TY - JOUR A1 - Eisert, Jens A1 - Gross, David T1 - Supersonic quantum communication N2 - When locally exciting a quantum lattice model, the excitation will propagate through the lattice. This effect is responsible for a wealth of nonequilibrium phenomena, and has been exploited to transmit quantum information. It is a commonly expressed belief that for local Hamiltonians, any such propagation happens at a finite "speed of sound". Indeed, the Lieb-Robinson theorem states that in spin models, all effects caused by a perturbation are essentially limited to a causal cone. We show that for meaningful translationally invariant bosonic models with nearest-neighbor interactions (addressing the challenging aspect of an experimental realization) this belief is incorrect: We prove that one can encounter accelerating excitations under the natural dynamics that allow for reliable transmission of information faster than any finite speed of sound. It also implies that the simulation of dynamics of strongly correlated bosonic models may be much harder than that of spin chains even in the low-energy sector. Y1 - 2009 UR - http://prl.aps.org/ U6 - https://doi.org/10.1103/Physrevlett.102.240501 SN - 0031-9007 ER - TY - JOUR A1 - Gross, David A1 - Liu, Yi-Kai A1 - Flammia, Steven T. A1 - Becker, Stephen A1 - Eisert, Jens T1 - Quantum state tomography via compressed sensing N2 - We establish methods for quantum state tomography based on compressed sensing. These methods are specialized for quantum states that are fairly pure, and they offer a significant performance improvement on large quantum systems. In particular, they are able to reconstruct an unknown density matrix of dimension d and rank r using O(rdlog(2)d) measurement settings, compared to standard methods that require d(2) settings. Our methods have several features that make them amenable to experimental implementation: they require only simple Pauli measurements, use fast convex optimization, are stable against noise, and can be applied to states that are only approximately low rank. The acquired data can be used to certify that the state is indeed close to pure, so no a priori assumptions are needed. Y1 - 2010 UR - http://prl.aps.org/ U6 - https://doi.org/10.1103/Physrevlett.105.150401 SN - 0031-9007 ER - TY - JOUR A1 - Gross, David A1 - Eisert, Jens T1 - Quantum computational webs N2 - We discuss the notion of quantum computational webs: These are quantum states universal for measurement-based computation, which can be built up from a collection of simple primitives. The primitive elements-reminiscent of building blocks in a construction kit-are (i) one-dimensional states (computational quantum wires) with the power to process one logical qubit and (ii) suitable couplings, which connect the wires to a computationally universal web. All elements are preparable by nearest-neighbor interactions in a single pass, of the kind accessible in a number of physical architectures. We provide a complete classification of qubit wires, a physically well-motivated class of universal resources that can be fully understood. Finally, we sketch possible realizations in superlattices and explore the power of coupling mechanisms based on Ising or exchange interactions. Y1 - 2010 UR - http://pra.aps.org/ U6 - https://doi.org/10.1103/Physreva.82.040303 SN - 1050-2947 ER - TY - JOUR A1 - Gross, David A1 - Flammia, Steven T. A1 - Eisert, Jens T1 - Most quantum states are too entangled to be useful as computational resources N2 - It is often argued that entanglement is at the root of the speedup for quantum compared to classical computation, and that one needs a sufficient amount of entanglement for this speedup to be manifest. In measurement- based quantum computing, the need for a highly entangled initial state is particularly obvious. Defying this intuition, we show that quantum states can be too entangled to be useful for the purpose of computation, in that high values of the geometric measure of entanglement preclude states from offering a universal quantum computational speedup. We prove that this phenomenon occurs for a dramatic majority of all states: the fraction of useful n-qubit pure states is less than exp(-n(2)). This work highlights a new aspect of the role entanglement plays for quantum computational speedups. Y1 - 2009 UR - http://prl.aps.org/ U6 - https://doi.org/10.1103/Physrevlett.102.190501 SN - 0031-9007 ER - TY - JOUR A1 - Gross, David A1 - Nesme, V. A1 - Vogts, H. A1 - Werner, Reinhard F. T1 - Index theory of one dimensional quantum walks and cellular automata JF - Communications in mathematical physics N2 - If a one-dimensional quantum lattice system is subject to one step of a reversible discrete-time dynamics, it is intuitive that as much "quantum information" as moves into any given block of cells from the left, has to exit that block to the right. For two types of such systems - namely quantum walks and cellular automata - we make this intuition precise by defining an index, a quantity that measures the "net flow of quantum information" through the system. The index supplies a complete characterization of two properties of the discrete dynamics. First, two systems S-1, S-2 can be "pieced together", in the sense that there is a system S which acts like S-1 in one region and like S-2 in some other region, if and only if S-1 and S-2 have the same index. Second, the index labels connected components of such systems: equality of the index is necessary and sufficient for the existence of a continuous deformation of S-1 into S-2. In the case of quantum walks, the index is integer-valued, whereas for cellular automata, it takes values in the group of positive rationals. In both cases, the map S bar right arrow. ind S is a group homomorphism if composition of the discrete dynamics is taken as the group law of the quantum systems. Systems with trivial index are precisely those which can be realized by partitioned unitaries, and the prototypes of systems with non-trivial index are shifts. Y1 - 2012 U6 - https://doi.org/10.1007/s00220-012-1423-1 SN - 0010-3616 VL - 310 IS - 2 SP - 419 EP - 454 PB - Springer CY - New York ER - TY - JOUR A1 - Müller, Markus P. A1 - Gross, David A1 - Eisert, Jens T1 - Concentration of Measure for Quantum States with a Fixed Expectation Value JF - Communications in mathematical physics N2 - Given some observable H on a finite-dimensional quantum system, we investigate the typical properties of random state vectors vertical bar psi >> that have a fixed expectation value < psi vertical bar H vertical bar psi > = E with respect to H. Under some conditions on the spectrum, we prove that this manifold of quantum states shows a concentration of measure phenomenon: any continuous function on this set is almost everywhere close to its mean. We also give a method to estimate the corresponding expectation values analytically, and we prove a formula for the typical reduced density matrix in the case that H is a sum of local observables. We discuss the implications of our results as new proof tools in quantum information theory and to study phenomena in quantum statistical mechanics. As a by-product, we derive a method to sample the resulting distribution numerically, which generalizes the well-known Gaussian method to draw random states from the sphere. Y1 - 2011 U6 - https://doi.org/10.1007/s00220-011-1205-1 SN - 0010-3616 VL - 303 IS - 3 SP - 785 EP - 824 PB - Springer CY - New York ER - TY - JOUR A1 - Gross, David A1 - Mueller, Markus A1 - Colbeck, Roger A1 - Dahlsten, Oscar C. O. T1 - All reversible dynamics in maximally nonlocal theories are trivial N2 - A remarkable feature of quantum theory is nonlocality ( Bell inequality violations). However, quantum correlations are not maximally nonlocal, and it is natural to ask whether there are compelling reasons for rejecting theories in which stronger violations are possible. To shed light on this question, we consider post-quantum theories in which maximally nonlocal states ( nonlocal boxes) occur. We show that reversible transformations in such theories are trivial: they consist solely of local operations and permutations of systems. In particular, no correlations can be created; nonlocal boxes cannot be prepared from product states and classical computers can efficiently simulate all such processes. Y1 - 2010 UR - http://prl.aps.org/ U6 - https://doi.org/10.1103/Physrevlett.104.080402 SN - 0031-9007 ER - TY - JOUR A1 - Colin, S A1 - Corbett, J A1 - Durt, T A1 - Gross, David T1 - About SICPOVMs and discrete Wigner distributions N2 - A set of d(2) vectors in a Hilbert space of dimension d is called equiangular if each pair of vectors encloses the same angle. The projection operators onto these vectors define a POVM which is distinguished by its high degree of symmetry. Measures of this kind are called symmetric informationally complete, or SIC POVMs for short, and could be applied for quantum state tomography. Despite its simple geometrical description, the problem of constructing SIC POVMs or even proving their existence seems to be very hard. It is our purpose to introduce two applications of discrete Wigner functions to the analysis of the problem at hand. First, we will present a method for identifying symmetries of SIC POVMs under Clifford operations. This constitutes an alternative approach to a structure described before by Zauner and Appleby. Further, a simple and geometrically motivated construction for an SIC POVM in dimensions two and three is given (which, unfortunately, allows no generalization). Even though no new structures are found, we hope that the re- formulation of the problem may prove useful for future inquiries Y1 - 2005 ER -