We introduce a quantum version for the statistical complexity measure, in the context of quantum information theory, and use it as a signalling function of quantum order-disorder transitions. We discuss the possibility for such transitions to characterize interesting physical phenomena, as quantum phase transitions, or abrupt variations in the correlation distributions. We apply our measure to two exactly solvable Hamiltonian models, namely: the $1D$-Quantum Ising Model and the Heisenberg XXZ spin-$1/2$ chain. We also compute this measure for one-qubit and two-qubit reduced states for the considered models, and analyse its behaviour across its quantum phase transitions for finite system sizes as well as in the thermodynamic limit by using Bethe ansatz.
In this work we show that one cannot use non-local resources for probabilistic signalling even if one can delete a quantum state with the help of probabilistic quantum deletion machine. Here we find that probabilistic quantum deletion machine is not going to help us in identifying two statistical mixture at remote location. Also we derive the bound on deletion probability from no-signalling condition.
We show that non-local resources cannot be used for probabilistic signalling even if one can produce exact clones with the help of a probabilistic quantum cloning machine (PQCM). We show that PQCM cannot help to distinguish two statistical mixtures at a remote location. Thus quantum theory rules out the possibility of sending superluminal signals not only deterministically but also probabilistically. We give a bound on the success probability of producing multiple clones in an entangled system.
Simple quantitative measures of indeterminism and signalling, $I$ and $S$, are defined for models of statistical correlations. It is shown that any such model satisfies a generalised Bell-type inequality, with tight upper bound B(I,S). This upper bound explicitly quantifies the complementary contributions required from indeterminism and signalling, for modelling any given violation of the standard Bell-CHSH inequality. For example, all models of the maximum quantum violation must either assign no more than 80% probability of occurrence to some underlying event, and/or allow a nonlocal change o…
In an earlier paper the author expounded an interferometer scheme to communicate classical data over an entangled quantum channel. We return to this concept to show that the laws of Quantum Mechanics are not violated and that the device is able to affect the statistical blend of the quantum states (and that this can be detected) but not the statistics - i.e. the physics of observables. The ideas of superluminal information transfer, discussed in the previous paper, are taken forward to develop a notion of absolute space, time and motion with relativistic effects ascribed to motion through an a…
In instantaneous nonlocal quantum computation, two parties cooperate in order to perform a quantum computation on their joint inputs, while being restricted to a single round of simultaneous communication. Previous results showed that instantaneous nonlocal quantum computation is possible, at the cost of an exponential amount of prior shared entanglement (in the size of the input). Here, we show that a linear amount of entanglement suffices, (in the size of the computation), as long as the parties share nonlocal correlations as given by the Popescu-Rohlich box. This means that communication is…
We apply the statistical measure of complexity, introduced by López-Ruiz, Mancini and Calbet to a hard-sphere dilute Fermi gas whose particles interact via a repulsive hard-core potential. We employ the momentum distribution of this system to calculate the information entropy, the disequilibrium and the statistical complexity. We examine possible connections between the particle correlations and energy of the system with those information and complexity measures. The hard-sphere model serves as a test bed for concepts about complexity.
In this work, we analyze the behavior of statistical complexity in several systems where two identical densities that travel in opposite direction cross each other. The crossing between two Gaussian, rectangular and triangular densities is studied in detail. For these three cases, the shape of the total density presenting an extreme value in complexity is found.
The Fisher-Shannon statistical measure of complexity is analyzed for a continuous manifold of quantum observables. It is probed then than calculating it only in the configuration and momentum spaces will not give a complete description for certain systems. Then a more general measure for the complexity of a quantum system by the integration of the usual Fisher-Shannon measure over all the parameter space is proposed. Finally, these measures are applied to the concrete case of a free particle in a box.
We study the information geometry and the entropic dynamics of a 3D Gaussian statistical model. We then compare our analysis to that of a 2D Gaussian statistical model obtained from the higher-dimensional model via introduction of an additional information constraint that resembles the quantum mechanical canonical minimum uncertainty relation. We show that the chaoticity (temporal complexity) of the 2D Gaussian statistical model, quantified by means of the Information Geometric Entropy (IGE) and the Jacobi vector field intensity, is softened with respect to the chaoticity of the 3D Gaussian st…
Harvard University（哈佛大学）；Institute of High Performance Computing, Agency for Science, Technology and Research (A*STAR)（新加坡科技研究局（A*STAR）高性能计算研究所）；Zhejiang Sci-Tech University（浙江理工大学）；Zhejiang University（浙江大学）
In theoretical machine learning, the statistical complexity is a notion that measures the richness of a hypothesis space. In this work, we apply a particular measure of statistical complexity, namely the Rademacher complexity, to the quantum circuit model in quantum computation and study how the statistical complexity depends on various quantum circuit parameters. In particular, we investigate the dependence of the statistical complexity on the resources, depth, width, and the number of input and output registers of a quantum circuit. To study how the statistical complexity scales with resourc…
Harvard University（哈佛大学）；Institute of High Performance Computing, Agency for Science, Technology and Research (A*STAR)（新加坡科技研究局高性能计算研究院）；Zhejiang Sci-Tech University（浙江理工大学）；Zhejiang University（浙江大学）
We investigate how the addition of quantum resources changes the statistical complexity of quantum circuits by utilizing the framework of quantum resource theories. Measures of statistical complexity that we consider include the Rademacher complexity and the Gaussian complexity, which are well-known measures in computational learning theory that quantify the richness of classes of real-valued functions. We derive bounds for the statistical complexities of quantum circuits that have limited access to certain resources and apply our results to two special cases: (1) stabilizer circuits that are …
The Fisher-Shannon information and a statistical measure of complexity are calculated in the position and momentum spaces for the wave functions of the quantum isotropic harmonic oscillator. We show that these magnitudes are independent of the strength of the harmonic potential. Moreover, for each level of energy, it is found that these two indicators take their minimum values on the orbitals that correspond to the classical (circular) orbits in the Bohr-like quantum image, just those with the highest orbital angular momentum.
Error-tolerant witnessing of divergences in classical and quantum statistical complexity
Griffith University（格里菲斯大学）；Nanyang Technological University（南洋理工大学）；National University of Singapore（新加坡国立大学）；Horizon Quantum Computing（Horizon量子计算公司）
The calculation of a statistical measure of complexity and the Fisher-Shannon information in nuclei is carried out in this work. We use the nuclear shell model in order to obtain the fractional occupation probabilities of nuclear orbitals. The increasing of both magnitudes, the statistical complexity and the Fisher-Shannon information, with the number of nucleons is observed. The shell structure is reflected by the behavior of the statistical complexity. The magic numbers are revealed by the Fisher-Shannon information.
A two-parameter family of complexity measures $\tilde{C}^{(α,β)}$ based on the Rényi entropies is introduced and characterized by a detailed study of its mathematical properties. This family is the generalization of a continuous version of the LMC complexity, which is recovered for $α=1$ and $β=2$. These complexity measures are obtained by multiplying two quantities bringing global information on the probability distribution defining the system. When one of the parameters, $α$ or $β$, goes to infinity, one of the global factors becomes a local factor. For this special case, the complexity is c…
We define a new query measure we call quantum distinguishing complexity, denoted QD(f) for a Boolean function f. Unlike a quantum query algorithm, which must output a state close to |0> on a 0-input and a state close to |1> on a 1-input, a "quantum distinguishing algorithm" can output any state, as long as the output states for any 0-input and 1-input are distinguishable. Using this measure, we establish a new relationship in query complexity: For all total functions f, Q_0(f)=O~(Q(f)^5), where Q_0(f) and Q(f) denote the zero-error and bounded-error quantum query complexity of f respectively, …
We study the distribution of the Schmidt coefficients of the reduced density matrix of a quantum system in a pure state. By applying general methods of statistical mechanics, we introduce a fictitious temperature and a partition function and translate the problem in terms of the distribution of the eigenvalues of random matrices. We investigate the appearance of two phase transitions, one at a positive temperature, associated to very entangled states, and one at a negative temperature, signalling the appearance of a significant factorization in the many-body wave function. We also focus on the…
The minimal memory required to model a given stochastic process - known as the statistical complexity - is a widely adopted quantifier of structure in complexity science. Here, we ask if quantum mechanics can fundamentally change the qualitative behaviour of this measure. We study this question in the context of the classical Ising spin chain. In this system, the statistical complexity is known to grow monotonically with temperature. We evaluate the spin chain's quantum mechanical statistical complexity by explicitly constructing its provably simplest quantum model, and demonstrate that this m…
It is conjectured that in the geometric formulation of quantum computing, one can study quantum complexity through classical entropy of statistical ensembles established non-relativistically in the group manifold of unitary operators. The kinetic and positional decompositions of statistical entropy are conjectured to correspond to the Kolmogorov complexity and computational complexity, respectively, of corresponding quantum circuits. In this paper, we claim that by applying the virial theorem to the group manifold, one can derive a generic relation between Kolmogorov complexity and computation…
The quest for quantum computers is motivated by their potential for solving problems that defy existing, classical, computers. The theory of computational complexity, one of the crown jewels of computer science, provides a rigorous framework for classifying the hardness of problems according to the computational resources, most notably time, needed to solve them. Its extension to quantum computers allows the relative power of quantum computers to be analyzed. This framework identifies families of problems which are likely hard for classical computers (``NP-complete'') and those which are likel…
