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arXiv:2609.40332

Tight Universal Bounds on Quantum Data Hiding with Multipartite Werner States

Oren Akresh, Jacob Beckey, Felix Leditzky

quant-phSubmitted Sep 30, 2026• Announced Oct 1, 2026• 0 citations
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Abstract

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 $n$-qudit Werner states is $O(n^2/d)$ 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 $n=O(d^{1/4})$ to $n=O(\sqrt d)$. Beyond data hiding, the same bound implies that testing any nontrivial unitarily invariant property with adaptive single-copy measurements requires $Ω(\sqrt d)$ 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.

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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.
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.
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 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 )).
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.
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.
We study the distinguishability of quantum states under local operations with classical communication (LOCC), separable, and positive-partial-transpose (PPT) measurements, focusing on locally diagonal orthogonally invariant (LDOI) states -- those invariant under local diagonal orthogonal twirling. This class includes many important families such as Werner states, isotropic states, X-states, and Dicke states. We show that optimal PPT and separable measurements for distinguishing LDOI states can always be taken to be LDOI, and the LOCC supremum can be approached by LDOI LOCC POVMs, enabling a dimensional reduction from n4n^4n4 to O(n2)O(n^2)O(n2) in the associated optimization problems. We establish efficiently computable bounds on the distinguishability of orthonormal LDOI bases and prove that for a broad class of such bases -- including all two-qubit cases -- the LOCC supremum equals the PPT and separable optima. More generally, we show the gap between PPT and LOCC distinguishability is at most (n−2)/(2n2)(n-2)/(2n^2)(n−2)/(2n2) for local dimension nnn.