We devise a protocol in which general nonclassical multipartite correlations produce a physically relevant effect, leading to the creation of bipartite entanglement. In particular, we show that the relative entropy of quantumness, which measures all nonclassical correlations among subsystems of a quantum system, is equivalent to and can be operationally interpreted as the minimum distillable entanglement generated between the system and local ancillae in our protocol. We emphasize the key role of state mixedness in maximizing nonclassicality: Mixed entangled states can be arbitrarily more nonclassical than separable and pure entangled states.PACS numbers: 03.65. Ud, 03.67.Ac, 03.67.Mn, 03.65.Ta The study of quantum correlations has traditionally focused on entanglement [1]. It is generally believed that entanglement is a necessary resource for quantum computers to outperform their classical counterparts. Indeed, it has been shown that for the setting of pure-state computation, the amount of entanglement present must grow with the system size for an exponential speed-up to occur [2]. In the context of mixedstate quantum information processing, however, there are computational and communication feats which are seemingly impossible to achieve with a classical computer, and yet can be attained with a quantum computer using little or no entanglement (e.g. [3,4]). For example, the Deterministic Quantum Computation with one Qubit (DQC1) model is believed to estimate the trace of a unitary matrix exponentially faster than any classical algorithm, yet with vanishing entanglement during the computation [5]. A second example is the ability for certain bipartite quantum systems to contain a large amount of "locked" classical correlations, which can then be "unlocked" with a disproportionately small amount of classical communication [4]. This task is impossible classically, yet the quantum states involved are separable, that is, unentangled. This raises the crucial question about which, if not entanglement, is the fundamental resource enabling such feats.One plausible explanation is associated with the presence in (generic [6]) quantum states of correlations which have nonclassical signatures that go beyond entanglement. Indeed, much attention has recently been devoted to understanding and quantifying such correlations for this very reason [6][7][8][9][10][11][12][13][14][15][16]. In particular, the separable quantum states of the systems involved in DQC1 and the locking protocol have been shown to possess non-zero amounts of such correlations [5,17], as measured by the quantum discord [7]. The latter strives to capture nonclassical correlations beyond entanglement and has recently received operational interpretations in terms of the quantum state merging protocol [18], but is unfortunately not a faithful measure [19]. A more accurate quantification of nonclassical correlations is provided by the so-called relative entropy of quantumness (REQ) [8,[10][11][12][13], defined as the minimum distance, in terms of relative entr...
We investigate signatures of non-classicality in quantum states, in particular, those involved in the DQC1 model of mixed-state quantum computation [Phys. Rev. Lett. 81, 5672 (1998)]. To do so, we consider two known non-classicality criteria. The first quantifies disturbance of a quantum state under locally noneffective unitary operations (LNU), which are local unitaries acting invariantly on a subsystem. The second quantifies measurement induced disturbance (MID) in the eigenbasis of the reduced density matrices. We study the role of both figures of non-classicality in the exponential speedup of the DQC1 model and compare them vis-a-vis the interpretation provided in terms of quantum discord. In particular, we prove that a non-zero quantum discord implies a non-zero shift under LNUs. We also use the MID measure to study the locking of classical correlations [Phys. Rev. Lett. 92, 067902 (2004)] using two mutually unbiased bases (MUB). We find the MID measure to exactly correspond to the number of locked bits of correlation. For three or more MUBs, it predicts the possibility of superior locking effects.
Given the density matrix $\rho$ of a bipartite quantum state, the quantum separability problem asks whether $\rho$ is entangled or separable. In 2003, Gurvits showed that this problem is NP-hard if $\rho$ is located within an inverse exponential (with respect to dimension) distance from the border of the set of separable quantum states. In this paper, we extend this NP-hardness to an inverse polynomial distance from the separable set. The result follows from a simple combination of works by Gurvits, Ioannou, and Liu. We apply our result to show (1) an immediate lower bound on the maximum distance between a bound entangled state and the separable set (assuming $\rm{P}\neq\rm{ NP}$), and (2) NP-hardness for the problem of determining whether a completely positive trace-preserving linear map is entanglement-breaking.
Constraint satisfaction problems are a central pillar of modern computational complexity theory. This survey provides an introduction to the rapidly growing field of Quantum Hamiltonian Complexity, which includes the study of quantum constraint satisfaction problems. Over the past decade and a half, this field has witnessed fundamental breakthroughs, ranging from the establishment of a "Quantum Cook-Levin Theorem" to deep insights into the structure of 1D low-temperature quantum systems via so-called area laws. Our aim here is to provide a computer science-oriented introduction to the subject in order to help bridge the language barrier between computer scientists and physicists in the field. As such, we include the following in this survey: (1) The motivations and history of the field, (2) a glossary of condensed matter physics terms explained in computer-science friendly language, (3) overviews of central ideas from condensed matter physics, such as indistinguishable particles, mean field theory, tensor networks, and area laws, and (4) brief expositions of selected computer science-based results in the area. For example, as part of the latter, we provide a novel information theoretic presentation of Bravyi's polynomial time algorithm for Quantum 2-SAT.Comment: v4: published version, 127 pages, introduction expanded to include brief introduction to quantum information, brief list of some recent developments added, minor changes throughou
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