Advances in Mathematical Physics

Volume 2018, Article ID 7060586, 8 pages

https://doi.org/10.1155/2018/7060586

## The Quantum Cheshire Cat Effect in the Presence of Decoherence

^{1}Institute of Physics, University of Silesia in Katowice, Katowice, Poland^{2}Silesian Center for Education and Interdisciplinary Research, University of Silesia in Katowice, Chorzów, Poland

Correspondence should be addressed to Jerzy Dajka; lp.ude.su@akjad.yzrej

Received 10 November 2017; Revised 12 March 2018; Accepted 28 March 2018; Published 9 May 2018

Academic Editor: Stephen C. Anco

Copyright © 2018 Monika Richter et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Even the subtle and apparently strange quantum effects can sometimes survive otherwise lethal influence of an omnipresent decoherence. We show that an archetypal quantum Cheshire Cat, a paradox of a separation between a position of a quantum particle, a photon, and its internal property, the polarization, in a two-path Mach–Zehnder setting, is robust to decoherence caused by a bosonic infinite bath locally coupled to the polarization of a photon. Decoherence affects either the cat or its grin depending on which of the two paths is noisy. For a pure decoherence, in an absence of photon–environment energy exchange, we provide exact results for weak values of the photon position and polarization indicating that the information loss affects the quantum Cheshire Cat only qualitatively and the paradox survives. We show that it is also the case beyond the pure decoherence for a small rate of dissipation.

#### 1. Introduction

Counterintuitive world of quanta is a stage of various phenomena which are sometimes classified as paradoxes [1]. Their paradoxial character very often originates from a difference between mathematical modelling required for macro and micro scale or from a very special character and role of the quantum measurement. For a long time it has been expected that because of an omnipresent decoherence being (one of possible) mechanisms for a quantum-to-classical passage [2] quantum paradoxes are hardly present in reality. Nowadays we know that it is not the case; many applications of entanglement can serve as spectacular examples of observing and utilizing essentially quantum properties. This is why it is so important for a given “paradox” to answer a natural question: does it remain “paradoxial” also in the presence of decoherence?

In this work we focus on the quantum Cheshire Cat, an effect recently added to a list of quantum “paradoxes” [3], which has attracted considerable interest of both theoretical and experimental physicists [4–6]. This is a paradox of separation of two properties of a quantum particle named in an analogy to the behaviour of the Cheshire Cat and its grin, a character in the novel* Alice’s Adventures in Wonderland* by Lewis Carroll. Alice, who (before she fell down in the Rabbit Hole) “has often seen a cat without a grin but never a grin without a cat,” would have been surprised seeing a photon, a cat, separated from its polarization, a grin.

The archetype of the quantum Cheshire Cat paradox proposed in [3] is a two-path Mach–Zehnder-like setting for a photon with an internal degree of freedom, the polarization. A state space of the considered system is a -dimensional Hilbert space , where and label the path “chosen” by the photon (either left or right ) and its polarization (horizontal or vertical ), respectively. The photon is prepared* (preselected)* in a stateand then detected* (postselected)* in a stateIdentification of the photon in one of the possible paths (arms) corresponds to a measurement related to the projectors:whereas a measurement of its polarization in a given (either left or right) arm requires the projectorswhere for . As detection of the Cheshire Cat requires simultaneous measurement for both paths of the photon, one applies there the* weak measurement* scheme [3, 7]. Both the interpretations and the broad possible applications of the quantum weak values are presented in [7]. Here, following [7], we simply define the th order weak value related to an observable asThere is a natural interpretation of the weak value if one considers unitary transformation generated by an observable . The quantity is a related transition amplitude from the preselected initial state into the final which can be then postselected. For sufficiently small one can Taylor-expand the amplitude with the expansion terms containing the weak values in (5) up to some finite order related to the magnitude of .

In the weak interaction regime, one can neglect higher order terms; that is, one can limit to . As the most of the experiments performed so far operate in this regime, further we limit to the first-order case and utilize the basic interpretation of as a change of the detection probability in the presence of (weak) interaction generated by . Let us remember that generically the weak value of an observable is a complex, yet measurable [7], quantity. For the quantum Cheshire Cat experiment, with given pre- (see (1)) and postselected (see (2)) states, one obtains [3]for “the cat” andfor its “grin,” respectively. According to the interpretation proposed in [3] and originating from* Alice in Wonderland* one obtains the cat residing in the left path separated from its grin which appears in the right path. Let us emphasize that our primary aim is to present that the Cheshire Cat effect, in its archetype formulated using quantum weak values, survives in the presence of decoherence of a certain, relatively general, type. Discussion of controversies concerning this effect such as those reported in [4, 8, 9] or attempts to extend our results to different treatments or formalism is beyond the assumed scope of our work.

The paper is organized as follows: (i) we present qualitative considerations showing how the predictions of [3] become modified by decoherence caused by an environment locally coupled to a photonic polarization. (ii) Further we exemplify our model and apply it to a simplest case of* pure decoherence* when one can neglect a photon–environment energy exchange. (iii) Our next step is to include dissipation. We are going to allow for a weak energy transfer between our system and its environment.

#### 2. The Cat, Its Grin, and the Noise

In this section, we discuss a general model of the quantum Cheshire Cat in the presence of decoherence caused by an environment coupled* locally* to the polarization of the photon. To formalize our discussion, we expand the state space into the triple , where the last term corresponds to an environment which needs to be included and which is going to be specified in the next section. We limit our attention to the simplest setting of* a noisy preselection* and we assume that the environment couples to photonic polarization only locally, that is, only in one of the two arms of the interferometer corresponding to either the sector or of the state space . We also assume that initially, prior to any interaction, the environment is in a pure state . In such a case there are two possible “noisy” preselected states: the first, where the polarization is affected by in a right pathand the second, for affecting polarization in the left path,The time -parameterized family of states results from the polarization–environment interaction after a time . A unitary operator , such that (the identity operator), describes an interaction between polarization and the environment. We assume that the postselection is not affected by the presence of the environment; that is,However, the weak values which are a figure of merit for the Cheshire Cat paradox become modified by decoherenceThe normalization factors read as follows:The weak value in (11) quantifies a weakly measured quantity of a physical system* coupled to* an environment. For the preselection given in (8) (indicated by the superscript below) the weak value of “the cat” position iswhereas for in (4) “the grin” isLet us notice that for the preselection in (8) decoherence affects the cat, originally residing in the (noiseless) -sector whereas the grin is solely confined to the -sector of the system. It is in an apparent contrast to what occurs for the preselection in (9). In that case (indicated by the superscript below), corresponding to a noisy -sector, originally occupied by the cat, the cat’s position remains confined to the -sector of the systemwhereas the grin becomes wiped off by decoherencethat is, it appears in both - and -sectors of the system. One can say that for the preselection in (8) decoherence attracts the cat whereas for the preselection in (9) decoherence attracts the grin.

The original quantum Cheshire Cat originates from a very peculiar asymmetry between initial preparation (preselection) of the system having internal degree of freedom and its postselection via a very specific measurement scheme. In our considerations, instead of studying more realistic schemes presented, for example, in [6], we limit our considerations to the archetype proposal schematically presented in Figure 1 of [3]. However, let us notice that an effect of decoherence in (9) and (8) is fully general as it incorporates a well established system–environment modelling [10] with a unitary evolution encoding both the time evolution of the system and the bath and its interaction which is going to be specified in the following part of the paper. The only thing assumed so far is (i) that the environment couples to the internal degree of freedom (the polarization) only and (ii) that the environment is* local*; that is, it is present in one of two arms of the interferometer only. Locality of the decoherence is justified since there is a spatial separation between photonic paths in the interferometer in particular if one limits to the polarization–environment interaction [11, 12].