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Frontiers of Physics

Temporal inequalities for sequential multi-time actions in quantum information processing

Marek Żukowski

Published Nov 26, 2013• 18 citations
Publisher

Abstract

A new kind of temporal inequalities are discussed, which apply to algorithmic processes, involving a finite memory processing unit. They are an alternative to the Leggett-Grag ones, as well as to the modified ones by Brukner et al . If one considers comparison of quantum and classical processes involving systems of finite memory (of the same capacity in both cases), the inequalities give a clear message why we can expect quantum speed-up. In a classical process one always has clearly defined values of possible measurements, or in terms of the information processing language, if we have a sequential computations of some function depending on data arriving at each step on an algorithm, the function always has a clearly defined value. In the quantum case only the final value, after the end of the algorithm, is defined. All intermediate values, in agreement with Bohr’s complementarity, cannot be ascribed a definite value.

DOI
10.1007/s11467-013-0400-2

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A new kind of temporal inequalities are discussed, which apply to algorithmic processes, involving a finite memory processing unit. They are an alternative to the Leggett-Grag ones, as well as to the modified ones by Brukner et al . If one considers comparison of quantum and classical processes involving systems of finite memory (of the same capacity in both cases), the inequalities give a clear message why we can expect quantum speed-up. In a classical process one always has clearly defined values of possible measurements, or in terms of the information processing language, if we have a sequential computations of some function depending on data arriving at each step on an algorithm, the function always has a clearly defined value. In the quantum case only the final value, after the end of the algorithm, is defined. All intermediate values, in agreement with Bohr’s complementarity, cannot be ascribed a definite value.
The usual formulation of Macrorealism is recast to make this notion fully concurrent with the basic ideas behind classical physics. The assumption of non-invasiveness of measurements is dropped. Instead, it is assumed that the current state of the system defines full initial conditions for its subsequent evolution. An example of a new family of temporal Bell inequalities is derived which can be applied to processes in which the state of the system undergoes arbitrarily many transformations (which was not the case in the original approach). An exponential (in terms of number of operations) violation of this inequality is demonstrated theoretically. Finally it is shown that such inequalities were indirectly tested in a 2005 experiment by the Weinfurter group.
时间的不平等的一种新类型被讨论，它适用于算法的进程，包含处理联合起来的有限记忆。他们是 Leggett-Grag，以及到由 Brukner 等的修改的一种选择。如果一个人考虑包含有限存储器的系统的量和古典进程的比较(在两个盒子中的一样的能力) ，不平等给我们能期望量加速的一条清楚的消息。在古典进程，一个人清楚地总是定义可能的大小的值，或以信息处理语言如果我们有一取决于到达各个的数据的某功能的顺序的计算走在一个算法上，函数总是有清楚地定义的值。在算法的结束以后，在量盒子中，仅仅最后的价值被定义。与 Bohrs 补充一致，所有中间的价值不能被归功于明确的价值。
All mental representations change with time. A baseline intuition is that mental representations have specific values at different time points, which may be more or less accessible, depending on noise, forgetting processes, etc. We present a radical alternative, motivated by recent research using the mathematics from quantum theory for cognitive modelling. Such cognitive models raise the possibility that certain possibilities or events may be incompatible, so that perfect knowledge of one necessitates uncertainty for the others. In the context of time-dependence, in physics, this issue is explored with the so-called temporal Bell (TB) or Leggett-Garg inequalities. We consider in detail the theoretical and empirical challenges involved in exploring the TB inequalities in the context of cognitive systems. One interesting conclusion is that we believe the study of the TB inequalities to be empirically more constrained in psychology than in physics. Specifically, we show how the TB inequalities, as applied to cognitive systems, can be derived from two simple assumptions: cognitive realism and cognitive completeness. We discuss possible implications of putative violations of the TB inequalities for cognitive models and our understanding of time in cognition in general. Overall, this paper provides a surprising, novel direction in relation to how time should be conceptualized in cognition.
We investigate the role of nonclassical temporal correlations in enhancing the performance of ticking clocks in a discrete-time scenario. We show that the problem of optimal models for ticking clocks is related to the violation of Leggett-Garg-type temporal inequalities formulated in terms of, possibly invasive, sequential measurements, but on a system with a bounded memory capacity. Ticking clocks inspire the derivation of a family of temporal inequalities showing a gap between classical and quantum correlations, despite involving no input. We show that quantum ticking-clock models achieving accuracy beyond the classical bound are also those violating Leggett-Garg-type temporal inequalities for finite sequences and we investigate their continuous-time limit. Interestingly, we show that optimal classical clock models in the discrete-time scenario do not have a well-defined continuous-time limit, a feature that is absent in quantum models.
We investigate the role of nonclassical temporal correlations in enhancing the performance of ticking clocks in a discrete-time scenario. We show that the problem of optimal models for ticking clocks is related to the violation of Leggett-Garg-type temporal inequalities formulated in terms of, possibly invasive, sequential measurements, but on a system with a bounded memory capacity. Ticking clocks inspire the derivation of a family of temporal inequalities showing a gap between classical and quantum correlations, despite involving no input. We show that quantum ticking-clock models achieving accuracy beyond the classical bound are also those violating Leggett-Garg-type temporal inequalities for finite sequences and we investigate their continuous-time limit. Interestingly, we show that optimal classical clock models in the discrete-time scenario do not have a well-defined continuous-time limit, a feature that is absent in quantum models.
Understanding temporal processes and their correlations in time is of paramount importance for the development of near-term technologies that operate under realistic conditions. Capturing the complete multi-time statistics that define a stochastic process lies at the heart of any proper treatment of memory effects. This is well understood in classical theory, where a hierarchy of joint probability distributions completely characterizes the process at hand. However, attempting to generalize this notion to quantum mechanics is problematic: observing realizations of a quantum process necessarily disturbs the state of the system, breaking an implicit, and crucial, assumption in the classical setting. This issue can be overcome by separating the experimental interventions from the underlying process, enabling an unambiguous description of the process itself and accounting for all possible multi-time correlations for any choice of interrogating instruments. In this paper, using a novel framework for the characterization of quantum stochastic processes, we first solve the long standing question of unambiguously describing the memory length of a quantum processes. This is achieved by constructing a quantum Markov order condition, which naturally generalizes its classical counterpart for the quantification of finite-length memory effects. As measurements are inherently invasive in quantum mechanics, one has no choice but to define Markov order with respect to the interrogating instruments that are used to probe the process at hand: different memory effects are exhibited depending on how one addresses the system, in contrast to the standard classical setting. We then fully characterize the structural constraints imposed on quantum processes with finite Markov order, shedding light on a variety of memory effects that can arise through various examples. Finally, we introduce an instrument-specific notion of memory strength that allows for a meaningful quantification of the temporal correlations between the history and the future of a process for a given choice of experimental intervention. These findings are directly relevant to both characterizing and exploiting memory effects that persist for a finite duration. In particular, immediate applications range from developing efficient compression and recovery schemes for the description of quantum processes with memory to designing coherent control protocols that efficiently perform information-theoretic tasks, amongst a plethora of others.