Entanglement Measures under Symmetry

We show how to simplify the computation of the entanglement of formation and the relative entropy of entanglement for states, which are invariant under a group of local symmetries. For several examples of groups we characterize the state spaces, which are invariant under these groups. For specific examples we calculate the entanglement measures. In particular, we derive an explicit formula for the entanglement of formation for UU-invariant states, and we find a counterexample to the additivity conjecture for the relative entropy of entanglement.Comment: RevTeX,16 pages,9 figures, reference added, proof of monotonicity corrected, results unchange

How many orthonormal bases are needed to distinguish all pure quantum states?

We collect some recent results that together provide an almost complete answer to the question stated in the title. For the dimension d=2 the answer is three. For the dimensions d=3 and d>4 the answer is four. For the dimension d=4 the answer is either three or four. Curiously, the exact number in d=4 seems to be an open problem

A Simple n-Dimensional Intrinsically Universal Quantum Cellular Automaton

We describe a simple n-dimensional quantum cellular automaton (QCA) capable of simulating all others, in that the initial configuration and the forward evolution of any n-dimensional QCA can be encoded within the initial configuration of the intrinsically universal QCA. Several steps of the intrinsically universal QCA then correspond to one step of the simulated QCA. The simulation preserves the topology in the sense that each cell of the simulated QCA is encoded as a group of adjacent cells in the universal QCA.Comment: 13 pages, 7 figures. In Proceedings of the 4th International Conference on Language and Automata Theory and Applications (LATA 2010), Lecture Notes in Computer Science (LNCS). Journal version: arXiv:0907.382

Unital Quantum Channels - Convex Structure and Revivals of Birkhoff's Theorem

The set of doubly-stochastic quantum channels and its subset of mixtures of unitaries are investigated. We provide a detailed analysis of their structure together with computable criteria for the separation of the two sets. When applied to O(d)-covariant channels this leads to a complete characterization and reveals a remarkable feature: instances of channels which are not in the convex hull of unitaries can return to it when either taking finitely many copies of them or supplementing with a completely depolarizing channel. In these scenarios this implies that a channel whose noise initially resists any environment-assisted attempt of correction can become perfectly correctable.Comment: 31 page

Test for entanglement using physically observable witness operators and positive maps

Motivated by the Peres-Horodecki criterion and the realignment criterion we develop a more powerful method to identify entangled states for any bipartite system through a universal construction of the witness operator. The method also gives a new family of positive but non-completely positive maps of arbitrary high dimensions which provide a much better test than the witness operators themselves. Moreover, we find there are two types of positive maps that can detect 2xN and 4xN bound entangled states. Since entanglement witnesses are physical observables and may be measured locally our construction could be of great significance for future experiments.Comment: 6 pages, 1 figure, revtex4 styl

Geometric measure of entanglement and applications to bipartite and multipartite quantum states

The degree to which a pure quantum state is entangled can be characterized by the distance or angle to the nearest unentangled state. This geometric measure of entanglement, already present in a number of settings (see Shimony 1995 and Barnum and Linden 2001), is explored for bipartite and multipartite pure and mixed states. The measure is determined analytically for arbitrary two-qubit mixed states and for generalized Werner and isotropic states, and is also applied to certain multipartite mixed states. In particular, a detailed analysis is given for arbitrary mixtures of three-qubit GHZ, W and inverted-W states. Along the way, we point out connections of the geometric measure of entanglement with entanglement witnesses and with the Hartree approximation method.Comment: 13 pages, 11 figures, this is a combination of three previous manuscripts (quant-ph/0212030, quant-ph/0303079, and quant-ph/0303158) made more extensive and coherent. To appear in PR

Construction of extremal local positive-operator-valued measures under symmetry

Published versio

Entanglement in SU(2)-invariant quantum spin systems

We analyze the entanglement of SU(2)-invariant density matrices of two spins $\vec S_{1}$, $\vec S_{2}$ using the Peres-Horodecki criterion. Such density matrices arise from thermal equilibrium states of isotropic spin systems. The partial transpose of such a state has the same multiplet structure and degeneracies as the original matrix with eigenvalue of largest multiplicity being non-negative. The case $S_{1}=S$, $S_{2}=1/2$ can be solved completely and is discussed in detail with respect to isotropic Heisenberg spin models. Moreover, in this case the Peres-Horodecki ciriterion turns out to be a sufficient condition for non-separability. We also characterize SU(2)-invariant states of two spins of length 1.Comment: 5 page

Characterizing the entanglement of symmetric many-particle spin-1/2 systems

Analyzing the properties of entanglement in many-particle spin-1/2 systems is generally difficult because the system's Hilbert space grows exponentially with the number of constituent particles, $N$. Fortunately, it is still possible to investigate many-particle entanglement when the state of the system possesses sufficient symmetry. In this paper, we present a practical method for efficiently computing various bipartite entanglement measures for states in the symmetric subspace and perform these calculations for $N\sim 10^3$. By considering all possible bipartite splits, we construct a picture of the multiscale entanglement in large symmetric systems. In particular, we characterize dynamically generated spin-squeezed states by comparing them to known reference states (e.g., GHZ and Dicke states) and new families of states with near-maximal bipartite entropy. We quantify the trade-off between the degree of entanglement and its robustness to particle loss, emphasizing that substantial entanglement need not be fragile.Comment: Updated version reflects changes made in January 200

A reversible theory of entanglement and its relation to the second law

We consider the manipulation of multipartite entangled states in the limit of many copies under quantum operations that asymptotically cannot generate entanglement. As announced in [Brandao and Plenio, Nature Physics 4, 8 (2008)], and in stark contrast to the manipulation of entanglement under local operations and classical communication, the entanglement shared by two or more parties can be reversibly interconverted in this setting. The unique entanglement measure is identified as the regularized relative entropy of entanglement, which is shown to be equal to a regularized and smoothed version of the logarithmic robustness of entanglement. Here we give a rigorous proof of this result, which is fundamentally based on a certain recent extension of quantum Stein's Lemma proved in [Brandao and Plenio, Commun. Math. 295, 791 (2010)], giving the best measurement strategy for discriminating several copies of an entangled state from an arbitrary sequence of non-entangled states, with an optimal distinguishability rate equal to the regularized relative entropy of entanglement. We moreover analyse the connection of our approach to axiomatic formulations of the second law of thermodynamics.Comment: 21 pages. revised versio