family of POVMs. These norms quantify how quantum measurements perform in the task of discriminating two quantum states. We address the following questions regarding their behaviour in the high-dimensional regime: (i) How many POVMs are needed to approximate the family of all POVMs on C^d? We show that the answer is exp(O(d^2)). (ii) How many outcomes are required for a POVM to emulate the uniform POVM on C^d? Improving on previous results, we show that O(d^2) outcomes are necessary and sufficient. Moreover, using the concept of zonoid and approximation results from Banach space theory, we show that this property extends (in a weaker form) to an arbitrary POVM. (iii) On a bipartite space, what are the relative strengths of standard classes of locally restricted measurements? We show that the class of PPT measurements typically performs almost as well as the class of all measurements whereas restricting to separable measurements implies a substantial loss.
Quantum data hiding concerns pairs of states which are highly distinguishable with global measurements, yet nearly indistinguishable when restricted to local operations and classical communication. More than two decades after Eggeling and Werner introduced a multipartite hiding scheme based on Werner states, the optimal dependence of its uniform security guarantee on the number of parties and local dimension remained unknown. We resolve this problem by showing that the distinguishing bias of any pair of nnn-qudit Werner states is O(n2/d)O(n^2/d)O(n2/d) under measurements whose effects remain positive under partial transposition (PPT) across every bipartition. An explicit pair attains this scaling using only nonadaptive local measurements, establishing worst-case optimality and showing that the PPT relaxation preserves the optimal dependence on both the number of parties and the local dimension. At fixed security, this extends the certified hiding regime from n=O(d1/4)n=O(d^{1/4})n=O(d1/4) to n=O(d)n=O(\sqrt d)n=O(d​). Beyond data hiding, the same bound implies that testing any nontrivial unitarily invariant property with adaptive single-copy measurements requires Ω(d)Ω(\sqrt d)Ω(d​) copies, yielding separations for any property with dimension-independent sample complexity under collective measurements. Our proof reduces the distinguishing bias to trace norms of partially transposed operators and analyzes them using mixed Schur-Weyl duality, demonstrating the utility of representation theoretic methods developed for port-based teleportation to data hiding and quantum property testing.
The unique features of entanglement and non-locality in quantum systems, where there are pairs of bipartite states perfectly distinguishable by general entangled measurements yet indistinguishable by local operations and classical communication, hold significant importance in quantum entanglement theory, distributed quantum information processing, and quantum data hiding. This paper delves into the entanglement cost for discriminating two bipartite quantum states, employing positive operator-valued measures (POVMs) with positive partial transpose (PPT) to achieve optimal success probability through general entangled measurements. We first introduce two quantities called the spectral PPT-distance and relative spectral PPT-distance of a POVM to quantify the localness of a general measurement. We show these quantities are related to the entanglement cost of optimal discrimination by PPT POVMs. Following this, we establish bounds and develop SDP hierarchies to estimate the entanglement cost of optimal discrimination by PPT POVMs for any pair of states. Leveraging these results, we show that a pure state can be optimally discriminated against any other state with the assistance of a single Bell state. This study advances our understanding of the pivotal role played by entanglement in quantum state discrimination, serving as a crucial element in unlocking quantum data hiding against locally constrained measurements.
The unique features of entanglement and non-locality in quantum systems, where there are pairs of bipartite states perfectly distinguishable by general entangled measurements yet indistinguishable by local operations and classical communication, hold significant importance in quantum entanglement theory, distributed quantum information processing, and quantum data hiding. This paper delves into the entanglement cost for discriminating two bipartite quantum states, employing positive operator-valued measures (POVMs) with positive partial transpose (PPT) to achieve optimal success probability through general entangled measurements. First, we introduce an efficiently computable quantity called the spectral PPT-distance of a POVM to quantify the localness of a general measurement. We show that it can be a lower bound for the entanglement cost of optimal discrimination by PPT POVMs. Second, we establish an upper bound on the entanglement cost of optimal discrimination by PPT POVMs for any pair of states. Leveraging this result, we show that a pure state can be optimally discriminated against any other state with the assistance of a single Bell state. This study advances our understanding of the pivotal role played by entanglement in quantum state discrimination, serving as a crucial element in unlocking quantum data hiding against locally constrained measurements.
Quantum data hiding asks how much distinguishing power can be lost when global measurements are restricted to local measurements and classical communication. In this work, we establish sharp results and improved bounds for several natural classes of restricted measurements. For bipartite systems on Cn⊗Cm\mathbb C^n\otimes\mathbb C^mCn⊗Cm, we prove that the optimal data-hiding ratios against separable and LOCC measurements are both min⁡{n,m}\min\{n,m\}min{n,m}. This result follows from a stronger result that, for every 2≤p≤∞2\le p\le\infty2≤p≤∞, the largest centered Schatten ppp-ball whose associated binary measurements are implementable by finite-round LOCC has radius min⁡{n,m}2/p−1\min\{n,m\}^{2/p-1}min{n,m}2/p−1. This strengthens the classic separable-ball theorems, while also providing an explicit finite-round LOCC implementation. For Alice-first one-way LOCC with Alice's local dimension equal to nnn, we prove that the optimal ratio is (1+o(1))n(1+o(1))n(1+o(1))n, with the upper bound obtained from a Gaussian rank-one POVM. For local operations without communication, we improve the universal upper bound to (π3/4+o(1))min⁡{n,m}(π\sqrt3/4+o(1))\min\{n,m\}(π3​/4+o(1))min{n,m}.
We prove a tight and close-to-optimal lower bound on the effectiveness of local quantum measurements (without classical communication) at discriminating any two bipartite quantum states. Our result implies, for example, that any two orthogonal quantum states of a nA×nBn_{A}\times n_{B}nA​×nB​ bipartite quantum system can be discriminated via local measurements with an error probability no larger than 12(1−1cmin⁡{nA,nB})\frac {1}2 \left ({1 - \frac {1}{c \min \{n_{A}, n_{B}\}} }\right)21​(1−cmin{nA​,nB​}1​) , where 1≤c≤221\leq c\leq 2\sqrt {2}1≤c≤22​ is a universal constant, and our bound scales provably optimally with the local dimensions nA,nBn_{A},n_{B}nA​,nB​ . Mathematically, this is achieved by showing that the distinguishability norm ∥⋅∥LO\|\cdot \|_{ \mathrm {LO}}∥⋅∥LO​ associated with local measurements satisfies that ∥⋅∥1≤22min⁡{nA,nB}∥⋅∥LO\|\cdot \|_{1}\leq 2\sqrt {2} \min \{n_{A},n_{B}\} \|\cdot \|_{ \mathrm {LO}}∥⋅∥1​≤22​min{nA​,nB​}∥⋅∥LO​ , where ∥⋅∥1\|\cdot \|_{1}∥⋅∥1​ is the trace norm.