Analysis of Quantum Entanglement in Quantum Programs using Stabilizer Formalism

Quantum entanglement plays an important role in quantum computation and communication. It is necessary for many protocols and computations, but causes unexpected disturbance of computational states. Hence, static analysis of quantum entanglement in quantum programs is necessary. Several papers studied the problem. They decided qubits were entangled if multiple qubits unitary gates are applied to them, and some refined this reasoning using information about the state of each separated qubit. However, they do not care about the fact that unitary gate undoes entanglement and that measurement may separate multiple qubits. In this paper, we extend prior work using stabilizer formalism. It refines reasoning about separability of quantum variables in quantum programs.Comment: In Proceedings QPL 2015, arXiv:1511.0118

Entanglement criterion for pure M⊗NM\otimes N bipartite quantum states

We propose a entanglement measure for pure $M \otimes N$ bipartite quantum states. We obtain the measure by generalizing the equivalent measure for a $2 \otimes 2$ system, via a $2 \otimes 3$ system, to the general bipartite case. The measure emphasizes the role Bell states have, both for forming the measure, and for experimentally measuring the entanglement. The form of the measure is similar to generalized concurrence. In the case of $2 \otimes 3$ systems, we prove that our measure, that is directly measurable, equals the concurrence. It is also shown that in order to measure the entanglement, it is sufficient to measure the projections of the state onto a maximum of $M(M-1)N(N-1)/2$ Bell states.Comment: 6 page

Separability, Locality, and Higher Dimensions in Quantum Mechanics

*A shortened version of this paper will appear in Current Controversies in Philosophy of Science, Dasgupta and Weslake, eds. Routledge.* This paper describes the case that can be made for a high-dimensional ontology in quantum mechanics based on the virtues of avoiding both nonseparability and non locality

Bipartite quantum systems: on the realignment criterion and beyond

Inspired by the `computable cross norm' or `realignment' criterion, we propose a new point of view about the characterization of the states of bipartite quantum systems. We consider a Schmidt decomposition of a bipartite density operator. The corresponding Schmidt coefficients, or the associated symmetric polynomials, are regarded as quantities that can be used to characterize bipartite quantum states. In particular, starting from the realignment criterion, a family of necessary conditions for the separability of bipartite quantum states is derived. We conjecture that these conditions, which are weaker than the parent criterion, can be strengthened in such a way to obtain a new family of criteria that are independent of the original one. This conjecture is supported by numerical examples for the low dimensional cases. These ideas can be applied to the study of quantum channels, leading to a relation between the rate of contraction of a map and its ability to preserve entanglement.Comment: 19 pages, 4 figures, improved versio

Entanglement and separability of quantum harmonic oscillator systems at finite temperature

In the present paper we study the entanglement properties of thermal (a.k.a. Gibbs) states of quantum harmonic oscillator systems as functions of the Hamiltonian and the temperature. We prove the physical intuition that at sufficiently high temperatures the thermal state becomes fully separable and we deduce bounds on the critical temperature at which this happens. We show that the bound becomes tight for a wide class of Hamiltonians with sufficient translation symmetry. We find, that at the crossover the thermal energy is of the order of the energy of the strongest normal mode of the system and quantify the degree of entanglement below the critical temperature. Finally, we discuss the example of a ring topology in detail and compare our results with previous work in an entanglement-phase diagram.Comment: 10 pages, 5 figure

Further results on entanglement detection and quantification from the correlation matrix criterion

The correlation matrix (CM) criterion is a recently derived powerful sufficient condition for the presence of entanglement in bipartite quantum states of arbitrary dimensions. It has been shown that it can be stronger than the positive partial transpose (PPT) criterion, as well as the computable cross norm or realignment (CCNR) criterion in different situations. However, it remained as an open question whether there existed sets of states for which the CM criterion could be stronger than both criteria simultaneously. Here, we give an affirmative answer to this question by providing examples of entangled states that scape detection by both the PPT and CCNR criteria whose entanglement is revealed by the CM condition. We also show that the CM can be used to measure the entanglement of pure states and obtain lower bounds for the entanglement measure known as tangle for general (mixed) states.Comment: 13 pages, no figures; added references, minor changes; section 4.3 added, to appear in J. Phys.