Designing quantum information processing via structural physical approximation

Abstract

In quantum information processing it may be possible to have efficient computation and secure communication beyond the limitations of classical systems. In a fundamental point of view, however, evolution of quantum systems by the laws of quantum mechanics is more restrictive than classical systems, identified to a specific form of dynamics, that is, unitary transformations and, consequently, positive and completely positive maps to subsystems. This also characterizes classes of disallowed transformations on quantum systems, among which positive but not completely maps are of particular interest as they characterize entangled states, a general resource in quantum information processing. Structural physical approximation offers a systematic way of approximating those nonphysical maps, positive but not completely positive maps, with quantum channels. Since it has been proposed as a method of detecting entangled states, it has stimulated fundamental problems on classifications of positive maps and the structure of Hermitian operators and quantum states, as well as on quantum measurement such as quantum design in quantum information theory. It has developed efficient and feasible methods of directly detecting entangled states in practice, for which proof-of-principle experimental demonstrations have also been performed with photonic qubit states. Here, we present a comprehensive review on quantum information processing with structural physical approximations and the related progress. The review mainly focuses on properties of structural physical approximations and their applications toward practical information applications.

Structural physical approximation for the realization of the optimal singlet fraction with two measurements

“…The above condition (13) is equivalent form of the condition (11) and hence the inequality implies that the eigenvalues of ρ 12 is less than 2 9 . Let V ≡ |φ φ| − 2 9 I.…”

“…where p * is the minimum value of p for which the approximated map P is a completely positive map [11]. * Electronic address: satyabrata@dtu.ac.in Partial transposition (PT) is another strong entanglement detection criterion given by Peres [12].…”

“…where q * = 16ν 1+16ν and ν = −min Q>0 T r[Q(id ⊗ T )|ψ + ψ + |], |ψ + = 1 √ 2 (|00 + |11 ). The map id ⊗ T is a non-physical map but its approximate map id ⊗ T is a completely positive map corresponds to a quantum channel that can be experimentally implementable [8,11]. Let σ 12 be a two qubit-state and the task is to determine whether it is an entangled state or separable state.…”

Mirrored entanglement witnesses

“…Mirrored structural physical approximations It turns out that our approach is directly related to the structural physical approximation (SPA) 14,15 of EWs. Recall, that for an EW W its SPA can be written as a non-negative observable…”

“…npj Quantum Information (2020)15 Published in partnership with The University of New South Wales1234567890():,;…”