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arXiv:1204.3912

X-matrices provide a lower bound of concurrence

Seyed Mohammad Hashemi Rafsanjani, Shantanu Agarwal

Published Apr 17, 2012• 3 citations
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Abstract

By focusing on the X-matrix part of a density matrix of two qubits we provide an algebraic lower bound for the concurrence. The lower bound is generalized for cases beyond two qubits and can serve as a sufficient condition for non-separability for bipartite density matrices of arbitrary dimension. Experimentally, our lower bound can be used to confirm non-separability without performing a complete state tomography.

DOI
10.48550/arxiv.1204.3912

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By focusing on the X-matrix part of a density matrix of two qubits we provide an algebraic lower bound for the concurrence. The lower bound is generalized for cases beyond two qubits and can serve as a sufficient condition for non-separability for bipartite density matrices of arbitrary dimension. Experimentally, our lower bound can be used to confirm non-separability without performing a complete state tomography.
We study the separability of bipartite quantum systems in arbitrary dimensions using the Bloch representation of their density matrix. This approach enables us to find an alternative characterization of the separability problem, from which we derive a necessary condition and sufficient conditions for separability. For a certain class of states the necessary condition and a sufficient condition turn out to be equivalent, therefore yielding a necessary and sufficient condition. The proofs of the sufficient conditions are constructive, thus providing decompositions in pure product states for the states that satisfy them. We provide examples that show the ability of these conditions to detect entanglement. In particular, the necessary condition is proved to be strong enough to detect bound entangled states.
Abstract Determining whether an unknown quantum state is entangled can be approached through quantum tomography and related algorithms. However, these methods are generally inefficient due to the NP-hardness of the problem. An alternative strategy is to treat it as a decision (or estimation) problem with a quantum data structure (Horodecki and Ekert 2002 Phys. Rev. Lett. 89 127902). Over the years, significant progress has been made: the two-qubit case is now fully understood, but for higher-dimensional systems only entanglement detection via negativity has been established. In this work, we propose schemes for measuring upper bounds on bipartite biconcurrence and concurrence, expressed as functions of a fixed (but arbitrary) unitary operation acting on a Hilbert space that is quadratically larger than the original state space. Independently, we provide a scheme for measuring a lower bound for concurrence. In the case of the states proportional to the projectors the schemes are getting significantly simpler and the upper bound on the biconcurrence, as well as the lower bound on the concurrence, become exact (up to experimental and numerical errors). This framework opens the door to tackling the separability problem using methods inspired by the variational quantum eigensolver.