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Quantum Information Processing

SLOCC orbit of rank-deficient two-qubit states: quantum entanglement, quantum discord and EPR steering

Paweł Caban, Jakub Rembieliński, Kordian A. Smoliński, Zbigniew Walczak

Published Jun 5, 2017• 6 citations
Publisher

Abstract

We study selected aspects of non-classical correlations of arbitrary states from the stochastic local operations and classical communication orbit of rank-deficient two-qubit states. In particular, we find explicitly entanglement of formation and quantum discord for these states. Moreover, we determine and analyze the Einstein–Podolsky–Rosen steering ellipsoids corresponding to these states.

DOI
10.1007/s11128-017-1626-7

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We study selected aspects of non-classical correlations of arbitrary states from the stochastic local operations and classical communication orbit of rank-deficient two-qubit states. In particular, we find explicitly entanglement of formation and quantum discord for these states. Moreover, we determine and analyze the Einstein–Podolsky–Rosen steering ellipsoids corresponding to these states.
Quantum nonlocality is an essential resource in quantum information and is commonly classified into three distinct forms: quantum entanglement, Einstein-Podolsky-Rosen (EPR) steering, and Bell's nonlocality. Gisin's theorem shows that pure-state entanglement implies Bell nonlocality, and it motivates the question of how entanglement and EPR steering are related beyond pure states. Here we prove that every entangled rank-2 two-qubit state is EPR steerable, with rank-1 states included as a special case. The proof combines a local-unitary parametrization of rank-2 two-qubit states, a necessary and sufficient separability condition, and a state-dependent nonlinear steering inequality. This gives an analytical characterization of EPR steerability for rank-2 two-qubit entangled states. Thus, rank-2 two-qubit entangled states provide certifiable EPR-steering resources in quantum information.
The question of which two-qubit states are steerable (i.e. permit a demonstration of EPR-steering) remains open. Here, a strong necessary condition is obtained for the steerability of two-qubit states having maximally-mixed reduced states, via the construction of local hidden state models. It is conjectured that this condition is in fact sufficient. Two provably sufficient conditions are also obtained, via asymmetric EPR-steering inequalities. Our work uses ideas from the quantum steering ellipsoid formalism, and explicitly evaluates the integral of n/(n⊺An)2\boldsymbol n/(\boldsymbol n^\intercal A\boldsymbol n)^2n/(n⊺An)2 over arbitrary unit hemispheres for any positive matrix AAA.
Quantum steering ellipsoid formalism provides a faithful representation of all two-qubit states and is useful for obtaining their correlation properties. The steering ellipsoids of two-qubit states that have undergone local operations on both the qubits, in order to bring the state to its canonical form, are the so-called canonical steering ellipsoids. The steering ellipsoids corresponding to the two-qubit subsystems of permutation-symmetric NNN-qubit states are considered here. We construct and analyze the geometric features of the canonical steering ellipsoids corresponding to pure permutation-symmetric NNN-qubit states with two distinct spinors. Depending on the degeneracy of the two spinors in the pure symmetric NNN-qubit state, several families arise which cannot be converted into one another through stochastic local operations and classical communication (SLOCC). The canonical steering ellipsoids of the two-qubit states drawn from the pure symmetric NNN-qubit states with two distinct spinors allow for a geometric visualization of the SLOCC-equivalent class of states. We show that the states belonging to the WWW class correspond to oblate spheroids centered at $(0,0,1/(N\ensuremath{-}1))$ with fixed semiaxis lengths $1/\sqrt{N\ensuremath{-}1}$ and $1/(N\ensuremath{-}1)$. The states belonging to all other SLOCC-inequivalent families correspond to ellipsoids centered at the origin of the Bloch sphere. We also explore volume monogamy relations of states belonging to these families, mainly the WWW class of states.
It is well known that quantum correlations such as quantum nonlocality and steering can be revealed by means of the implementation of local filters or stochastic local operation and classical communication (SLOCC). This means that bipartite quantum states that are entangled but local (unsteerable), can be driven locally to quantum nonlocal (steerable) states. However, there does not exist a criterion telling us whether the revealed quantum correlation is accessible by locally implementing on a single party, or both parties are required to cooperate by implementing local filters. In this work, we proposed a criterion based on the quantum steering ellipsoid and provided examples of two-qubit states with locally inaccessible quantum correlations.
Canonical forms of two-qubits under the action of stochastic local operations and classical communications (SLOCC) offer great insight for understanding nonlocality and entanglement shared by them. They also enable geometric picture of two-qubit states within the Bloch ball. It has been shown [Phys. Rev. A 64, 010101(R) (2001)] that an arbitrary two-qubit state gets transformed under SLOCC into one of the two different canonical forms. One of these happens to be the Bell diagonal form of two-qubit states and the other a nondiagonal canonical form is obtained for a family of rank deficient two-qubit states. The method employed by Verstraete et al. [Phys. Rev. A 64, 010101(R) (2001)] required highly nontrivial results on matrix decompositions in nnn-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.
Quantum nonlocality manifests in multipartite systems through entanglement, Bell's nonlocality, and Einstein-Podolsky-Rosen (EPR) steering. While Peres's positive-partial-transpose criterion provides a simple and powerful test for entanglement, a comparably elegant spectral criterion for detecting EPR steering remains an open challenge. In this work, we systematically explore whether a Peres-type criterion can be established for EPR steering in the two-qubit system. Focusing on rank-2 (including rank-1) states and the two-qubit Werner state, we analyze the eigenvalues of their partially transposed density matrices and construct a significant steering criterion based on symmetric combinations of these eigenvalues. We prove that this criterion serves as a necessary and sufficient condition for steerability for the Werner state, all two-qubit pure states, all two-qubit rank-2 states. Furthermore, we validate the criterion for higher-rank states (rank-3 and rank-4) and show that the results align with known steering inequalities. Our findings suggest a more unified framework for detecting quantum nonlocality via partial transposition and open avenues for further theoretical and numerical investigations into steering detection.