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arXiv:1809.04999

Relativity of quantum superpositions

Magdalena Zych, Fabio Costa, Timothy C. Ralph

Published Sep 13, 2018• 14 citations
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Abstract

In modern physics only relative quantities are considered to have physical significance. For example, position assigned to a system depends on the choice of coordinates, and only relative distances between different systems have physical relevance. On the other hand, in quantum theory the scenario where one system, A, is localised around certain position while another system B is in a spatial superposition is considered to be physically different from the scenario where A is in a superposition, while B is localised. Different physical effects are anticipated in the two scenarios especially when the two systems have widely different masses. Here we show that for any superposed amplitudes that are related by a symmetry transformation, the above scenarios are physically equivalent: probabilities for any measurement are the same whether we assign the superposition state to system A or to system B. More generally, linearity of quantum theory guarantees that if a theory is invariant under some symmetry transformations it is necessarily also invariant under their arbitrary `superpositions'. Thus the notion of a superposition turns out to be relative to the choice of coordinates, once it is realised that relations between coordinates do not need to be classical.

DOI
10.48550/arxiv.1809.04999

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In modern physics only relative quantities are considered to have physical significance. For example, position assigned to a system depends on the choice of coordinates, and only relative distances between different systems have physical relevance. On the other hand, in quantum theory the scenario where one system, A, is localised around certain position while another system B is in a spatial superposition is considered to be physically different from the scenario where A is in a superposition, while B is localised. Different physical effects are anticipated in the two scenarios especially when the two systems have widely different masses. Here we show that for any superposed amplitudes that are related by a symmetry transformation, the above scenarios are physically equivalent: probabilities for any measurement are the same whether we assign the superposition state to system A or to system B. More generally, linearity of quantum theory guarantees that if a theory is invariant under some symmetry transformations it is necessarily also invariant under their arbitrary `superpositions'. Thus the notion of a superposition turns out to be relative to the choice of coordinates, once it is realised that relations between coordinates do not need to be classical.
Under the principle that quantum mechanical observables are invariant under relevant symmetry transformations, we explore how the usual, non-invariant quantities may capture measurement statistics. Using a relativisation mapping, viewed as the incorporation of a quantum reference frame, we show that the usual quantum description approximates the relative one precisely when the reference system admits an appropriate localisable quantity and a localised state. From this follows a new perspective on the nature and reality of quantum superpositions and optical coherence.
Under the principle that quantum-mechanical observables are invariant under relevant symmetry transformations, we explore how the usual, non-invariant quantities may capture measurement statistics. Using a relativisation mapping, viewed as the incorporation of a quantum reference frame, we show that the usual quantum description approximates the relative one precisely when the reference system admits an appropriate localisable quantity and a localised state. From this follows a new perspective on the nature and reality of quantum superpositions and optical coherence.
Decoherence may not solve all of the measurement problems of quantum mechanics. It is proposed that a solution to these problems may be to allow that superpositions describe physically real systems in the following sense. Each quantum system "carries" around a local spacetime in whose terms other quantum systems may take on nonlocal states. Each quantum system forms a physically valid coordinate frame. The laws of physics should be formulated to be invariant under the group of allowed transformations among such frames. A transformation of relatively superposed spatial coordinates that allows an electron system to preserve the de Broglie Relation in describing a double-slit laboratory system-in analogy to a Minkowskian Transformation-is proposed. In general, "quantum relativity" says is invariant under transformations among quantum reference frames. Some conjectures on how this impacts gravity and gauge invariance are made.
The study of quantum reference frames (QRFs) is motivated by the idea of taking into account the quantum properties of the reference frames used, explicitly or implicitly, in our description of physical systems. Like classical reference frames, QRFs can be used to define physical quantities relationally. Unlike their classical analogue, they relativise the notions of superposition and entanglement. Here, we explain this feature by examining how configurations or locations are identified across different branches in superposition. We show that, in the presence of symmetries, whether a system is in "the same" or "different" configurations across the branches depends on the choice of QRF. Hence, sameness and difference - and thus superposition and entanglement - lose their absolute meaning. We apply these ideas to the context of semi-classical spacetimes in superposition and use coincidences of four scalar fields to construct a comparison map between spacetime points in the different branches. This reveals that the localisation of an event is frame-dependent. We discuss the implications for indefinite causal order and the locality of interaction and conclude with a generalisation of Einstein's hole argument to the quantum context.
In physics, every observation is made with respect to a frame of reference. Although reference frames are usually not considered as degrees of freedom, in all practical situations it is a physical system which constitutes a reference frame. Can a quantum system be considered as a reference frame and, if so, which description would it give of the world? Here, we introduce a general method to quantise reference frame transformations, which generalises the usual reference frame transformation to a "superposition of coordinate transformations". We describe states, measurement, and dynamical evolution in different quantum reference frames, without appealing to an external, absolute reference frame, and find that entanglement and superposition are frame-dependent features. The transformation also leads to a generalisation of the notion of covariance of dynamical physical laws, to an extension of the weak equivalence principle, and to the possibility of defining the rest frame of a quantum system.
One of the most profound and difficult problems of physics in the last century is the construction of a simple theory which unites the ideas of quantum mechanics and relativity. The concepts which are used in quantum mechanics related to the measurements of positions, momenta, energy and time, are the same concepts in terms of which the special relativistic postulates are formulated. Hence, it is at least possible to formulate the requirement of special relativistic invariance for quantum theories and to ascertain whether these requirements are satisfied. The fact that the answer is nearer to no than to yes, that quantum mechanics has not yet been fully adjusted to the postulates of the special theory, is perhaps irritating. In the general theory of relativity, the situation is different: the space-time coordinates are only auxiliary quantities to which arbitrary values for every event could be given. Hence, the measurement of position, that is, of the space coordin ates, is certainly not a significant measurement if the postulates of the general theory are adopted: any arbitrary value can be given to the coordinates. The same holds for momenta. The purpose of this brief paper is to make in principle some personal and particular remarks on the fundamental concepts of time and mass. Some comments about the consequences on the relativistic dynamics of quantum systems are also formulated.