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.
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 )).
Standard approaches to quantum statistical inference rely on measurements that induce a collapse of the wave function, effectively consuming the quantum state to extract information. In this work, we investigate the fundamental limits of \emph{locally-gentle} quantum state certification, where the learning algorithm is constrained to perturb the state by at most
α in trace norm, thereby allowing for the reuse of samples. We analyze the hypothesis testing problem of distinguishing whether an unknown state
ρ is equal to a reference
ρ0 or
ε-far from it. We derive the minimax sample complexity for this problem, quantifying the information-theoretic price of non-destructive measurements. Specifically, by constructing explicit measurement operators, we show that the constraint of
α-gentleness imposes a sample size penalty of
α2d, yielding a total sample complexity of
n=Θ(ε2α2d3). Our results clarify the trade-off between information extraction and state disturbance, and highlight deep connections between physical measurement constraints and privacy mechanisms in quantum learning. Crucially, we find that the sample size penalty incurred by enforcing
α-gentleness scales linearly with the Hilbert-space dimension
d rather than the number of parameters
d2−1 typical for high-dimensional private estimation.
The present paper studies an operator norm that captures the distinguishability of quantum strategies in the same sense that the trace norm captures the distinguishability of quantum states or the diamond norm captures the distinguishability of quantum channels. Characterizations of its unit ball and dual norm are established via strong duality of a semidefinite optimization problem. A full, formal proof of strong duality is presented for the semidefinite optimization problem in question. This norm and its properties are employed to generalize a state discrimination result of Ref. [GW05]. The generalized result states that for any two convex sets S,T of strategies there exists a fixed interactive measurement scheme that successfully distinguishes any choice of s in S from any choice of t in T with bias proportional to the minimal distance between the sets S and T as measured by this norm. A similar discrimination result for channels then follows as a special case.