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 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.
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.
We firstly investigate the multipartite entanglement features of the quantum states, including the iteration states achieved by repeated application of Grover iteration and the Oracle ones into which the above iteration states evolve by applying single Oracle operation, employed in Grover's search algorithm by means of the separable degree and the entanglement measure. Then we give the quantitative and qualitative descriptions of the entanglement dynamics in Grover's search algorithm. Our results show that for most instances (i) the separable degrees of these states and ranges of their maximum Schmidt numbers are invariable by following the dynamics of Grover's search algorithm; (ii) the dynamics of Grover's search algorithm is almost "filled" by the fully entangled states.
Grover's quantum search algorithm is analyzed for the case in which the initial state is an arbitrary pure quantum state $|\ensuremath{\varphi}〉$ of n qubits. It is shown that the optimal time to perform the measurement is independent of $|\ensuremath{\varphi}〉,$ namely, it is identical to the optimal time in the original algorithm in which $|\ensuremath{\varphi}〉=|0〉,$ with the same number of marked states, r. The probability of success Ps is obtained in terms of the amplitudes of the state $|\ensuremath{\varphi}〉$ and is shown to be independent of r. A class of states, which includes fixed points and cycles of the Grover iteration operator, is identified. The relevance of these results in the context of using the success probability as an entanglement measure is discussed. In particular, the Groverian entanglement measure, previously limited to a single marked state, is generalized to the case of several marked states.
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 ).
The main topic of this compilation thesis is the investigation of multipartite entanglement of finite dimensional systems. We developed a numerical algorithm that detects if a multipartite state is entangled or separable in a finite number of steps of a semi-definite optimization task. This method is an extension of previously known semi-definite methods, which are inconclusive when the state is separable. In our case, if the state is separable, an explicit decomposition into a mixture of separable states can be extracted. This was achieved by mapping the entanglement problem onto the mathematically well studied truncated moment problem. Additionally, a new way of writing the partially transposed state for symmetric multi-qubit states was developed which simplifies many results previously known in entanglement theory. This new way of writing the partial transpose criterion unifies different interpretations and alternative formulations of the partial transpose criterion and it is also a part in the aforementioned semi-definite algorithm. The geometric properties of entangled symmetric states of two qubits were studied in detail: We could answer the question of how far a given pure state is from the convex hull of symmetric separable states, as measured by the Hilbert-Schmidt distance, by giving an explicit formula for the closest separable symmetric state. For mixed states we could provide a numerical upper and analytical lower bound for this distance. For a larger number of qubits we investigated the ball of absolutely classical states, i.e. symmetric multi-qubit states that stay separable under any unitary transformation. We found an analytical lower bound for the radius of this ball around the maximally mixed symmetric state and gave a numerical upper bound on this radius, by searching for an entangled state as close as possible to the maximally mixed symmetric state. The tensor representation of a symmetric multi-qubit state, or spin-j state, allowed us to study entanglement properties based on the spectrum of the tensor via tensor eigenvalues. The definiteness of this tensor relates to the entanglement of the state it represents and, hence, the smallest tensor eigenvalue can be used to detect entanglement. However, the tensor eigenvalues are more difficult to determine than the familiar matrix eigenvalues which made the investigation computationally more challenging. The relationship between the value of the smallest tensor eigenvalue and the amount of entanglement in the state was also investigated. It turned out that they are strongly correlated for small system sizes, i.e. for up to six qubits. However, to investigate this correlation we needed an independent way to gauge the amount of entanglement of a state and in order to do so we improved existing numerical methods to determine the distance of a state to the set of separable symmetric states, using a combination of linear and quadratic programming. The tensor representation of symmetric multi-qubit states was also used to formally define a new tensor class of regularly decomposable tensors that corresponds to the set of separable symmetric multi-qubit states.
The Groverian entanglement measure, introduced earlier for pure quantum states of multiple qubits [O. Biham, M.A. Nielsen, and T. Osborne, Phys. Rev. A 65, 062312 (2002)], is generalized to the case of mixed states. The Groverian measure of a mixed state of n qubits is obtained by a purification procedure into a pure state of 2n qubits, followed by an optimization process, before the resulting state is fed into Grover's search algorithm. It is expressed in terms of the maximal success probability of the algorithm and in this sense provides an operational measure of entanglement.