Method for classifying multiqubit states via the rank of the coefficient matrix and its application to four-qubit states

Abstract

We construct coefficient matrices of size 2^l by 2^{n-l} associated with pure n-qubit states and prove the invariance of the ranks of the coefficient matrices under stochastic local operations and classical communication (SLOCC). The ranks give rise to a simple way of partitioning pure n-qubit states into inequivalent families and distinguishing degenerate families from one another under SLOCC. Moreover, the classification scheme via the ranks of coefficient matrices can be combined with other schemes to build a more refined classification scheme. To exemplify we classify the nine families of four qubits introduced by Verstraete et al. [Phys. Rev. A 65, 052112 (2002)] further into inequivalent subfamilies via the ranks of coefficient matrices, and as a result, we find 28 genuinely entangled families and all the degenerate classes can be distinguished up to permutations of the four qubits. We also discuss the completeness of the classification of four qubits into nine families.

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Cited by in corpus (13)

• SLOCC classification of n qubits invoking the proportional relationships for spectrums and for standard Jordan normal forms
• Entanglement classification of 2 X 2 X 2 X d quantum systems via the ranks of the multiple coefficient matrices
• Classification of arbitrary-dimensional multipartite pure states under stochastic local operations and classical communication using the rank of coefficient matrix
• Quantifying entanglement of arbitrary-dimensional multipartite pure states in terms of the singular values of coefficient matrices
• Stochastic local operations and classical communication (SLOCC) and local unitary operations (LU) classifications of n qubits via ranks and singular values of the spin-flipping matrices
• Detecting genuine multipartite correlations in terms of the rank of coefficient matrix
• Criterion for SLOCC Equivalence of Multipartite Quantum States
• Polynomial invariants of degree 4 for even-$n$ qubits and their applications in entanglement classification
• Quantum entanglement in finite-dimensional Hilbert spaces
• Criteria for SLOCC and LU Equivalence of Generic Multi-qudit States
• Freudenthal ranks: GHZ vs. W
• Entanglement classification via integer partitions
• Classification of the Entangled States of $2\times L\times M\times N\times H$