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arXiv:0912.1085 · International Journal of Quantum Information

ALTERNATIVE DECOMPOSITION OF TWO-QUTRIT PURE STATES AND ITS RELATION WITH ENTANGLEMENT INVARIANTS

RUI-JUAN GU, FU-LIN ZHANG, SHAO-MING FEI, JING-LING CHEN

Published Aug 25, 2011• 1 citation
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Abstract

Based on maximally entangled states in the full- and sub-spaces of two-qutrits, we present an alternative decomposition of two-qutrit pure states in a form [Formula: see text]. Similar to the Schmidt decomposition, all two-qutrit pure states can be transformed into the alternative decomposition under local unitary transformations, and the parameter p1 is shown to be an entanglement invariant.

DOI
10.1142/s0219749911008040

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Based on maximally entangled states in the full- and sub-spaces of two-qutrits, we present an alternative decomposition of two-qutrit pure states in a form [Formula: see text]. Similar to the Schmidt decomposition, all two-qutrit pure states can be transformed into the alternative decomposition under local unitary transformations, and the parameter p1 is shown to be an entanglement invariant.
Abstract We present Schmidt decomposition formulas for mutually orthogonal two-qubit pure states and classify orthonormal sets according to their entanglement structure. We begin by presenting explicit formulas for the Schmidt decomposition of arbitrary two-qubit pure states. Building on this, we construct and analyze all possible configurations of two mutually orthogonal pure states, providing corresponding Schmidt decomposition expressions for each case. For orthonormal sets of three and four states, we derive explicit formulas for selected representative types and discuss the difficulties in obtaining closed-form expressions for more general cases. In particular, we focus on special classes of orthonormal bases that include one or two maximally entangled states. Finally, we prove that no orthonormal basis can contain three product states and one entangled state, revealing a structural constraint that restricts the composition of entangled orthonormal sets.
It is well known that the Schmidt decomposition exists for all pure states of a two-party quantum system. We demonstrate that there are two ways to obtain an analogous decomposition for arbitrary rank-1 operators acting on states of a bipartite finite-dimensional Hilbert space. These methods amount to joint Schmidt-type decompositions of two pure states where the two sets of coefficients and local bases depend on the properties of either state, however, at the expense of the local bases not all being orthonormal and in one case the complex-valuedness of the coefficients. With these results we derive several generally valid purity-type formulas for one-party reductions of rank-1 operators, and we point out relevant relations between the Schmidt decomposition and the Bloch representation of bipartite pure states.
Entanglement between three or more parties exhibits a realm of properties unknown to two-party states. Bipartite states are easily classified using the Schmidt decomposition. The Schmidt coefficients of a bipartite pure state encompass all the nonlocal properties of the state and can be ``seen'' by looking at one party's density matrix only. Pure states of three and more parties, however, lack such a simple form. They have more invariants under local unitary transformations than any one party can ``see'' on their subsystem. These ``hidden nonlocalities'' will allow us to exhibit a class of multipartite states that cannot be distinguished from each other by any party. Generalizing a result of Bennett, Popescu, Rohrlich, Smolin, and Thapliyal, and using a recent result by Nielsen, we will show that these states cannot be transformed into each other by local actions and classical communication. Furthermore, we will use an orthogonal subset of such states to hint at applications to cryptography and illustrate an extension to quantum secret sharing [using recently suggested ((n,k))((n,k))((n,k))-threshold schemes].
A complete analysis of entangled bipartite qutrit pure states is carried out based on a simple entanglement measure. An analysis of all possible extremally entangled pure bipartite qutrit states is shown to reduce, with the help of SLOCC transformations, to three distinct types. The analysis and the results should be helpful for finding different entanglement types in multipartite pure state systems. PACS: 03.67.-a; 03.67.Mn; 03.65.Ud; 03.65.Bz Quantum entanglement is an important resource to be implemented in quantum information processing and quantum communication protocols. [1] The interests in studying multi-dimensional entangled states come from both the foundations of quantum mechanics and the development of new protocols in quantum communication. For example, it was shown that maximally entangled states of two quantum systems in a high-dimensional Hilbert space, qudits, violate local realism stronger than qubits, and entangled qudits are less affected by noise than entangled qubits. [2,3] In quantum cryptography [4], the use of entangled qutrits [4,5] or qudits [6,7] instead of qubits is more secure against eavesdropping attacks. Furthermore, the protocols for quantum teleportation or for quantum cryptography, work best with maximally entangled states. All these facts motivate the development of techniques to generate entangled states among quantum systems in higher dimensional Hilbert space with a good quality of entanglement.
We report an exhaustive numerical analysis of violations of local realism by two qutrits in all possible pure entangled states. In Bell-type experiments we allow any pair of local unitary U(3) transformations to define the measurement bases. Surprisingly, Schmidt-rank-2 states, resembling pairs of maximally entangled qubits, lead to the most noise-robust violations of local realism. The phenomenon seems to be even more pronounced for four- and five-dimensional systems, for which we tested a few interesting examples.