The quantum strategy (or quantum combs ) framework is a useful tool for reasoning about interactions among entities that process and exchange quantum information over the course of multiple turns. We prove a time-reversal property for a class of linear functions, defined on quantum strategy representations within this framework, that corresponds to the set of rank-one positive semidefinite operators on a certain space. This time-reversal property states that the maximum value obtained by such a function over all valid quantum strategies is also obtained when the direction of time for the function is reversed, despite the fact that the strategies themselves are generally not time reversible. An application of this fact is an alternative proof of a known relationship between the conditional min- and max-entropy of bipartite quantum states, along with generalizations of this relationship.
The quantum strategy (or quantum combs ) framework is a useful tool for reasoning about interactions among entities that process and exchange quantum information over the course of multiple turns. We prove a time-reversal property for a class of linear functions, defined on quantum strategy representations within this framework, that corresponds to the set of rank-one positive semidefinite operators on a certain space. This time-reversal property states that the maximum value obtained by such a function over all valid quantum strategies is also obtained when the direction of time for the function is reversed, despite the fact that the strategies themselves are generally not time reversible. An application of this fact is an alternative proof of a known relationship between the conditional min- and max-entropy of bipartite quantum states, along with generalizations of this relationship.
We study properties of quantum strategies, which are complete specifications of a given party's actions in any multiple-round interaction involving the exchange of quantum information with one or more other parties. In particular, we focus on a representation of quantum strategies that generalizes the Choi-Jamiołkowski representation of quantum operations. This new representation associates with each strategy a positive semidefinite operator acting only on the tensor product of its input and output spaces. Various facts about such representations are established, and two applications are discussed: the first is a new and conceptually simple proof of Kitaev's lower bound for strong coin-flipping, and the second is a proof of the exact characterization QRG = EXP of the class of problems having quantum refereed games.
This thesis is divided into two parts. In Part I we introduce a new formalism for quantum strategies, which specify the actions of one party in any multi-party interaction involving the exchange of multiple quantum messages among the parties. This formalism associates with each strategy a single positive semidefinite operator acting only upon the tensor product of the input and output message spaces for the strategy. We establish three fundamental properties of this new representation for quantum strategies and we list several applications, including a quantum version of von Neumann's celebrated 1928 Min-Max Theorem for zero-sum games and an efficient algorithm for computing the value of such a game. In Part II we establish several properties of a class of quantum operations that can be implemented locally with shared quantum entanglement or classical randomness. In particular, we establish the existence of a ball of local operations with shared randomness lying within the space spanned by the no-signaling operations and centred at the completely noisy channel. The existence of this ball is employed to prove that the weak membership problem for local operations with shared entanglement is strongly NP-hard. We also provide characterizations of local operations in terms of linear functionals that are positive and "completely" positive on a certain cone of Hermitian operators, under a natural notion of complete positivity appropriate to that cone. We end the thesis with a discussion of the properties of no-signaling quantum operations.
Abstract We study properties of quantum strategies, which are complete specifications of a givenparty's actions in any multiple-round interaction involving the exchange of quantum information with one or more other parties. In particular, we focus on a representation of quantumstrategies that generalizes the Choi-Jamiol/kowski representation of quantum operations. This new representation associates with each strategy a positive semidefinite operator acting onlyon the tensor product of its input and output spaces. Various facts about such representations are established, and two applications are discussed: the first is a new and conceptually simpleproof of Kitaev's lower bound for strong coin-flipping, and the second is a proof of the exact characterization QRG = EXP of the class of problems having quantum refereed games. 1 Introduction The theory of games provides a general structure within which both cooperation and competitionamong independent entities may be modeled, and provides powerful tools for analyzing these models. Applications of this theory have fundamental importance in many areas of science.This paper considers games in which the players may exchange and process quantum information. We focus on competitive games, and within this context the types of games we consider arevery general. For instance, they allow multiple rounds of interaction among the players involved, and place no restrictions on players ' strategies beyond those imposed by the theory of quantuminformation. While classical games can be viewed as a special case of quantum games, it is important tostress that there are fundamental differences between general quantum games and classical games. For example, the two most standard representations of classical games, namely the normal formand extensive form representations, are not directly applicable to general quantum games. This is due to the nature of quantum information, which admits a continuum of pure (meaning extremal)
One aspect of complexity for quantum many-body systems resides in their ability to exhibit entanglement, uniquely quantum correlations central to quantum computation. To quantitatively explore the diminishing effect that quantum noise has on entanglement resources, we focus on the negativity witness, an entanglement measure based on local time reversal transformations that provides a perfect witness in simple physical contexts. Beyond calculation of the negativity, we analyze the associated eigenvalue spectra of partially transposed density matrices for qudit-qudit bipartitions, finding that higher negativity rank provides greater(reduced) robustness to targeted(depolarizing) noise. In particular, we extend the maximum number of separable states that can be arbitrarily mixed into an entangled pure ensemble while retaining negativity to be exactly equal to the original negativity rank. Furthermore, we report how understanding the role of the negativity rank with respect to the completely depolarizing channel allows direct calculation of (1) certain noise boundaries for the Absolutely Positive Partial Transpose (APPT) subspace beyond which no unitary operator is capable of generating negativity and (2) the maximum qutrit-qutrit APPT purity. Beyond proving tight bounds and contributing to the ongoing search for APPT entanglement, these results expand our understanding of the nuanced relationship between computation with entanglement and practical environments for operating quantum devices.
In a unified framework, we estimate the following quantities of interest in quantum information theory: (1) the minimum-error distinguishability of arbitrary ensembles of mixed quantum states; (2) the approximate reversibility of quantum dynamics in terms of entanglement fidelity (This is referred to as “channel-adapted quantum error recovery” when applied to the composition of an encoding operation and a noise channel.); (3) the maximum overlap between a bipartite pure quantum state and a bipartite mixed-state that may be achieved by applying a local quantum operation to one part of the mixed-state; and (4) the conditional min-entropy of bipartite quantum states. A refined version of the author’s techniques [J. Tyson, J. Math. Phys. 50, 032016 (2009)] for bounding the first quantity is employed to give two-sided estimates of the remaining three quantities. We obtain a closed-form approximate reversal channel. Using a state-dependent Kraus decomposition, our reversal may be interpreted as a quadratically weighted version of that of Barnum and Knill [J. Math. Phys. 43, 2097 (2002)]. The relationship between our reversal and Barnum and Knill’s is therefore similar to the relationship between Holevo’s asymptotically optimal measurement [A. S. Kholevo, Theor. Probab. Appl. 23, 411 (1978)] and the “pretty good” measurement of Belavkin [Stochastics 1, 315 (1975)] and Hausladen and Wootters [J. Mod. Opt. 41, 2385 (1994)]. In particular, we obtain relatively simple reversibility estimates without negative matrix-powers at no cost in tightness of our bounds. Our recovery operation is found to significantly outperform the so-called “transpose channel” in the simple case of depolarizing noise acting on half of a maximally entangled state. Furthermore, our overlap results allow the entangled input state and the output target state to differ, thus obtaining estimates in a somewhat more general setting. Using a result of König et al. [IEEE Trans. Inf. Theory 55, 4337 (2009)], our maximum overlap estimate is used to bound the conditional min-entropy of arbitrary bipartite states. Our primary tool is “small angle” initialization of an abstract generalization of the iterative schemes of Ježek et al. [Phys. Rev. A 65, 060301 (2002)], Ježek et al. [Phys. Rev. A 68, 012305 (2003)], and Reimpell and Werner [Phys. Rev. Lett. 94, 080501 (2005)]. The monotonicity result of Reimpell [Ph.D. thesis, Technishe Universität, 2007] follows in greater generality.
The dynamics of an open quantum system can be described by a quantum operation: A linear, complete positive map of operators. Here, I exhibit a compact expression for the time reversal of a quantum operation, which is closely analogous to the time reversal of a classical Markov transition matrix. Since open quantum dynamics are stochastic, and not, in general, deterministic, the time reversal is not, in general, an inversion of the dynamics. Rather, the system relaxes toward equilibrium in both the forward and reverse time directions. The probability of a quantum trajectory and the conjugate, time reversed trajectory are related by the heat exchanged with the environment.