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For a system randomly prepared in a number of quantum states, we present a lower bound for the distinguishability of the quantum states, that is, the success probability of determining the states in the form of entropy. When the states are all pure, acquiring the entropic lower bound requires only the density operator and the number of the possible states. This entropic bound shows a relation between the von Neumann entropy and the distinguishability.

Quantum mechanics does not allow determining the state of a system by measuring a single copy of the ensemble. Nevertheless, if some prior information is known, it is possible to guess the state with a certain degree of confidence even by a single measurement. Given some prior information, to what extent quantum states can be distinguished is an intriguing issue from both fundamental and practical points of view. For example, this problem is closely related to efficiencies of quantum communication [

There are different approaches to the distinguishability of quantum states [

One may pay attention to the von Neumann entropy as a quantity related to the distinguishability in light of the capacity of a quantum state for embodying quantum information. When a system is probabilistically prepared in one of a certain number of quantum states, its state of the statistical mixture is described by a density operator. According to the quantum source theorem [

We here present a lower bound for the distinguishability, that is, the optimal success probabilities of distinguishing between quantum states, as a function of entropies of the system. For a system prepared in one of

Consider a quantum system prepared in one of

An equivalent way of describing the scenario is to consider a classical-quantum system

One may expect an entropic lower bound from the intuition that the correlation of

For the quantum case, we can still obtain a random variable

For a set of quantum states with preparation probabilities

For the proof, we employ the conditional min-entropy [

Let us take a closer look into the form of the lower bound. Its form is exactly of (

On the assumption that the system is prepared in one of

Therefore, we see that the larger von Neumann entropy guarantees the better distinguishability. Notwithstanding the missing information on the component states and the preparation probabilities, the density operator of a system alone can provide a lower bound for distinguishability of

In this section, we compare the entropic lower bound to other previously known bounds. One is the lower bound given by the square-root measurement [

Another one is the pairwise-overlap bound. For an ensemble of pure states

We now consider a few exemplary sets of pure states

(a) Lower bounds for the optimal discrimination probability for the states in (

The next example of the component states is four 2-dimensional states with equal probabilities,

Finally, we consider a discrimination problem where the component states are not given, but the density operator and the number of possible states are given. Assume that a system is prepared in one of

We have presented an entropic lower bound for the optimal success probability of distinguishing quantum states. It provides a connection between the optimal discrimination probability and quantum entropy, that is, between a practically relevant quantity and a primary function in quantum information theory. When the quantum states are all pure, the entropic bound is reduced to a form that requires less information for its evaluation, namely, the density operator and the number of the possible states. It shows that the von Neumann entropy can lower bound the distinguishability of

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (no. 2010-0018295) and the Center for Theoretical Physics at Seoul National University.