arXiv subjects
Publications and source records attributed to N. J. Cerf.
We suggest that the framework of quantum information theory, which has been developing rapidly in recent years due to intense activity in quantum computation and quantum communication, is a reasonable starting point to study non-equilibrium quantum statistical phenomena. As an application, we discuss the non-equilibrium quantum thermodynamics of black hole formation and evaporation.
A quantum algorithm is known that solves an unstructured search problem in a number of iterations of order $\sqrt{d}$, where $d$ is the dimension of the search space, whereas any classical algorithm necessarily scales as $O(d)$. It is shown here that an improved quantum search algorithm can be devised that exploits the structure of a tree search problem by nesting this standard search algorithm. The number of iterations required to find the solution of an average instance of a constraint satisfaction problem scales as $\sqrt{d^α}$, with a constant $α<1$ depending on the nesting depth and the problem considered. When applying a single nesting level to a problem with constraints of size 2 such as the graph coloring problem, this constant $α$ is estimated to be around 0.62 for average instances of maximum difficulty. This corresponds to a square-root speedup over a classical nested search algorithm, of which our presented algorithm is the quantum counterpart.
An investigation of Einstein's ``physical'' reality and the concept of quantum reality in terms of information theory suggests a solution to quantum paradoxes such as the Einstein-Podolsky-Rosen (EPR) and the Schroedinger-cat paradoxes. Quantum reality, the picture based on unitarily evolving wavefunctions, is complete, but appears incomplete from the observer's point of view for fundamental reasons arising from the quantum information theory of measurement. Physical reality, the picture based on classically accessible observables is, in the worst case of EPR experiments, unrelated to the quantum reality it purports to reflect. Thus, quantum information theory implies that only correlations, not the correlata, are physically accessible: the mantra of the Ithaca interpretation of quantum mechanics.
We present a constructive method to translate small quantum circuits into their optical analogues, using linear components of present-day quantum optics technology only. These optical circuits perform precisely the computation that the quantum circuits are designed for, and can thus be used to test the performance of quantum algorithms. The method relies on the representation of several quantum bits by a single photon, and on the implementation of universal quantum gates using simple optical components (beam splitters, phase shifters, etc.). The optical implementation of Brassard et al.'s teleportation circuit, a non-trivial 3-bit quantum computation, is presented as an illustration.
The traveling salesman problem (TSP) consists of finding the length of the shortest closed tour visiting N ``cities''. We consider the Euclidean TSP where the cities are distributed randomly and independently in a d-dimensional unit hypercube. Working with periodic boundary conditions and inspired by a remarkable universality in the kth nearest neighbor distribution, we find for the average optimum tour length = beta_E(d) N^{1-1/d} [1+O(1/N)] with beta_E(2) = 0.7120 +- 0.0002 and beta_E(3) = 0.6979 +- 0.0002. We then derive analytical predictions for these quantities using the random link approximation, where the lengths between cities are taken as independent random variables. From the ``cavity'' equations developed by Krauth, Mezard and Parisi, we calculate the associated random link values beta_RL(d). For d=1,2,3, numerical results show that the random link approximation is a good one, with a discrepancy of less than 2.1% between beta_E(d) and beta_RL(d). For large d, we argue that the approximation is exact up to O(1/d^2) and give a conjecture for beta_E(d), in terms of a power series in 1/d, specifying both leading and subleading coefficients.
A framework for a quantum mechanical information theory is introduced that is based entirely on density operators, and gives rise to a unified description of classical correlation and quantum entanglement. Unlike in classical (Shannon) information theory, quantum (von Neumann) conditional entropies can be negative when considering quantum entangled systems, a fact related to quantum non-separability. The possibility that negative (virtual) information can be carried by entangled particles suggests a consistent interpretation of quantum informational processes.
We analyze the properties of the conditional amplitude operator, the quantum analog of the conditional probability which has been introduced in [quant-ph/9512022]. The spectrum of the conditional operator characterizing a quantum bipartite system is invariant under local unitary transformations and reflects its inseparability. More specifically, it is shown that the conditional amplitude operator of a separable state cannot have an eigenvalue exceeding 1, which results in a necessary condition for separability. This leads us to consider a related separability criterion based on the positive map $Γ:ρ\to (Tr ρ) - ρ$, where $ρ$ is an Hermitian operator. Any separable state is mapped by the tensor product of this map and the identity into a non-negative operator, which provides a simple necessary condition for separability. In the special case where one subsystem is a quantum bit, $Γ$ reduces to time-reversal, so that this separability condition is equivalent to partial transposition. It is therefore also sufficient for $2\times 2$ and $2\times 3$ systems. Finally, a simple connection between this map and complex conjugation in the "magic" basis is displayed.
A systematic method for simulating small-scale quantum circuits by use of linear optical devices is presented. It relies on the representation of several quantum bits by a single photon, and on the implementation of universal quantum gates using simple optical components (beam splitters, phase shifters, etc.). This suggests that the optical realization of small quantum networks is reasonable given the present technology in quantum optics, and could be a useful technique for testing simple quantum algorithms or error-correction schemes. The optical circuit for quantum teleportation is presented as an illustration.
We discuss the capacity of quantum channels for information transmission and storage. Quantum channels have dual uses: they can be used to transmit known quantum states which code for classical information, and they can be used in a purely quantum manner, for transmitting or storing quantum entanglement. We propose here a definition of the von Neumann capacity of quantum channels, which is a quantum mechanical extension of the Shannon capacity and reverts to it in the classical limit. As such, the von Neumann capacity assumes the role of a classical or quantum capacity depending on the usage of the channel. In analogy to the classical construction, this capacity is defined as the maximum von Neumann mutual entropy processed by the channel, a measure which reduces to the capacity for classical information transmission through quantum channels (the "Kholevo capacity") when known quantum states are sent. The quantum mutual entropy fulfills all basic requirements for a measure of information, and observes quantum data-processing inequalities. We also derive a quantum Fano inequality relating the quantum loss of the channel to the fidelity of the quantum code. The quantities introduced are calculated explicitly for the quantum "depolarizing" channel. The von Neumann capacity is interpreted within the context of superdense coding, and an "extended" Hamming bound is derived that is consistent with that capacity.
The many-body dynamics of a quantum computer can be reduced to the time evolution of non-interacting quantum bits in auxiliary fields by use of the Hubbard-Stratonovich representation of two-bit quantum gates in terms of one-bit gates. This makes it possible to perform the stochastic simulation of a quantum algorithm, based on the Monte Carlo evaluation of an integral of dimension polynomial in the number of quantum bits. As an example, the simulation of the quantum circuit for the Fast Fourier Transform is discussed.
An analysis of quantum measurement is presented that relies on an information-theoretic description of quantum entanglement. In a consistent quantum information theory of entanglement, entropies (uncertainties) conditional on measurement outcomes can be negative, implying that measurement can be described via unitary, entropy-conserving, interactions, while still producing randomness in a measurement device. In such a framework, quantum measurement is not accompanied by a wave-function collapse, or a quantum jump. The theory is applied to the measurement of incompatible variables, giving rise to a stronger entropic uncertainty relation than heretofore known. It is also applied to standard quantum measurement situations such as the Stern-Gerlach and double-slit experiments to illustrate how randomness, inherent in the conventional quantum probabilities, arises in a unitary framework. Finally, the present view clarifies the relationship between classical and quantum concepts.
We derive entropic Bell inequalities from considering entropy Venn diagrams. These entropic inequalities, akin to the Braunstein-Caves inequalities, are violated for a quantum mechanical Einstein-Podolsky-Rosen pair, which implies that the conditional entropies of Bell variables must be negative in this case. This suggests that the satisfaction of entropic Bell inequalities is equivalent to the non-negativity of conditional entropies as a necessary condition for separability.
The amount of information that can be accessed via measurement of a quantum system prepared in different states is limited by the Kholevo bound. We present a simple proof of this theorem and its extension to sequential measurements based on the properties of quantum conditional and mutual entropies. The proof relies on a minimal physical model of the measurement which does not assume environmental decoherence, and has an intuitive diagrammatic representation.
We present a novel quantum Monte Carlo method based on a path integral in Fock space, which allows to compute finite-temperature properties of a many-body nuclear system with a monopole pairing interaction in the canonical ensemble. It enables an exact calculation of the thermodynamic variables such as the internal energy, the entropy, or the specific heat, from the measured moments of the number of hops in a path of nuclear configurations. Monte Carlo calculations for a single-shell $(h_{11/2})^6$ model are consistent with an exact calculation from the many-body spectrum in the seniority model.