Laser Physics

Stochastic local operations and classical communication invariants via square matrix

Xin-Wei Zha, Irfan Ahmed, Da Zhang, Feng Wen, Yanpeng Zhang

Published Jan 4, 2019• 3 citations

Abstract

Abstract In this paper, we raise a few corrections from the Li et al paper (2015 Phys. Rev . A 91 012302). We obtain some stochastic local operations and classical communication (SLOCC) invariants from second degree to sixth degree by defining square matrices. The relation of SLOCC invariants and the character polynomial of the square matrix are defined for three and four qubit states. This method comes as a useful resource in determining SLOCC invariants for n-qubit states.

DOI: 10.1088/1555-6611/aaf637

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Laser Physics

Stochastic local operations and classical communication invariants via square matrix

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Published Jan 4, 2019• 3 citations

Publisher

Abstract

DOI: 10.1088/1555-6611/aaf637

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n

-qubit SLOCC invariants, complete descriptions of these invariants exist. However, these descriptions give infinite sets; of great interest is finding generating sets of invariants. In this regard, degree bounds are highly sought after to limit the possible sizes of such generating sets. In this paper we give new upper bounds on the degrees of the invariants, both for a certain complete set of local unitary invariants as well as the

n

-qubit SLOCC invariants. We show that there exists a complete set of local unitary invariants of density operators in a Hilbert space

\mathcal{H}

, of dimension

d

, which are generated by invariants of degree at most

d^4

. This in turn allows us to show that the

n

-qubit SLOCC invariants are generated by invariants of degree at most

2^{4n}

n

-dimensional spaces with an indefinite metric. Here we employ an entirely different approach---inspired by the methods developed by Rao et al. [J. Mod. Opt. 45, 955 (1998)] in classical polarization optics---which leads naturally towards the identification of two inequivalent SLOCC invariant canonical forms for two-qubit states. In addition, our approach results in a simple geometric visualization of two-qubit states in terms of their SLOCC canonical forms.

We propose a method of classifying

n

-qubit states into stochastic local operations and classical communication inequivalent families in terms of the rank of the square matrix $C{(i{\ensuremath{\sigma}}_{y})}^{\ensuremath{\bigotimes}k}{C}^{T}$, where

C

is the rectangular coefficient matrix of the state and ${\ensuremath{\sigma}}_{y}$ is the Pauli operator. The rank of the square matrix $C{(i{\ensuremath{\sigma}}_{y})}^{\ensuremath{\bigotimes}k}{C}^{T}$ is capable of distinguishing between

n

-qubit Greenberger-Horne-Zeilinger and

W

states. The determinant of the matrix gives rise to a family of polynomial invariants for

n

qubits which include as special cases well-known polynomial invariants in the literature. In addition, explicit expressions can be given for these polynomial invariants and this allows us to investigate the properties of entanglement measures built upon the absolute values of polynomial invariants for product states.