Canonical forms of two-qubit states under local operations

Abstract

Canonical forms of two-qubits under the action of stochastic local operations and classical communications (SLOCC) offer great insight for understanding non-locality and entanglement shared by them. They also enable geometric picture of two-qubit states within the Bloch ball. It has been shown (Verstraete et.al. {Phys. Rev. A, 64, 010101(R) (2001)) that an arbitrary two-qubit state gets transformed under SLOCC into one of the {\em two} different canonical forms. One of these happens to be the Bell diagonal form of two-qubit states and the other non-diagonal canonical form is obtained for a family of rank deficient two-qubit states. The method employed by Verstraete et.al. required highly non-trivial results on matrix decompositions in $n$ dimensional spaces with 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.

arXiv comment14 pages, two figures; proof of the theorem in page 4 is outlined in the Appendix A of this revised version; accepted for publication in Phys. Rev. A

References in corpus (16)

• Information-theoretic aspects of quantum inseparability of mixed states
• The Bloch Vector for N-Level Systems
• Local filtering operations on two qubits
• Quantum Steering Ellipsoids
• Einstein-Podolsky-Rosen steering and the steering ellipsoid
• Entangled Bloch Spheres: Bloch Matrix and Two Qubit State Space
• Geometry of the generalized Bloch sphere for qutrits
• Quantum steering ellipsoids, extremal physical states and monogamy
• On the optimal measurement for quantum discord of two-qubit states
• Necessary and sufficient condition for steerability of two-qubit states by the geometry of steering outcomes
• Steering Bell-diagonal states
• Geometric picture of quantum discord for two-qubit quantum states
• Quantum correlations of two-qubit states with one maximally mixed marginal
• Non-separability and steerability of two-qubit states from the geometry of steering outcomes
• Entanglement and the geometry of two qubits
• Visualizing Two Qubits

Cited by in corpus (10)

• Role of Bell-CHSH violation and local filtering in quantum key distribution
• Canonical steering ellipsoids of pure symmetric multiqubit states with two distinct spinors and volume monogamy of steering
• Geometric picture for SLOCC classification of pure permutation symmetric three-qubit states
• Emergence of maximal hidden quantum correlations and its trade-off with the filtering probability in dissipative two-qubit systems
• Multiphoton, multimode state classification for nonlinear optical circuits
• Canonical structures of $A$ and $B$ forms
• Swapping of quantum correlations and the role of local filtering operations
• Lorentz invariants of pure three-qubit states
• Lorentz canoncial forms of two-qubit states
• Entanglement and volume monogamy features of permutation symmetric N-qubit pure states with N-distinct spinors: GHZ and WWbar states