Upper bounds on the error probabilities and asymptotic error exponents in quantum multiple state discrimination

Abstract

We consider the multiple hypothesis testing problem for symmetric quantum state discrimination between r given states σ 1 , . . . , σ r . By splitting up the overall test into multiple binary tests in various ways we obtain a number of upper bounds on the optimal error probability in terms of the binary error probabilities. These upper bounds allow us to deduce various bounds on the asymptotic error rate, for which it has been hypothesized that it is given by the multi-hypothesis quantum Chernoff bound (or Chernoff divergence) C(σ 1 , . . . , σ r ), as recently introduced by Nussbaum and Szko la in analogy with Salikhov's classical multi-hypothesis Chernoff bound. This quantity is defined as the minimum of the pairwise binary Chernoff divergences min j<k C(σ j , σ k ). It was known already that the optimal asymptotic rate must lie between C/3 and C, and that for certain classes of sets of states the bound is actually achieved. It was known to be achieved, in particular, when the state pair that is closest together in Chernoff divergence is more than 6 times closer than the next closest pair. Our results improve on this in two ways. Firstly, we show that the optimal asymptotic rate must lie between C/2 and C. Secondly, we show that the Chernoff bound is already achieved when the closest state pair is more than 2 times closer than the next closest pair. We also show that the Chernoff bound is achieved when at least r − 2 of the states are pure, improving on a previous result by Nussbaum and Szko la. Finally, we indicate a number of potential pathways along which a proof (or disproof) may eventually be found that the multi-hypothesis quantum Chernoff bound is always achieved.

Discriminating quantum states: The multiple Chernoff distance

“…In [1], Audenaert et al employed the Holevo?Helstrom tests {ρ ⊗n 1 − ρ ⊗n 2 > 0}, 1 1 − {ρ ⊗n 1 − ρ ⊗n 2 > 0} to achieve the binary Chernoff distance in testing ρ ⊗n 1 versus ρ ⊗n 2 . However, to date we do not have a way to generalize the method of Audenaert et al to deal with the r > 2 cases, even though there is the multiple generalization of the Holevo?Helstrom tests [20,41]; see discussions in [32] and [2] on this issue. Here, using a conceptually different method, we derive a new upper bound for the optimal error probability of equation (2).…”

“…see also [22] and [2] for alternative formulations. However, the fact that equation (39) involves an optimization problem itself, makes it difficult to apply this formula directly.…”

Thermodynamic Capacity of Quantum Processes

“…Now letω B achieve the maximum for the candidate smoothed max-entropy H max (A | B)ρ = 2 ln F (ρ AB , 1 A ⊗ω B ). By invoking [64,Lemma 4.9], which states that…”

Inequalities for the quantum privacy

“…An asymptotic theory of quantum inference [21] has also been developed. Recent developments in asymptotic quantum hypothesis testing have been obtained [22][23][24]. In particular, the multiple hypothesis testing problem for symmetric quantum state discrimination was addresses and upper bounds on asymptotic error exponents were derived [24].…”

“…Recent developments in asymptotic quantum hypothesis testing have been obtained [22][23][24]. In particular, the multiple hypothesis testing problem for symmetric quantum state discrimination was addresses and upper bounds on asymptotic error exponents were derived [24]. If we want to fully know the density operator ρ, we need an infinite ensemble of quantum systems prepared in the same state, which is impossible in pratice [21].…”