The analysis of classical consensus algorithms relies on contraction properties of Markov matrices with respect to the Hilbert semi-norm (infinitesimal version of Hilbert's projective metric) and to the total variation norm. We generalize these properties to the case of operators on cones. This is motivated by the study of “non-commutative consensus”, i.e., of the dynamics of linear maps leaving invariant cones of positive semi-definite matrices. Such maps appear in quantum information (Kraus maps), and in the study of matrix means. We give a characterization of the contraction rate of an abstract Markov operator on a cone, which extends classical formulæ obtained by Dœblin and Dobrushin in the case of Markov matrices. In the special case of Kraus maps, we relate the absence of contraction to the positivity of the “zero-error capacity” of a quantum channel. We finally show that a number of decision problems concerning the contraction rate of Kraus maps reduce to finding a rank one matrix in linear spaces satisfying certain conditions and discuss complexity issues.
The analysis of classical consensus algorithms relies on contraction properties of Markov matrices with respect to the Hilbert semi-norm (infinitesimal version of Hilbert's projective metric) and to the total variation norm. We generalize these properties to the case of operators on cones. This is motivated by the study of “non-commutative consensus”, i.e., of the dynamics of linear maps leaving invariant cones of positive semi-definite matrices. Such maps appear in quantum information (Kraus maps), and in the study of matrix means. We give a characterization of the contraction rate of an abstract Markov operator on a cone, which extends classical formulæ obtained by Dœblin and Dobrushin in the case of Markov matrices. In the special case of Kraus maps, we relate the absence of contraction to the positivity of the “zero-error capacity” of a quantum channel. We finally show that a number of decision problems concerning the contraction rate of Kraus maps reduce to finding a rank one matrix in linear spaces satisfying certain conditions and discuss complexity issues.
We introduce and apply Hilbert's projective metric in the context of quantum information theory. The metric is induced by convex cones such as the sets of positive, separable or positive partial transpose operators. It provides bounds on measures for statistical distinguishability of quantum states and on the decrease of entanglement under protocols involving local quantum operations and classical communication or under other cone-preserving operations. The results are formulated in terms of general cones and base norms and lead to contractivity bounds for quantum channels, for instance, improving Ruskai's trace-norm contraction inequality. A new duality between distinguishability measures and base norms is provided. For two given pairs of quantum states we show that the contraction of Hilbert's projective metric is necessary and sufficient for the existence of a probabilistic quantum operation that maps one pair onto the other. Inequalities between Hilbert's projective metric and the Chernoff bound, the fidelity and various norms are proven.
There are various notions of positivity for matrices and linear matrix-valued maps that play important roles in quantum information theory. The cones of positive semidefinite matrices and completely positive linear maps, which represent quantum states and quantum channels respectively, are the most ubiquitous positive cones. There are also many natural cones that can been regarded as "more" or "less" positive than these standard examples. In particular, entanglement theory deals with the cones of separable operators and entanglement witnesses, which satisfy very strong and weak positivity properties respectively. Rather complementary to the various cones that arise in entanglement theory are norms. The trace and operator norms for operators and the diamond and completely bounded norms for superoperators are the typical norms that are seen throughout quantum information theory. In this work our main goal is to develop a family of norms that play a role analogous to the cone of entanglement witnesses. We investigate the basic mathematical properties of these norms, including their relationships with other well-known norms, their isometry groups, and their dual norms. We also make the place of these norms in entanglement theory rigorous by showing that entanglement witnesses arise from minimal operator systems, and analogously our norms arise from minimal operator spaces. Finally, we connect the various cones and norms considered here to several seemingly unrelated problems from other areas. We characterize the problem of whether or not non-positive partial transpose bound entangled states exist in terms of one of our norms, and provide evidence in favour of their existence. We also characterize the minimum gate fidelity of a quantum channel, the maximum output purity and its completely bounded counterpart, and the geometric measure of entanglement in terms of these norms.
This thesis is the fruit of investigations on the extension of ideas of Markov chain mixing to the quantum setting, and its application to problems of dissipative engineering. A Markov chain describes a statistical process where the probability of future events depends only on the state of the system at the present point in time, but not on the history of events. Very many important processes in nature are of this type, therefore a good understanding of their behavior has turned out to be very fruitful for science. Markov chains always have a non-empty set of limiting distributions (stationary states). The aim of Markov chain mixing is to obtain (upper and/or lower) bounds on the number of steps it takes for the Markov chain to reach a stationary state. The natural quantum extensions of these notions are density matrices and quantum channels. We set out to develop a general mathematical framework for studying quantum Markov chain mixing. We introduce two new distance measures into the quantum setting; the quantum χ2divergence and Hilbert’s projective metric. The quantum χ2 divergence allows us to extend the class of functional techniques called Poincare inequalities to the quantum setting. In the process, we identify the appropriate framework for discussing notions such as detailed balance and χ2-mixing: monotone Riemannian metrics on matrix spaces. This insight allows us to characterize the connection between spectral properties of a (primitive) quantum channel and its mixing time. Within the same framework we also derive a restricted quantum version of the celebrated conductance bound (Cheeger’s inequality). We then consider Hilbert’s projective metric a well known tool in Perron-Frobenius theory in the context of quantum information theory. As a classic tool for analyzing convergence of positive maps on cones, Hilbert’s projective metric is especially good at providing existence proofs of maps between two spaces (or cones). We relate this measure to distinguishability measures on restricted cones of operators, and analyze the observable loss of information after the application of a quantum channel under a restrict set of measurements. Various notions of contractivity of quantum channels are revealed through Hilbert’s metric. Still on the topic of quantum Markov chains, we introduce the notion of cutoff phenomenon to the quantum setting. The cutoff phenomenon describes the situation when a Markov chain does not converge for a potentially long time, and then at a specific point in time abruptly converges to equilibrium. In the thermodynamic limit, the convergence profile will then look like a step function. We show that this type of convergence behavior occurs, as measured in trace-norm, when a system is subject to a product channel, and the noise on each product is modeled by a one-parameter semigroup of quantum channels. Finally, we consider three independent tasks of dissipative engineering. The first task, which is aimed at experimental realization in the near future, consists in dissipatively
Abstract The max-relative entropy and the conditional min-entropy of a quantum state plays a central role in one-shot and zero-error quantum information theory. One attractive feature of this quantity is that it can be expressed as an optimization over the cone of positive semidefinite operators. Recently, it was shown that when replacing this cone with the cone of separable operators, a new type of conditional min-entropy emerges that admits an operational interpretation in terms of communicating classical information over a quantum channel. In this work, we explore more deeply the idea of building information-theoretic quantities from different base cones and determine which results in quantum information theory rely upon the positive semidefinite cone and which can be generalized. In terms of asymptotic information processing, we find that the standard equipartition properties break down if a given cone fails to approximate the positive semidefinite cone sufficiently well. We also show that the near-equivalence of the smooth max and Hartley entropies breaks down in this setting. We present parallel results for the extended conditional min-entropy, which requires extending the notion of k-superpositive channels to superchannels. On the other hand, we show that for classical-quantum states the separable cone is sufficient to re-cover the asymptotic theory, thereby drawing a strong distinction between the fully and partial quantum settings. We also present operational uses of this framework. We show that the cone restricted min-entropy of a Choi operator captures a measure of entanglement-assisted noiseless classical communication using restricted measurements. We also introduce a novel min-entropy-like quantity that captures the conditions for when one quantum channel can be transformed into another using bistochastic pre-processing. Lastly, we relate this framework to general conic norms and their non-additivity. Throughout this work, we concretely study generalized entropies in resource theories that capture locality and resource theories of coherence/Abelian symmetries.
Drawing on results of Choi, Størmer and Woronowicz, we present a nearly complete characterization of certain important classes of positive maps. In particular, we construct a general class of positive linear maps acting between two matrix algebras ℬ( ) and ℬ( ), where and are finite-dimensional Hilbert spaces. It turns out that elements of this class are characterized by operators from the dual cone of the set of all separable states on ℬ( ⊗ ). Subsequently, the relation between entanglements and positive maps is described. Finally, a new characterization of the cone ℬ( ) + ⊗ℬ( ) + is given.
The classical Schrödinger bridge seeks the most likely probability law for a diffusion process, in path space, that matches marginals at two end points in time; the likelihood is quantified by the relative entropy between the sought law and a prior. Jamison proved that the new law is obtained through a multiplicative functional transformation of the prior. This transformation is characterised by an automorphism on the space of endpoints probability measures, which has been studied by Fortet, Beurling, and others. A similar question can be raised for processes evolving in a discrete time and space as well as for processes defined over non-commutative probability spaces. The present paper builds on earlier work by Pavon and Ticozzi and begins by establishing solutions to Schrödinger systems for Markov chains. Our approach is based on the Hilbert metric and shows that the solution to the Schrödinger bridge is provided by the fixed point of a contractive map. We approach, in a similar manner, the steering of a quantum system across a quantum channel. We are able to establish existence of quantum transitions that are multiplicative functional transformations of a given Kraus map for the cases where the marginals are either uniform or pure states. As in the Markov chain case, and for uniform density matrices, the solution of the quantum bridge can be constructed from the fixed point of a certain contractive map. For arbitrary marginal densities, extensive numerical simulations indicate that iteration of a similar map leads to fixed points from which we can construct a quantum bridge. For this general case, however, a proof of convergence remains elusive.