Some problems in Quantum state discrimination[HBNI Th97]

Abstract

The problem of Quantum State Discrimination (QSD) is integral to Quantum Information Theory, for e.g. it finds application in many experimental setups which requires one to distinguish among the incoming signal or in the context of communication etc. The basic premise is that one is given an unknown quantum state from a known quantum-state ensemble from which each state can occur with some apriori probability, and one is then tasked with finding which state was given using a quantum measurement. There are many variants of QSD, of which I take up two in this thesis. I consider two problems in each of these variants of QSD, thus four problems totally. The first of these variants is the well known problem of Minimum Error Discrimination (MED) of quantum states. Firstly, I examine the MED problem for ensembles of linearly independent (LI) pure states. While the analytic solution for MED of such ensembles isn't known, the following structure of the problem is known: there exists a bijection from the set of LI pure state ensembles to itself, and this bijection is such that the Pretty Good Measurement (PGM) of the image under this bijection of an ensemble is the optimal POVM for the MED of said ensemble. While the existence of such a bijection is known, its functional form (how it acts on an ensemble of LI pure states) isn't. That said, the action of its inverse is known. I used the implicit function theorem to obtain the solution using the inverse of this bijection. In the second problem I show that this structure (i.e., such a bijection) can be extended to the case of ensembles of LI mixed states, and that this structure can then be subsequently used in the same fashion to compute the optimal POVM for MED of these ensembles. It is seen that this algorithm is as efficient as popular SDP algorithms which are in use to solve this problem. The second variant of QSD concerns the perfect local distinguishability of pairwise orthogonal bipartite quantum states. Firstly, a necessary condition for the local distinguishability of pairwise orthogonal Maximally Entangled States, which goes beyond the known orthogonality preserving condition, is derived. This is done using the Holevo-like upper bound for local distinguishability of ensembles of bipartite states – a result derived by Badziag et al. in Phys. Rev. Lett., vol. 91, p. 117901. It is seen that for sets of d MES in dxd systems, this necessary condition acquires a particularly special form so that it can be checked by solving a set of simultaneous linear equations. This condition is then tested on sets of four MES in 4x4 systems, where these MES are of the form of the Generalized Bell basis states. It is found that for these sets of states this necessary condition is also sufficient to determine the local (in)distinguishability, which shows the strength of this necessary condition. In the next problem I propose a framework for the one-way local distinguishability of sets of pairwise orthogonal states. This is shown through the following sequence of steps: 1.) I borrow a result which states that a set of pairwise orthogonal states is locally distinguishable if and only if the first party can perform an orthogonality preserving rank-one measurement, 2.) I show that the set of orthogonality preserving self-adjoint operators on the first party's side forms a vector space, for which a basis can be constructed, 3.) the existence of an orthogonality preserving rank-one measurement implies and is implied by the existence of a Maximally Abelian Subspace (MAS) in this vector space. Hence I reduce the problem of finding a one-way LOCC protocol to finding a MAS in this vector space. It is shown that the problem of finding such an MAS depends on the dimension of the vector space, for e.g., the result that any two pure states can be perfectly distinguished by one-way LOCC corresponds to the case where the dimension is d^2-2. This framework also gives the necessary and sufficient condition for the one-way LOCC distinguishability of almost all sets of d orthogonal bipartite pure states. Hence it is argued that a deeper study of this structure is likely to prove rewarding for progress in this topic.