Compressed sensing performance bounds under Poisson noise

This paper describes performance bounds for compressed sensing (CS) where the underlying sparse or compressible (sparsely approximable) signal is a vector of nonnegative intensities whose measurements are corrupted by Poisson noise. In this setting, standard CS techniques cannot be applied directly for several reasons. First, the usual signal-independent and/or bounded noise models do not apply to Poisson noise, which is non-additive and signal-dependent. Second, the CS matrices typically considered are not feasible in real optical systems because they do not adhere to important constraints, such as nonnegativity and photon flux preservation. Third, the typical $\ell_2$--$\ell_1$ minimization leads to overfitting in the high-intensity regions and oversmoothing in the low-intensity areas. In this paper, we describe how a feasible positivity- and flux-preserving sensing matrix can be constructed, and then analyze the performance of a CS reconstruction approach for Poisson data that minimizes an objective function consisting of a negative Poisson log likelihood term and a penalty term which measures signal sparsity. We show that, as the overall intensity of the underlying signal increases, an upper bound on the reconstruction error decays at an appropriate rate (depending on the compressibility of the signal), but that for a fixed signal intensity, the signal-dependent part of the error bound actually grows with the number of measurements or sensors. This surprising fact is both proved theoretically and justified based on physical intuition.Comment: 12 pages, 3 pdf figures; accepted for publication in IEEE Transactions on Signal Processin

Operational distance and fidelity for quantum channels

We define and study a fidelity criterion for quantum channels, which we term the minimax fidelity, through a noncommutative generalization of maximal Hellinger distance between two positive kernels in classical probability theory. Like other known fidelities for quantum channels, the minimax fidelity is well-defined for channels between finite-dimensional algebras, but it also applies to a certain class of channels between infinite-dimensional algebras (explicitly, those channels that possess an operator-valued Radon--Nikodym density with respect to the trace in the sense of Belavkin--Staszewski) and induces a metric on the set of quantum channels which is topologically equivalent to the CB-norm distance between channels, precisely in the same way as the Bures metric on the density operators associated with statistical states of quantum-mechanical systems, derived from the well-known fidelity (`generalized transition probability') of Uhlmann, is topologically equivalent to the trace-norm distance.Comment: 26 pages, amsart.cls; improved intro, fixed typos, added a reference; accepted by J. Math. Phy

Entropy production rates of bistochastic strictly contractive quantum channels on a matrix algebra

We derive, for a bistochastic strictly contractive quantum channel on a matrix algebra, a relation between the contraction rate and the rate of entropy production. We also sketch some applications of our result to the statistical physics of irreversible processes and to quantum information processing.Comment: 7 pages; revised version submitted to J. Phys.

Radon-Nikodym derivatives of quantum operations

Given a completely positive (CP) map $T$, there is a theorem of the Radon-Nikodym type [W.B. Arveson, Acta Math. {\bf 123}, 141 (1969); V.P. Belavkin and P. Staszewski, Rep. Math. Phys. {\bf 24}, 49 (1986)] that completely characterizes all CP maps $S$ such that $T-S$ is also a CP map. This theorem is reviewed, and several alternative formulations are given along the way. We then use the Radon-Nikodym formalism to study the structure of order intervals of quantum operations, as well as a certain one-to-one correspondence between CP maps and positive operators, already fruitfully exploited in many quantum information-theoretic treatments. We also comment on how the Radon-Nikodym theorem can be used to derive norm estimates for differences of CP maps in general, and of quantum operations in particular.Comment: 22 pages; final versio

Strongly Incompatible Quantum Devices

The fact that there are quantum observables without a simultaneous measurement is one of the fundamental characteristics of quantum mechanics. In this work we expand the concept of joint measurability to all kinds of possible measurement devices, and we call this relation compatibility. Two devices are incompatible if they cannot be implemented as parts of a single measurement setup. We introduce also a more stringent notion of incompatibility, strong incompatibility. Both incompatibility and strong incompatibility are rigorously characterized and their difference is demonstrated by examples.Comment: 27 pages (AMSart), 6 figure

Strictly contractive quantum channels and physically realizable quantum computers

We study the robustness of quantum computers under the influence of errors modelled by strictly contractive channels. A channel $T$ is defined to be strictly contractive if, for any pair of density operators $\rho,\sigma$ in its domain, $\| T\rho - T\sigma \|_1 \le k \| \rho-\sigma \|_1$ for some $0 \le k < 1$ (here $\| \cdot \|_1$ denotes the trace norm). In other words, strictly contractive channels render the states of the computer less distinguishable in the sense of quantum detection theory. Starting from the premise that all experimental procedures can be carried out with finite precision, we argue that there exists a physically meaningful connection between strictly contractive channels and errors in physically realizable quantum computers. We show that, in the absence of error correction, sensitivity of quantum memories and computers to strictly contractive errors grows exponentially with storage time and computation time respectively, and depends only on the constant $k$ and the measurement precision. We prove that strict contractivity rules out the possibility of perfect error correction, and give an argument that approximate error correction, which covers previous work on fault-tolerant quantum computation as a special case, is possible.Comment: 14 pages; revtex, amsfonts, amssymb; made some changes (recommended by Phys. Rev. A), updated the reference

Generation and manipulation of squeezed states of light in optical networks for quantum communication and computation

We analyze a fiber-optic component which could find multiple uses in novel information-processing systems utilizing squeezed states of light. Our approach is based on the phenomenon of photon-number squeezing of soliton noise after the soliton has propagated through a nonlinear optical fiber. Applications of this component in optical networks for quantum computation and quantum cryptography are discussed.Comment: 12 pages, 2 figures; submitted to Journal of Optics

Minimum error discrimination of Pauli channels

We solve the problem of discriminating with minimum error probability two given Pauli channels. We show that, differently from the case of discrimination between unitary transformations, the use of entanglement with an ancillary system can strictly improve the discrimination, and any maximally entangled state allows to achieve the optimal discrimination. We also provide a simple necessary and sufficient condition in terms of the structure of the channels for which the ultimate minimum error probability can be achieved without entanglement assistance. When such a condition is satisfied, the optimal input state is simply an eigenstate of one of the Pauli matrices.Comment: 8 pages, no figure

The Generalized Lyapunov Theorem and its Application to Quantum Channels

We give a simple and physically intuitive necessary and sufficient condition for a map acting on a compact metric space to be mixing (i.e. infinitely many applications of the map transfer any input into a fixed convergency point). This is a generalization of the "Lyapunov direct method". First we prove this theorem in topological spaces and for arbitrary continuous maps. Finally we apply our theorem to maps which are relevant in Open Quantum Systems and Quantum Information, namely Quantum Channels. In this context we also discuss the relations between mixing and ergodicity (i.e. the property that there exist only a single input state which is left invariant by a single application of the map) showing that the two are equivalent when the invariant point of the ergodic map is pure.Comment: 13 pages, 3 figure

Nested Invariance Pooling and RBM Hashing for Image Instance Retrieval

The goal of this work is the computation of very compact binary hashes for image instance retrieval. Our approach has two novel contributions. The first one is Nested Invariance Pooling (NIP), a method inspired from i-theory, a mathematical theory for computing group invariant transformations with feed-forward neural networks. NIP is able to produce compact and well-performing descriptors with visual representations extracted from convolutional neural networks. We specifically incorporate scale, translation and rotation invariances but the scheme can be extended to any arbitrary sets of transformations. We also show that using moments of increasing order throughout nesting is important. The NIP descriptors are then hashed to the target code size (32-256 bits) with a Restricted Boltzmann Machine with a novel batch-level reg-ularization scheme specifically designed for the purpose of hashing (RBMH). A thorough empirical evaluation with state-of-the-art shows that the results obtained both with the NIP descriptors and the NIP+RBMH hashes are consistently outstanding across a wide range of datasets