Using postmeasurement information in state discrimination

Abstract

We consider a special form of state discrimination in which after the measurement we are given additional information that may help us identify the state. This task plays a central role in the analysis of quantum cryptographic protocols in the noisy-storage model, where the identity of the state corresponds to a certain bit string, and the additional information is typically a choice of encoding that is initially unknown to the cheating party. We first provide simple optimality conditions for measurements for any such problem and show upper and lower bounds on the success probability. For a certain class of problems, we furthermore provide tight bounds on how useful postmeasurement information can be. In particular, we show that for this class finding the optimal measurement for the task of state discrimination with postmeasurement information does in fact reduce to solving a different problem of state discrimination without such information. However, we show that for the corresponding classical state discrimination problems with postmeasurement information such a reduction is impossible, by relating the success probability to the violation of Bell inequalities. This suggests the usefulness of postmeasurement information as another feature that distinguishes the classical from a quantum world.

Quantifying Quantum Resources with Conic Programming

“…A typical instance of this problem is called state discrimination with post-measurement information [18,19]. The ensemble E = {p a , a } a∈I can be partitioned into nonempty disjoint ensembles E x = {p a , a } a∈Ix , where…”

Witnessing incompatibility of quantum channels

“…A particular instance of the scheme introduced in this section is the discrimination task with pre-and postmeasurement information described and studied in [12,35,36,37]. In the latter task, Bob is asked to retrieve Alice's label z ∈ X 1 ∪ X 2 by simply performing a measurement N on the received state a z , without making any processing of a z before that.…”

State discrimination with postmeasurement information and incompatibility of quantum measurements

“…In this task, Alice encodes classical information in quantum states and Bob then performs a measurement to guess the correct state, but Alice announces some partial information on her encoding before Bob must make his guess. This task was further studied in [5], and it was suggested that the usefulness of post-measurement information distinguishes the quantum from the classical world. In this work we reveal a link between the task of state discrimination with post-measurement information and the incompatibility of quantum measurements.…”