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Journal of Physics A Mathematical and Theoretical

The geometry of multi-qubit entanglement

Toshihiro Iwai

Published Sep 18, 2007• 10 citations
Publisher

Abstract

This paper, a continuation of a previous one ( Iwai 2007 J. Phys. A: Math. Theor. 40 1361 ), studies the geometry of multi-qubit entanglement with respect to bipartite partitions. An n -qubit system ( C 2 ) ⊗ n is isomorphic with , the linear space of 2 ℓ × 2 m complex matrices, where ℓ + m = n . According to the isomorphism, the local transformation group U (2 ℓ ) × U (2 m ) acts on the space of normalized states in . Let M and G denote the space of normalized states and the local transformation group acting on M , respectively. According to the orbit types, M is stratified into strata, among which a principal stratum and the sets of separable states and maximally entangled states will be identified. For , a function F ( C ) = det( I − CC *) proves to serve as a measure of entanglement, where I denotes the 2 ℓ × 2 ℓ identity matrix. The F ( C ) attains the minimal and the maximal values, respectively, on the sets of separable states and maximally entangled states, and further takes no extremal values on the principal stratum. The F ( C ) projects to a function on the factor space G \ M . A naturally defined metric on M also projects to that on G \ M , which serves to measure the distance between the separable states, F −1 (0), and the states, F −1 ( k ), of prescribed value k of measure. To be precise, one has to restrict M to the principal stratum, when projecting the metric. Three- and four-qubit systems will be studied intensively, and then multi-qubit systems discussed.

DOI
10.1088/1751-8113/40/40/011

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This paper, a continuation of a previous one ( Iwai 2007 J. Phys. A: Math. Theor. 40 1361 ), studies the geometry of multi-qubit entanglement with respect to bipartite partitions. An n -qubit system ( C 2 ) ⊗ n is isomorphic with , the linear space of 2 ℓ × 2 m complex matrices, where ℓ + m = n . According to the isomorphism, the local transformation group U (2 ℓ ) × U (2 m ) acts on the space of normalized states in . Let M and G denote the space of normalized states and the local transformation group acting on M , respectively. According to the orbit types, M is stratified into strata, among which a principal stratum and the sets of separable states and maximally entangled states will be identified. For , a function F ( C ) = det( I − CC *) proves to serve as a measure of entanglement, where I denotes the 2 ℓ × 2 ℓ identity matrix. The F ( C ) attains the minimal and the maximal values, respectively, on the sets of separable states and maximally entangled states, and further takes no extremal values on the principal stratum. The F ( C ) projects to a function on the factor space G \ M . A naturally defined metric on M also projects to that on G \ M , which serves to measure the distance between the separable states, F −1 (0), and the states, F −1 ( k ), of prescribed value k of measure. To be precise, one has to restrict M to the principal stratum, when projecting the metric. Three- and four-qubit systems will be studied intensively, and then multi-qubit systems discussed.
It is widely recognized that concurrence can be regarded as a measure of two-qubit entanglement. This paper studies the geometry of concurrence for a two-qubit system to show that the concurrence serves as a coordinate of the factor space , where is the space of normalized two-qubit states, and where G = U (1) × SU (2) × SU (2). Any monotonically increasing function of the concurrence can serve as a measure for entanglement. From the viewpoint of Riemannian geometry, a state with concurrence r is shown to be distant from the separable states by , where r ranges over 0 ⩽ r ⩽ 1. In addition, measures of entanglement for n -qubits are discussed on the basis of a bipartite decomposition with ℓ + m = n . They are invariant under the local unitary transformation group U (2 ℓ ) × U (2 m ).
Quantum entanglement is a fundamental resource in quantum information theory, yet its general characterization and quantification remain challenging, especially in multipartite systems. In this work we investigate entanglement from a geometric perspective, focusing on the Riemannian structure induced by the Fubini--Study metric on the projective Hilbert space of multi-qubit quantum states. By exploiting the local-unitary invariance of this metric, we derive the entanglement distance (ED), a geometric measure that quantifies entanglement as an obstruction to locally minimizing the sum of squared Fubini--Study distances generated by local operations. We analyze the properties of ED for pure multi-qubit states and discuss its behavior under local operations and classical communication. In particular, we show that ED reproduces established entanglement measures in well-defined and restricted settings. For pure states of two qubits, ED reduces to an exact monotone function of the concurrence and, independently, to an explicit monotone function of the entropy of entanglement. These results provide a clear geometric interpretation of standard bipartite entanglement measures within the present framework, while highlighting the limitations of such correspondences beyond the two-qubit case.
A measure of entanglement with respect to a bipartite partition of n -qubit has been defined and studied from the viewpoint of Riemannian geometry (Iwai 2007 J. Phys. A: Math. Theor. 40 12161 ). This paper has two aims. One is to study further the geometry of entanglement, and the other is to investigate Grover's search algorithms, both the original and the fixed-point ones, in reference with entanglement. As the distance between the maximally entangled states and the separable states is known already in the previous paper, this paper determines the set of maximally entangled states nearest to a typical separable state which is used as an initial state in Grover's search algorithms, and to find geodesic segments which realize the above-mentioned distance. As for Grover's algorithms, it is already known that while the initial and the target states are separable, the algorithms generate sequences of entangled states. This fact is confirmed also in the entanglement measure proposed in the previous paper, and then a split Grover algorithm is proposed which generates sequences of separable states only with respect to the bipartite partition.
A geometric approach to investigation of quantum entanglement is advocated. Analyzing the space of pure states of a bipartite system we show how an entanglement measure of a gives state can be related to its distance to the closest separable state. We study geometry of the (N2 - 1)-dimensional convex body of mixed quantum states acting on an N-dimensional Hilbert space and demonstrate that it belongs to the class of sets of a constant height. The same property characterizes the set of all separable states of a two-qubit system. These results contribute to our understanding of quantum entanglement and its dynamics.
Quantum entanglement was first recognized as a feature of quantum mechanics in the famous paper of Einstein, Podolsky, and Rosen. Recently it has been realized that quantum entanglement is a key ingredient in quantum computation, quantum communication, and quantum cryptography. In this paper, we introduce algebraic sets, which are determinantal varieties in the complex projective spaces or the products of complex projective spaces, for the mixed states on bipartite or multipartite quantum systems as their invariants under local unitary transformations. These invariants are naturally arised from the physical consideration of measuring mixed states by separable pure states. Our construction has applications in the following important topics in quantum information theory: (1) separability criterion, it is proved that the algebraic sets must be a union of the linear subspaces if the mixed states are separable; (2) simulation of Hamiltonians, it is proved that the simulation of semipositive Hamiltonians of the same rank implies the projective isomorphisms of the corresponding algebraic sets; (3) construction of bound entangled mixed states, examples of the entangled mixed states which are invariant under partial transpositions (thus PPT bound entanglement) are constructed systematically from our new separability criterion.
Complex Hopf fibration and quaternionic Hopf fibration provide distinct ways of describing the state space for two-qubit quantum states. In this research, we have studied the geometry of quantum states for a two-level quantum system, along with the correspondence between complex Hopf bundle and quaternionic Hopf bundle. In the first part of our study, we investigate the behaviour of local coordinates for both Hopf bundles under different degree of entanglement such as entangled states and non-entangled states. Fubini-Study metric for complex projective space is also obtained. Its form suggests that for the intermediate entangled states, complex projective space CP3 can be described as a set of flat three-tori parametrized by a three-sphere. The local inhomogeneous coordinate of CP3 (base space of complex Hopf fibration) is found to carry the description of both subsystems A and B, whereas in the case of maximally entangled state, basis elements of local coordinates CP3 is not linearly independent of each other. Next, we construct a base space map between CP3 and S4, which is denoted as η map. After the mapping, we obtained phases in the base space manifold, different sections and coordinate charts are related by transition functions. We found that there is an inherent symmetry of coordinate transformation corresponds to different sections of CP3, which is expressed in terms of transition functions having the U(1) group structure. Also under η map, phases and transition function in S4 is doubled over that of CP3, indicating subtle symmetric changes after the mapping. The base space coordinates of quaternionic Hopf bundle are consist of two parts, whereby the first part is invariant to the coordinate transformation in CP3 but sensitive to the coordinate transformation in S4.