To probe a bipartite quantum system, one may use arbitrary global operators or restrict to product operators acting separately on the two subsystems. We determine the sharp universal comparison between the resulting norms. For every $z\in M_n\otimes M_m$, we prove $\|z\|_1\leq\sqrt{2}\min\{n,m\}\|z\|_\varepsilon$, where $\|\cdot\|_1$ is the trace norm and $\|\cdot\|_\varepsilon$ is the injective tensor norm associated with the trace norms on $M_n$ and $M_m$. To prove the upper bound, we establish an $L_1$ noncommutative Khintchine inequality whose random coefficients are the entries of a Haar unitary. We also show that the coefficient $\sqrt{2}$ is sharp. As applications, we show that the same sharp constant governs the gap between bipartite correlation measured in trace norm and that measured by a correlation function, and obtain an improved universal upper bound for quantum data hiding. The upper bound has also been formalized and machine-checked in Lean.
To probe a bipartite quantum system, one may use arbitrary global operators or restrict to product operators acting separately on the two subsystems. We determine the sharp universal comparison between the resulting norms. For every z∈Mn⊗Mm, we prove ∥z∥1≤2min{n,m}∥z∥ε, where ∥⋅∥1 is the trace norm and ∥⋅∥ε is the injective tensor norm associated with the trace norms on Mn and Mm. To prove the upper bound, we establish an L1 noncommutative Khintchine inequality whose random coefficients are the entries of a Haar unitary. We also show that the coefficient 2 is sharp. As applications, we show that the same sharp constant governs the gap between bipartite correlation measured in trace norm and that measured by a correlation function, and obtain an improved universal upper bound for quantum data hiding. The upper bound has also been formalized and machine-checked in Lean.
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×nB bipartite quantum system can be discriminated via local measurements with an error probability no larger than 21(1−cmin{nA,nB}1) , where 1≤c≤22 is a universal constant, and our bound scales provably optimally with the local dimensions nA,nB . Mathematically, this is achieved by showing that the distinguishability norm ∥⋅∥LO associated with local measurements satisfies that ∥⋅∥1≤22min{nA,nB}∥⋅∥LO , where ∥⋅∥1 is the trace norm.
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, 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≤p≤∞, 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 (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 n-qudit Werner states is 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) to 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) 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.
In this work, we present an in-depth study of average-case quantum distances introduced in Maciejewski et al. (2022). The average-case distances approximate, up to the relative error, the average Total-Variation (TV) distance between measurement outputs of two quantum processes, in which quantum objects of interest (states, measurements, or channels) are intertwined with random quantum circuits. Contrary to conventional distances, such as trace distance or diamond norm, they quantifyaverage-casestatistical distinguishability via random quantum circuits. We prove that once a family of random circuits forms an$\delta −approximate4−design,with\delta =o(d^{-8}),thentheaverage−casedistancescanbeapproximatedbysimpleexplicitfunctionsthatcanbeexpressedviasimpledegreetwopolynomialsinobjectsofinterest.Forsystemsofmoderatedimension,theycanbeeasilyexplicitlycomputed–nooptimizationisneededasopposedtodiamondnormdistancebetweenchannelsoroperationaldistancebetweenmeasurements.Weprovethatthosefunctions,whichwecallquantumaverage−casedistances,haveaplethoraofdesirableproperties,suchassubadditivityw.r.t.tensorproducts,jointconvexity,and(restricted)data−processinginequalities.Notably,alldistancesutilizetheHilbert−Schmidt(HS)norm,whichprovidesthisnormwithanewoperationalinterpretation.Wealsoprovideupperboundsonthemaximalratiobetweenworst−caseandaverage−casedistances,andforeachofthem,weprovideanexamplethatsaturatesthebound.Specifically,weshowthatforeachdimensiondthisratioisatmostd^{\frac {1}{2}}, d, d^{\frac {3}{2}}$for states, measurements, and channels, respectively. To support the practical usefulness of our findings, we study multiple examples in which average-case quantum distances can be calculated analytically.
Multipartite entanglement is the premier resource for quantum technologies. Yet, its exact quantification in the laboratory is notoriously challenging, typically requiring the full knowledge of high dimensional quantum states. Here, we construct observable bounds to multipartite entanglement for systems of arbitrary size, which are functions of the local and global state purities, and correlation functions. First, we derive experimentally accessible upper and lower limits to both the bipartite entanglement of formation and the squashed entanglement of bipartite systems, by leveraging cornerstone results of quantum information theory: the entropy strong subadditivity inequality and the Koashi-Winter monogamy relation. Then, we convert them into bounds to the entanglement up to degree k for arbitrary states, and to the genuine k-partite entanglement, by employing a recently proposed method. Finally, we analytically and numerically test these results, by bounding the multipartite entanglement of several relevant states and mixtures, including the important classes of GHZ, Dicke, W states, and random pure states.
Abstract We define and study XOR games in the framework of general probabilistic theories, which encompasses all physical models whose predictive power obeys minimal requirements. The bias of an XOR game under local or global strategies is shown to be given by a certain injective or projective tensor norm, respectively. The intrinsic (i.e. model-independent) advantage of global over local strategies is thus connected to a universal functionr(n, m) called ‘projective–injective ratio’. This is defined as the minimal constant ρ ρ such that ∥⋅∥X⊗πY⩽ρ∥⋅∥X⊗εY ‖·‖X⊗πY⩽ρ‖·‖X⊗εY holds for all Banach spaces of dimensions dimX=n dimX=n and dimY=m dimY=m , where X⊗πY X⊗πY and X⊗εY X⊗εY are the projective and injective tensor products. By requiring that X=Y X=Y , one obtains a symmetrised version of the above ratio, denoted by rs(n) rs(n) . We prove that r(n,m)⩾19/18 r(n,m)⩾19/18 for all n,m⩾2 n,m⩾2 , implying that injective and projective tensor products are never isometric. We then study the asymptotic behaviour ofr(n, m) and rs(n) rs(n) , showing that, up to log factors: rs(n) rs(n) is of the order n n (which is sharp);r(n, n) is at least of the order n1/6 n1/6 ; andr(n, m) grows at least as min{n,m}1/8 min{n,m}1/8 . These results constitute our main contribution to the theory of tensor norms. In our proof, a crucial role is played by an ‘ ℓ1/ℓ2/ℓ∞ ℓ1/ℓ2/ℓ∞ trichotomy theorem’ based on ideas by Pisier, Rudelson, Szarek, and Tomczak-Jaegermann. The main operational consequence we draw is that there is a universal gap between local and global strategies in general XOR games, and that this grows as a power of the minimal local dimension. In the quantum case, we are able to determine this gap up to universal constants. As a corollary, we obtain an improved bound on the scaling of the maximal quantum data hiding efficiency against local measurements.