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arXiv:2608.06308

On Optimal Quantum Data Hiding and Maximal Separable Ball

Zhi Li

quant-phSubmitted Aug 6, 2026• Announced Aug 7, 2026• 0 citations
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Abstract

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 $\mathbb C^n\otimes\mathbb C^m$, we prove that the optimal data-hiding ratios against separable and LOCC measurements are both $\min\{n,m\}$. This result follows from a stronger result that, for every $2\le p\le\infty$, the largest centered Schatten $p$-ball whose associated binary measurements are implementable by finite-round LOCC has radius $\min\{n,m\}^{2/p-1}$. 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 $n$, we prove that the optimal ratio is $(1+o(1))n$, with the upper bound obtained from a Gaussian rank-one POVM. For local operations without communication, we improve the universal upper bound to $(π\sqrt3/4+o(1))\min\{n,m\}$.

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quant-pharXiv:2609.40332•Oct 1, 2026•0 citations

On the Chernoff distance for asymptotic LOCC discrimination of bipartite quantum states

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(2,m)-threshold quantum data hiding

Donghoon Ha, Jeong San Kim

We consider multiparty quantum state discrimination and present a multiparty quantum data-hiding scheme for one classical bit to be shared among multiple parties. In the proposed scheme, any pair of parties can collaborate to perfectly recover the hidden bit through a joint measurement, whereas measurements based on local operations and classical communication(LOCC) performed even by all parties reveal only an arbitrarily small amount of information. We further provide bounds on the optimal LOCC discrimination of multiparty quantum states. The proposed scheme can be implemented using only separable states of low-dimensional quantum systems, enhancing its practical feasibility.

quant-pharXiv:2607.08070•Journal of Physics A Mathematical and Theoretical•Jul 10, 2026•0 citations
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}.
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.
Motivated by the recent discovery of a quantum Chernoff theorem for asymptotic state discrimination, we investigate the distinguishability of two bipartite mixed states under the constraint of local operations and classical communication (LOCC), in the limit of many copies. While for two pure states a result of Walgate et al. shows that LOCC is just as powerful as global measurements, data hiding states (DiVincenzo et al.) show that locality can impose severe restrictions on the distinguishability of even orthogonal states. Here we determine the optimal error probability and measurement to discriminate many copies of particular data hiding states (extremal d times d Werner states) by a linear programming approach. Surprisingly, the single-copy optimal measurement remains optimal for n copies, in the sense that the best strategy is measuring each copy separately, followed by a simple classical decision rule. We also put a lower bound on the bias with which states can be distinguished by separable operations. This is a shortened version of a paper [1] recently submitted to Communications in Mathematical Physics; here the proofs have been omitted.
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 )).
Entanglement and nonlocality play a fundamental role in quantum computing. To understand the interplay between these phenomena, researchers have considered the model of local operations and classical communication, or LOCC for short, which is a restricted subset of all possible operations that can be performed on a multipartite quantum system. The task of distinguishing states from a set that is known a priori to all parties is one of the most basic problems among those used to test the power of LOCC protocols, and it has direct applications to quantum data hiding, secret sharing and quantum channel capacity. The focus of this thesis is on state distinguishability problems for classes of quantum operations that are more powerful than LOCC, yet more restricted than global operations, namely the classes of separable and positive-partial-transpose (PPT) measurements. We \nbuild a framework based on convex optimization (on cone programming, in particular) to study such problems. Compared to previous approaches to the problem, the method described in this thesis provides precise numerical bounds and quantitative analytic results. By combining the duality theory of cone programming with the channel-state duality, we also establish a novel connection between the state distinguishability problem and the study of positive linear maps, which is a topic of independent interest in quantum information theory. \n \nWe apply our framework to several questions that were left open in previous works regarding the distinguishability of maximally entangled states and unextendable product sets. First, we exhibit small sets of orthogonal maximally entangled states in that are not perfectly distinguishable by LOCC. As a consequence of this, we show a gap between the power of PPT and separable measurements for the task of distinguishing sets consisting only of maximally entangled states. Furthermore, we prove tight bounds on the entanglement cost that is necessary to distinguish any sets of Bell states, thus showing that quantum teleportation is optimal for this task. Finally, we provide an easily checkable characterization of when an unextendable product set is perfectly discriminated by separable measurements, along with the first known example of an unextendable product set that cannot be perfectly discriminated by separable measurements.
Suppose we want to benchmark a quantum device held by a remote party, e.g., by testing its ability to carry out challenging quantum measurements outside of a free set of measurements M\mathcal{M}M. A very simple way to do so is to set up a binary-state discrimination task that cannot be solved efficiently by means of free measurements. If we can find pairs of orthogonal states that become arbitrarily indistinguishable under measurements in M\mathcal{M}M, in the sense that the error probability in discrimination approaches that of a random guess, we say that there are data hiding against M\mathcal{M}M. Here we investigate data hiding in the context of continuous-variable quantum systems. First, we look at the case where M\mathcal{M}M denotes the set of measurements implementable with local operations and classical communication. While previous studies have placed upper bounds on the maximum efficiency of data hiding in terms of the local dimension and are thus not applicable to continuous-variable systems, we establish more general bounds that rely solely on the local mean photon number of the states employed. Along the way, we perform a rigorous quantitative analysis of the error introduced by the nonideal Braunstein-Kimble quantum teleportation protocol, determining how much squeezing and local detection efficiency are needed in order to teleport an arbitrary multimode local state of known mean energy with a prescribed accuracy. Finally, following a seminal proposal by Sabapathy and Winter, we look at data hiding against Gaussian operations assisted by the feedforward of measurement outcomes, providing an example of a relatively simple scheme that works with a single mode only.
We consider multiparty quantum state discrimination and present a multiparty quantum data-hiding scheme for one classical bit to be shared among multiple parties. In the proposed scheme, any pair of parties can collaborate to perfectly recover the hidden bit through a joint measurement, whereas measurements based on local operations and classical communication(LOCC) performed even by all parties reveal only an arbitrarily small amount of information. We further provide bounds on the optimal LOCC discrimination of multiparty quantum states. The proposed scheme can be implemented using only separable states of low-dimensional quantum systems, enhancing its practical feasibility.