Operational Algorithms for Separable Qubit X States

Demosthenes Ellinas

doi:10.3390/condmat4030064

Abstract

This work motivates and applies operational methodology to simulation of quantum statistics of separable qubit X states. Three operational algorithms for evaluating separability probability distributions are put forward. Building on previous findings, the volume function characterizing the separability distribution is determined via quantum measurements of multi-qubit observables. Three measuring states, one for each algorithm are generated via (i) a multi-qubit channel map, (ii) a unitary operator generated by a Hamiltonian describing a non-uniform hypergraph configuration of interactions among 12 qubits, and (iii) a quantum walk CP map in a extended state space. Higher order CZ gates are the only tools of the algorithms hence the work associates itself computationally with the Instantaneous Quantum Polynomial-time Circuits (IQP), while wrt possible implementation the work relates to the Lechner-Hauke-Zoller (LHZ) architecture of higher order coupling. Finally some uncertainty aspects of the quantum measurement observables are discussed together with possible extensions to non-qubit separable bipartite systems.

Highlights

The framework and previous work: An extensive literature exist of mainly numerical studies of the quantum entanglement found in the density matrices of bipartite quantum systems, via certain matrix distance measures
Promoting the manifold of parameters determining the density matrices of the total and the reduced quantum systems into a statistical event space endowed with a distance measure, the “separability probability” can be cast into a geometric probability given e.g., by the ratio of the corresponding volume of separable marginal systems to the total volume of bipartite system in the manifold of parameters
It proceeds to puts forward three operational algorithms for evaluating separability probability distributions for X states of qubits

Summary

Introduction

The framework and previous work: An extensive literature exist of mainly numerical studies of the quantum entanglement found in the density matrices of bipartite quantum systems, via certain matrix distance measures. Outline of paper’s contribution: This work starts providing a motivation for applying operational methodology to simulation quantum statistics of separable states. It proceeds to puts forward three operational algorithms for evaluating separability probability distributions for X states of qubits. Based on the analogous situation of the laser quantum simulation, similar general ideas can be specified in the present case of simulating the statistics of separable X states To this end, this work puts forward some hitherto unknown simulator that employs a lattice of 12 qubits and provides versions of some operational algorithms that derive the conjectured probability distribution of pairs of separable qubits in an X form density matrix. The operational algorithmic methodology put forward here is grounded in a well developed operational methodology of constructing quantum observables and quantum measurements with desired properties—see, e.g., [25] for a general theory, [26] for related quantum optical problems, and [27,28] for problems formulated in quantum mechanical phase space

X States Operational Framework

Hypergraph Hamiltonian Coupling

Quantum Walk Simulation of Separability

Aspects of Computation and Implementation

Conclusions