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International Journal of Quantum Information

Linear entropy of two-qubit gates and majorization

S. Balakrishnan

Published Aug 1, 2014• 0 citations
Publisher

Abstract

Linear entropy is widely used as a measure of operator entanglement of two-qubit gates. In this paper, we characterize the two-qubit gates of same linear entropy using the majorization criteria of the Schmidt coefficients of the gates. It is found that the gates with same linear entropy must have set of same Schmidt coefficients or set of different Schmidt coefficients which are not ordered according to the majorization relation.

DOI
10.1142/s0219749914500294

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Linear entropy is widely used as a measure of operator entanglement of two-qubit gates. In this paper, we characterize the two-qubit gates of same linear entropy using the majorization criteria of the Schmidt coefficients of the gates. It is found that the gates with same linear entropy must have set of same Schmidt coefficients or set of different Schmidt coefficients which are not ordered according to the majorization relation.
Two different measures of operator entanglement of two-qubit gates, namely, Schmidt strength and linear entropy, are studied. While these measures are shown to have a one-to-one relation between them for the Schmidt number 2 class of gates, no such relation exists for the Schmidt number 4 class, implying that the measures are inequivalent in general. Further, we establish a simple relation between linear entropy and local invariants of two-qubit gates. The implication of the relation is discussed.
Accurate characterization of two-qubit gates will be critical for any realization of quantum computation. We discuss a range of measurements for characterizing two-qubit gates. These measures are architecture-independent and span a range of complexity from simple measurement routines to full quantum-state and process tomography. Simple indicative measures, which flag but do not quantify gate operation in the quantum regime, include the fringe visibility, parity, Bell-state fidelity, and entanglement witnesses. Quantitative measures of gate output states include linear entropy and tangle; measures of, and error bounds to, whole-gate operation are provided by metrics such as process fidelity, process distance, and average gate fidelity. We discuss which measures are appropriate, given the stage of development of the gate, and highlight connections between them.
Abstract We show that the geometrical features of the unimodular squared eigenvalues of the nonlocal part of a two-qubit gate in the complex plane can be used to study the nonlocal characteristics of the two-qubit gate. It has been shown that the entangling power of a two-qubit gate can be expressed in terms of the mean squared length of the chords connecting the squared eigenvalues of the nonlocal part with each other. In this paper, we provide similar expressions for the gate typicality and the linear entropy of two-qubit gates. The expression of gate typicality involves the squared lengths of three chords connecting a squared eigenvalue of the nonlocal part with the complex conjugate of the other three squared eigenvalues. The expression of linear entropy is obtained from the expressions of entangling power and gate typicality. We also express them in terms of the linear entropies of certain two-qubit states. In addition, we show that the normalized entangling power of a two-qubit gate can be provided exact description in the Argand diagram. Finally, we analyze the simplices enclosing the origin in the convex hull of the squared eigenvalues of the nonlocal part of the perfect entangling two-qubit gates to construct a pair of orthonormal product states that can be transformed into maximally entangled states, and divide the region of perfect entanglers in the Weyl chamber into different regions based on the uniqueness of the simplices enclosing the origin.
Nonlocal two‐qubit gates are represented by canonical decomposition or equivalently by operator‐Schmidt decomposition. The former decomposition results in geometrical representation such that all the two‐qubit gates form tetrahedron within which perfect entanglers form a polyhedron. On the other hand, it is known from the later decomposition that Schmidt number of nonlocal gates can be either 2 or 4. In this work, all the edges of tetrahedron are characterized using Schmidt strength, a measure of operator entanglement. It is found that one edge of the tetrahedron, which includes SWAP and Double‐CNOT, possesses the maximum Schmidt strength. It implies that all the gates in the edge are maximally entangled.
We consider the Schmidt decomposition of a bipartite density operator induced by the Hilbert-Schmidt scalar product, and we study the relation between the Schmidt coefficients and entanglement. First, we define the Schmidt equivalence classes of bipartite states. Each class consists of all the density operators (in a given bipartite Hilbert space) sharing the same set of Schmidt coefficients. Next, we review the role played by the Schmidt coefficients with respect to the separability criterion known as the 'realignment' or 'computable cross norm' criterion; in particular, we highlight the fact that this criterion relies only on the Schmidt equivalence class of a state. Then, the relevance — with regard to the characterization of entanglement — of the 'symmetric polynomials' in the Schmidt coefficients and a new family of separability criteria that generalize the realignment criterion are discussed. Various interesting open problems are proposed.
We consider the Schmidt decomposition of a bipartite density operator induced by the Hilbert-Schmidt scalar product, and we study the relation between the Schmidt coefficients and entanglement. First, we define the Schmidt equivalence classes of bipartite states. Each class consists of all the density operators (in a given bipartite Hilbert space) sharing the same set of Schmidt coefficients. Next, we review the role played by the Schmidt coefficients with respect to the separability criterion known as the ‘realignment ’ or ‘computable cross norm ’ criterion; in particular, we highlight the fact that this criterion relies only on the Schmidt equivalence class of a state. Then, the relevance — with regard to the characterization of entanglement — of the ‘symmetric polynomials ’ in the Schmidt coefficients and a new family of separability criteria that generalize the realignment criterion are discussed. Various interesting open problems are proposed. 1