Phase-space characterization of complexity in quantum many-body dynamics

We propose a phase-space Wigner harmonics entropy measure for many-body quantum dynamical complexity. This measure, which reduces to the well known measure of complexity in classical systems and which is valid for both pure and mixed states in single-particle and many-body systems, takes into account the combined role of chaos and entanglement in the realm of quantum mechanics. The effectiveness of the measure is illustrated in the example of the Ising chain in a homogeneous tilted magnetic field. We provide numerical evidence that the multipartite entanglement generation leads to a linear increase of entropy until saturation in both integrable and chaotic regimes, so that in both cases the number of harmonics of the Wigner function grows exponentially with time. The entropy growth rate can be used to detect quantum phase transitions. The proposed entropy measure can also distinguish between integrable and chaotic many-body dynamics by means of the size of long term fluctuations which become smaller when quantum chaos sets in.Comment: 10 pages, 9 figure

Entanglement, randomness and chaos

Entanglement is not only the most intriguing feature of quantum mechanics, but also a key resource in quantum information science. The entanglement content of random pure quantum states is almost maximal; such states find applications in various quantum information protocols. The preparation of a random state or, equivalently, the implementation of a random unitary operator, requires a number of elementary one- and two-qubit gates that is exponential in the number n_q of qubits, thus becoming rapidly unfeasible when increasing n_q. On the other hand, pseudo-random states approximating to the desired accuracy the entanglement properties of true random states may be generated efficiently, that is, polynomially in n_q. In particular, quantum chaotic maps are efficient generators of multipartite entanglement among the qubits, close to that expected for random states. This review discusses several aspects of the relationship between entanglement, randomness and chaos. In particular, I will focus on the following items: (i) the robustness of the entanglement generated by quantum chaotic maps when taking into account the unavoidable noise sources affecting a quantum computer; (ii) the detection of the entanglement of high-dimensional (mixtures of) random states, an issue also related to the question of the emergence of classicality in coarse grained quantum chaotic dynamics; (iii) the decoherence induced by the coupling of a system to a chaotic environment, that is, by the entanglement established between the system and the environment.Comment: Review paper, 40 pages, 7 figures, added reference

Statistical mechanics of multipartite entanglement

We characterize the multipartite entanglement of a system of n qubits in terms of the distribution function of the bipartite purity over all balanced bipartitions. We search for those (maximally multipartite entangled) states whose purity is minimum for all bipartitions and recast this optimization problem into a problem of statistical mechanics.Comment: final versio

Two-message quantum interactive proofs and the quantum separability problem

Suppose that a polynomial-time mixed-state quantum circuit, described as a sequence of local unitary interactions followed by a partial trace, generates a quantum state shared between two parties. One might then wonder, does this quantum circuit produce a state that is separable or entangled? Here, we give evidence that it is computationally hard to decide the answer to this question, even if one has access to the power of quantum computation. We begin by exhibiting a two-message quantum interactive proof system that can decide the answer to a promise version of the question. We then prove that the promise problem is hard for the class of promise problems with "quantum statistical zero knowledge" (QSZK) proof systems by demonstrating a polynomial-time Karp reduction from the QSZK-complete promise problem "quantum state distinguishability" to our quantum separability problem. By exploiting Knill's efficient encoding of a matrix description of a state into a description of a circuit to generate the state, we can show that our promise problem is NP-hard with respect to Cook reductions. Thus, the quantum separability problem (as phrased above) constitutes the first nontrivial promise problem decidable by a two-message quantum interactive proof system while being hard for both NP and QSZK. We also consider a variant of the problem, in which a given polynomial-time mixed-state quantum circuit accepts a quantum state as input, and the question is to decide if there is an input to this circuit which makes its output separable across some bipartite cut. We prove that this problem is a complete promise problem for the class QIP of problems decidable by quantum interactive proof systems. Finally, we show that a two-message quantum interactive proof system can also decide a multipartite generalization of the quantum separability problem.Comment: 34 pages, 6 figures; v2: technical improvements and new result for the multipartite quantum separability problem; v3: minor changes to address referee comments, accepted for presentation at the 2013 IEEE Conference on Computational Complexity; v4: changed problem names; v5: updated references and added a paragraph to the conclusion to connect with prior work on separability testin

Quantum de Finetti Theorems under Local Measurements with Applications

Quantum de Finetti theorems are a useful tool in the study of correlations in quantum multipartite states. In this paper we prove two new quantum de Finetti theorems, both showing that under tests formed by local measurements one can get a much improved error dependence on the dimension of the subsystems. We also obtain similar results for non-signaling probability distributions. We give the following applications of the results: We prove the optimality of the Chen-Drucker protocol for 3-SAT, under the exponential time hypothesis. We show that the maximum winning probability of free games can be estimated in polynomial time by linear programming. We also show that 3-SAT with m variables can be reduced to obtaining a constant error approximation of the maximum winning probability under entangled strategies of O(m^{1/2})-player one-round non-local games, in which the players communicate O(m^{1/2}) bits all together. We show that the optimization of certain polynomials over the hypersphere can be performed in quasipolynomial time in the number of variables n by considering O(log(n)) rounds of the Sum-of-Squares (Parrilo/Lasserre) hierarchy of semidefinite programs. As an application to entanglement theory, we find a quasipolynomial-time algorithm for deciding multipartite separability. We consider a result due to Aaronson -- showing that given an unknown n qubit state one can perform tomography that works well for most observables by measuring only O(n) independent and identically distributed (i.i.d.) copies of the state -- and relax the assumption of having i.i.d copies of the state to merely the ability to select subsystems at random from a quantum multipartite state. The proofs of the new quantum de Finetti theorems are based on information theory, in particular on the chain rule of mutual information.Comment: 39 pages, no figure. v2: changes to references and other minor improvements. v3: added some explanations, mostly about Theorem 1 and Conjecture 5. STOC version. v4, v5. small improvements and fixe