- We introduce a class of so called Markovian marginals, which gives a natural framework for constructing solutions to the quantum marginal problem. We consider a set of marginals that possess a certain internal quantum Markov chain structure. If they are equipped with such a structure and are locally consistent on their overlapping supports, there exists a global state that is consistent with all the marginals. The proof is constructive, and relies on a reduction of the marginal problem to a certain combinatorial problem. By employing an entanglement entropy scaling law, we give a physical argument that the requisite structure exists in any states with finite correlation lengths. This includes topologically ordered states as well as finite temperature Gibbs states.
- Sep 28 2016 quant-ph arXiv:1609.08595v1It is a fundamental property of quantum mechanics that information is lost as a result of performing measurements. Indeed, with every quantum measurement one can associate a number -- its POVM norm constant -- that quantifies how much the distinguishability of quantum states degrades in the worst case as a result of the measurement. This raises the obvious question which measurements preserve the most information in these sense of having the largest norm constant. While a number of near-optimal schemes have been found (e.g. the uniform POVM, or complex projective 4-designs), they all seem to be difficult to implement in practice. Here, we analyze the distinguishability of quantum states under measurements that are orbits of the Clifford group. The Clifford group plays an important role e.g. in quantum error correction, and its elements are considered simple to implement. We find that the POVM norm constants of Clifford orbits depend on the effective rank of the states that should be distinguished, as well as on a quantitative measure of the "degree of localization in phase space" of the vectors in the orbit. The most important Clifford orbit is formed by the set of stabilizer states. Our main result implies that stabilizer measurements are essentially optimal for distinguishing pure quantum states. As an auxiliary result, we use the methods developed here to prove new entropic uncertainty relations for stabilizer measurements. This paper is based on a very recent analysis of the representation theory of tensor powers of the Clifford group.
- Sep 28 2016 quant-ph arXiv:1609.08170v1We develop an efficient quantum implementation of an important signal processing algorithm for line spectral estimation: the matrix pencil method, which determines the frequencies and damping factors of signals consisting of finite sums of exponentially damped sinusoids. Our algorithm provides a quantum speedup in a natural regime where the sampling rate is much higher than the number of sinusoid components. Along the way, we develop techniques that are expected to be useful for other quantum algorithms as well - consecutive phase estimations to efficiently make products of asymmetric low rank matrices classically accessible and an alternative method to efficiently exponentiate non-Hermitian matrices. Our algorithm features an efficient quantum-classical division of labor: The time-critical steps are implemented in quantum superposition, while an interjacent step, requiring only exponentially few parameters, can operate classically. We show that frequencies and damping factors can be obtained in time logarithmic in the number of sampling points, exponentially faster than known classical algorithms.
- Sep 28 2016 quant-ph arXiv:1609.08592v1We give a capacity formula for the classical information transmission over a noisy quantum channel, with separable encoding by the sender and limited resources provided by the receiver's pre-shared ancilla. Instead of a pure state, we consider the signal-ancilla pair in a mixed state, purified by a "witness". Thus, the signal-witness correlation limits the resource available from the signal-ancilla correlation. Our formula characterizes the utility of different forms of resources, including noisy or limited entanglement assistance, for classical communication. With separable encoding, the sender's signals between different channel uses are still allowed to be entangled, yet our capacity formula is additive. In particular, for generalized covariant channels our capacity formula has a simple closed-form. Moreover, our additive capacity formula upper bounds the general coherent attack's information gain in various two-way quantum key distribution protocols. For Gaussian protocols, the additivity of the formula indicates that the collective Gaussian attack is the most powerful.
- Sep 28 2016 quant-ph arXiv:1609.08487v1Device-independent quantum cryptography allows security even if the devices used to execute the protocol are untrusted - whether this is due to unknown imperfections in the implementation, or because the adversary himself constructed them to subvert the security of the protocol. While device-independence has seen much attention in the domain of quantum key distribution, relatively little is known for general protocols. Here we introduce a new model for device-independence for two-party protocols and position verification in the noisy-storage model. For the first time, we show that such protocols are secure in the most general device-independent model in which the devices may have arbitrary memory, states and measurements. In our analysis, we make use of a slight modification of a beautiful new tool developed in [arXiv:1607.01796] called "Entropy Accumulation Theorem". What's more, the protocols we analyze use only simple preparations and measurements, and can be realized using any experimental setup able to perform a CHSH Bell test. Specifically, security can be attained for any violation of the CHSH inequality, where a higher violation merely leads to a reduction in the amount of rounds required to execute the protocol.
- Sep 28 2016 quant-ph arXiv:1609.08188v1The Clifford group is a fundamental structure in quantum information with a wide variety of applications. We discuss the tensor representations of the $q$-qubit Clifford group, which is defined as the normalizer of the $q$-qubit Pauli group in $U(2^q)$. In particular, we characterize all irreducible subrepresentations of the two-copy representation $\varphi^{\otimes2}$ of the Clifford group on the matrix space $\mathbb{C}^{d\times d}\otimes \mathbb{C}^{d\times d}$ with $d=2^q$. In an upcoming companion paper we applied this result to cut down the number of samples necessary to perform randomised benchmarking, a method for characterising quantum systems.
- Using the notion of TRO's (ternary ring of operators) and independence from operator algebra theory, we discover a new class of channels which allow single-letter bounds for their quantum and private capacity, as well as strong converse rates. This class goes beyond degradable channels. The estimate are based on a "local comparison theorem" for sandwiched Rényi relative entropy and complex interpolation. As an application, we discover new small dimensional examples which admit an easy formula for quantum and private capacities.
- Sep 28 2016 quant-ph arXiv:1609.08571v1The Feynman-Kitaev Hamiltonian used in the proof of QMA-completeness of the local Hamiltonian problem has a ground state energy which scales as $\Omega((1-\sqrt{\epsilon}) T^{-3})$ when it is applied to a circuit of size $T$ and maximum acceptance probability $\epsilon$. We refer to this quantity as the quantum UNSAT penalty, and using a modified form of the Feynman Hamiltonian with a non-uniform history state as its ground state we improve its scaling to $\Omega((1-\sqrt{\epsilon})T^{-2})$, without increasing the number of local terms or their operator norms. As part of the proof we show how to construct a circuit Hamiltonian for any desired probability distribution on the time steps of the quantum circuit (which, for example, can be used to increase the probability of measuring a history state in the final step of the computation). Next we show a tight $\mathcal{O}(T^{-2})$ upper bound on the product of the spectral gap and ground state overlap with the endpoints of the computation for any clock Hamiltonian that is tridiagonal in the time register basis, which shows that the scaling of the quantum UNSAT penalty achieved by our construction cannot be further improved within this framework. Our proof of the upper bound applies a quantum-to-classical mapping for arbitrary tridiagonal Hermitian matrices combined with a sharp bound on the spectral gap of birth-and-death Markov chains. In the context of universal adiabatic computation we show how to reduce the number of qubits required to represent the clock by a constant factor over the standard construction, but show that it is otherwise already optimal in the sense we consider and cannot be further improved with tridiagonal clock Hamiltonians, which agrees with a similar upper bound from a previous study.
- Sep 28 2016 quant-ph arXiv:1609.08172v1A unitary t-design is a set of unitaries that is "evenly distributed" in the sense that the average of any t-th order polynomial over the design equals the average over the entire unitary group. In various fields -- e.g. quantum information theory -- one frequently encounters constructions that rely on matrices drawn uniformly at random from the unitary group. Often, it suffices to sample these matrices from a unitary t-design, for sufficiently high t. This results in more explicit, derandomized constructions. The most prominent unitary t-design considered in quantum information is the multi-qubit Clifford group. It is known to be a unitary 3-design, but, unfortunately, not a 4-design. Here, we give a simple, explicit characterization of the way in which the Clifford group fails to constitute a 4-design. Our results show that for various applications in quantum information theory and in the theory of convex signal recovery, Clifford orbits perform almost as well as those of true 4-designs. Technically, it turns out that in a precise sense, the 4th tensor power of the Clifford group affords only one more invariant subspace than the 4th tensor power of the unitary group. That additional subspace is a stabilizer code -- a structure extensively studied in the field of quantum error correction codes. The action of the Clifford group on this stabilizer code can be decomposed explicitly into previously known irreps of the discrete symplectic group. We give various constructions of exact complex projective 4-designs or approximate 4-designs of arbitrarily high precision from Clifford orbits. Building on results from coding theory, we give strong evidence suggesting that these orbits actually constitute complex projective 5-designs.
- We propose an extension of the sandwiched Rényi relative $\alpha$-entropy for states on arbitrary von Neumann algebra, for the values $\alpha>1$. For this, we use Kosaki's definition of noncommutative $L_p$-spaces with respect to a state. Some properties of these extensions are proved, in particular the data processing inequality with respect to positive trace preserving maps. It is also shown that equality in data processing inequality characterizes sufficiency of quantum channels.
- Sep 28 2016 quant-ph physics.comp-ph arXiv:1609.08584v1The de Finetti representation theorem for continuous variable quantum system is first developed to approximate an N-partite continuous variable quantum state with a convex combination of independent and identical subsystems, which requires the original state to obey permutation symmetry conditioned on successful experimental verification on k of N subsystems. We generalize the de Finetti theorem to include asymmetric bounds on the variance of canonical observables and biased basis selection during the verification step. Our result thereby enables application of infinite-dimensional de Finetti theorem to situations where two conjugate measurements obey different statistics, such as the security analysis of quantum key distribution protocols based on squeezed state against coherent attack.
- Sep 28 2016 astro-ph.CO arXiv:1609.08554v1Understanding of the observed structure in the universe can be reached only in the theoretical framework of dark matter. N-body simulations are indispensable for the analysis of the formation and evolution of the dark matter web. Two primary fields - density and velocity fields - are used in most of studies. However dark matter provides two additional fields which are unique for collisionless media only. These are the multi- stream field in Eulerian space and flip-flop field in Lagrangian space. The flip-flop field represents the number of sign reversals of an elementary volume of each collisionless fluid element. This field can be estimated by counting the sign reversals of the Jacobian at each particle at every time step of the simulation. The Jacobian is evaluated by numerical differentiation of the Lagrangian submanifold, i.e., the three-dimensional dark matter sheet in the six-dimensional space formed by three Lagrangian and three Eulerian coordinates. We present the results of the statistical study of the evolution of the flip-flop field from z = 50 to the present time z = 0. A number of statistical characteristics show that the pattern of the flip-flop field remains remarkably stable from z = 30 to the present time. As a result the flip-flop field evaluated at z = 0 stores a wealth of information about the dynamical history of the dark matter web. In particular one of the most intriguing properties of the flip-flop is a unique capability to preserve the information about the merging history of dark matter haloes.
- Sep 28 2016 quant-ph arXiv:1609.08526v1The roles of Lie groups in Feynman's path integrals in non-relativistic quantum mechanics are discussed. Dynamical as well as geometrical symmetries are found useful for path integral quantization. Two examples having the symmetry of a non-compact Lie group are considered. The first is the free quantum motion of a particle on a space of constant negative curvature. The system has a group SO(d,1) associated with the geometrical structure, to which the technique of harmonic analysis on a homogeneous space is applied. As an example of a system having a non-compact dynamical symmetry, the d-dimensional harmonic oscillator is chosen, which has the non-compact dynamical group SU(1,1) besides its geometrical symmetry SO(d). The radial path integral is seen as a convolution of the matrix functions of a compact group element of SU(1,1) on the continuous basis.
- Sep 28 2016 astro-ph.CO arXiv:1609.08510v1We use the Fisher matrix formalism to study the expansion and growth history of the Universe using galaxy clustering with 2D angular cross-correlation tomography in spectroscopic or high resolution photometric redshift surveys. The radial information is contained in the cross correlations between narrow redshift bins. We show how multiple tracers with redshift space distortions cancel sample variance and arbitrarily improve the constraints on the dark energy equation of state $\omega(z)$ and the growth parameter $\gamma$ in the noiseless limit. The improvement for multiple tracers quickly increases with the bias difference between the tracers, up to a factor $\sim4$ in $\text{FoM}_{\gamma\omega}$. We model a magnitude limited survey with realistic density and bias using a conditional luminosity function, finding a factor 1.3-9.0 improvement in $\text{FoM}_{\gamma\omega}$ -- depending on global density -- with a split in a halo mass proxy. Partly overlapping redshift bins improve the constraints in multiple tracer surveys a factor $\sim1.3$ in $\text{FoM}_{\gamma\omega}$. This findings also apply to photometric surveys, where the effect of using multiple tracers is magnified. We also show large improvement on the FoM with increasing density, which could be used as a trade-off to compensate some possible loss with radial resolution.
- Sep 28 2016 cs.CV arXiv:1609.08399v1Most existing automatic house price estimation systems rely only on some textual data like its neighborhood area and the number of rooms. The final price is estimated by a human agent who visits the house and assesses it visually. In this paper, we propose extracting visual features from house photographs and combining them with the house's textual information. The combined features are fed to a fully connected multilayer Neural Network (NN) that estimates the house price as its single output. To train and evaluate our network, we have collected the first houses dataset (to our knowledge) that combines both images and textual attributes. The dataset is composed of 535 sample houses from the state of California, USA. Our experiments showed that adding the visual features increased the R-value by a factor of 3 and decreased the Mean Square Error (MSE) by one order of magnitude compared with textual-only features. Additionally, when trained on the benchmark textual-only features housing dataset, our proposed NN still outperformed the existing model published results.
- Nash equilibrium is not guaranteed in finite quantum games. In this letter, we revisit this fact using John Nash's original approach of countering sets and Kakutani's fixed point theorem. To the best of our knowledge, this mathematically formal approach has not been explored before in the context of quantum games. We use this approach to draw conclusions about Nash equilibrium states in quantum informational processes such as quantum computing and quantum communication protocols.
- Sep 28 2016 astro-ph.CO arXiv:1609.08247v1Peculiar velocity surveys present a very promising route to measuring the growth rate of large-scale structure and its scale dependence. However, individual peculiar velocity surveys suffer from large statistical errors due to the intrinsic scatter in the relations used to infer a galaxy's true distance. In this context we use a Fisher Matrix formalism to investigate the statistical benefits of combining multiple peculiar velocity surveys. We find that for all cases we consider there is a marked improvement on constraints on the linear growth rate $f\sigma_{8}$. For example, the constraining power of only a few peculiar velocity measurements is such that the addition of the 2MASS Tully-Fisher survey (containing only $\sim2,000$ galaxies) to the full redshift and peculiar velocity samples of the 6-degree Field Galaxy Survey (containing $\sim 110,000$ redshifts and $\sim 9,000$ velocities) can improve growth rate constraints by $\sim20\%$. Furthermore, the combination of the future TAIPAN and WALLABY+WNSHS surveys has the potential to reach a $\sim3\%$ error on $f\sigma_{8}$, which will place tight limits on possible extensions to General Relativity. We then turn to look at potential systematics in growth rate measurements that can arise due to incorrect calibration of the peculiar velocity zero-point and from scale-dependent spatial and velocity bias. For next generation surveys, we find that neglecting velocity bias in particular has the potential to bias constraints on the growth rate by over $5\sigma$, but that an offset in the zero-point has negligible impact on the velocity power spectrum.
- Sep 28 2016 gr-qc arXiv:1609.08189v1In this letter, cosmology of a simple NMDC gravity with $\xi R \phi_{,\mu}\phi^{,\mu}$ term and a free kinetic term is considered in flat geometry and in presence of dust matter. A logarithm field transformation $\phi' = \mu \ln \phi$ is proposed phenomenologically to ensures domination of the NMDC term at small field values. Assuming slow-roll approximation, equation of motion, scalar field solution and potential are derived as function of kinematic variables. The field solution and potential are found straightforwardly for power-law, de-Sitter and super-acceleration expansions.
- Sep 28 2016 quant-ph arXiv:1609.08148v1Although the notion of superdeterminism can, in principle, account for the violation of the Bell inequalities, this potential explanation has been roundly rejected by the quantum foundations community. The arguments for rejection, one of the most substantive coming from Bell himself, are critically reviewed. In particular, analysis of Bell's argument reveals an implicit unwarranted assumption: that the Euclidean metric is the appropriate yardstick for measuring distances in state space. Bell's argument is largely negated if this yardstick is instead based on the alternative $p$-adic metric. Such a metric, common in number theory, arises naturally when describing chaotic systems which evolve precisely on self-similar invariant sets in their state space. A locally-causal realistic model of quantum entanglement is developed, based on the premise that the laws of physics ultimately derive from an invariant-set geometry in the state space of a deterministic quasi-cyclic mono-universe. Based on this, the notion of a complex Hilbert vector is reinterpreted in terms of an uncertain selection from a finite sample space of states, leading to a novel form of `consistent histories' based on number-theoretic properties of the transcendental cosine function. This leads to novel realistic interpretations of position/momentum non-commutativity, EPR, the Bell Theorem and the Tsirelson bound. In this inherently holistic theory - neither conspiratorial, retrocausal, fine tuned nor nonlocal - superdeterminism is not invoked by fiat but is emergent from these `consistent histories' number-theoretic constraints. Invariant set theory provides new perspectives on many of the contemporary problems at the interface of quantum and gravitational physics, and, if correct, may signal the end of particle physics beyond the Standard Model.
- Sep 28 2016 physics.gen-ph quant-ph arXiv:1609.08427v1A physical picture for Quantum Mechanics which permits to conciliate it with the usual common sense is proposed. The picture agrees with the canonical Copenhagen interpretation making more clear its statements.
- Sep 28 2016 astro-ph.GA arXiv:1609.08605v1
- Sep 28 2016 astro-ph.HE arXiv:1609.08606v1
- Sep 28 2016 math.NT arXiv:1609.08603v1
- Sep 28 2016 math.NT arXiv:1609.08602v1
- Sep 28 2016 math.DG arXiv:1609.08601v1
- Sep 28 2016 math.CV arXiv:1609.08600v1
- Sep 28 2016 math.DG arXiv:1609.08599v1
- Sep 28 2016 math.AP arXiv:1609.08597v1
- Sep 28 2016 math.RT arXiv:1609.08593v1
- Sep 28 2016 physics.atom-ph arXiv:1609.08591v1
- Sep 28 2016 quant-ph physics.atom-ph arXiv:1609.08589v1
- Sep 28 2016 cs.DS arXiv:1609.08588v1
- Sep 28 2016 cs.NI arXiv:1609.08583v1
- Sep 28 2016 physics.chem-ph arXiv:1609.08577v1
- Sep 28 2016 math.CA arXiv:1609.08575v1
- Sep 28 2016 cs.DC arXiv:1609.08574v1
- Sep 28 2016 cond-mat.stat-mech arXiv:1609.08573v1
- Sep 28 2016 astro-ph.CO arXiv:1609.08572v1