Multipartite Quantum Correlation and Communication Complexities

The concepts of quantum correlation complexity and quantum communication complexity were recently proposed to quantify the minimum amount of resources needed in generating bipartite classical or quantum states in the single-shot setting. The former is the minimum size of the initially shared state $\sigma$ on which local operations by the two parties (without communication) can generate the target state $\rho$, and the latter is the minimum amount of communication needed when initially sharing nothing. In this paper, we generalize these two concepts to multipartite cases, for both exact and approximate state generation. Our results are summarized as follows. (1) For multipartite pure states, the correlation complexity can be completely characterized by local ranks of sybsystems. (2) We extend the notion of PSD-rank of matrices to that of tensors, and use it to bound the quantum correlation complexity for generating multipartite classical distributions. (3) For generating multipartite mixed quantum states, communication complexity is not always equal to correlation complexity (as opposed to bipartite case). But they differ by at most a factor of 2. Generating a multipartite mixed quantum state has the same communication complexity as generating its optimal purification. But for correlation complexity of these two tasks can be different (though still related by less than a factor of 2). (4) To generate a bipartite classical distribution $P(x,y)$ approximately, the quantum communication complexity is completely characterized by the approximate PSD-rank of $P$. The quantum correlation complexity of approximately generating multipartite pure states is bounded by approximate local ranks.Comment: 19 pages; some typos are correcte

Quantum mutual information and quantumness vectors for multi-qubit systems

We introduce a new information theoretic measure of quantum correlations for multiparticle systems. We use a form of multivariate mutual information -- the interaction information and generalize it to multiparticle quantum systems. There are a number of different possible generalizations. We consider two of them. One of them is related to the notion of quantum discord and the other to the concept of quantum dissension. This new measure, called dissension vector, is a set of numbers -- quantumness vector. This can be thought of as a fine-grained measure, as opposed to measures that quantify some average quantum properties of a system. These quantities quantify/characterize the correlations present in multiparticle states. We consider some multiqubit states and find that these quantities are responsive to different aspects of quantumness, and correlations present in a state. We find that different dissension vectors can track the correlations (both classical and quantum), or quantumness only. As physical applications, we find that these vectors might be useful in several information processing tasks. We consider the role of dissension vectors -- (a) in deciding the security of BB84 protocol against an eavesdropper and (b) in determining the possible role of correlations in the performance of Grover search algorithm. Especially, in the Grover search algorithm, we find that dissension vectors can detect the correlations and show the maximum correlations when one expects.Comment: 18 pages 8 figures. Updated. Comments are welcom

Classical and quantum partition bound and detector inefficiency

We study randomized and quantum efficiency lower bounds in communication complexity. These arise from the study of zero-communication protocols in which players are allowed to abort. Our scenario is inspired by the physics setup of Bell experiments, where two players share a predefined entangled state but are not allowed to communicate. Each is given a measurement as input, which they perform on their share of the system. The outcomes of the measurements should follow a distribution predicted by quantum mechanics; however, in practice, the detectors may fail to produce an output in some of the runs. The efficiency of the experiment is the probability that the experiment succeeds (neither of the detectors fails). When the players share a quantum state, this gives rise to a new bound on quantum communication complexity (eff*) that subsumes the factorization norm. When players share randomness instead of a quantum state, the efficiency bound (eff), coincides with the partition bound of Jain and Klauck. This is one of the strongest lower bounds known for randomized communication complexity, which subsumes all the known combinatorial and algebraic methods including the rectangle (corruption) bound, the factorization norm, and discrepancy. The lower bound is formulated as a convex optimization problem. In practice, the dual form is more feasible to use, and we show that it amounts to constructing an explicit Bell inequality (for eff) or Tsirelson inequality (for eff*). We give an example of a quantum distribution where the violation can be exponentially bigger than the previously studied class of normalized Bell inequalities. For one-way communication, we show that the quantum one-way partition bound is tight for classical communication with shared entanglement up to arbitrarily small error.Comment: 21 pages, extended versio