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A Relational Approach to Quantum Reference Frames for Spins

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A Relational Approach to Quantum Reference Frames for Spins A relational approach to quantum reference frames for spins Jacques Pienaar In the literature on quantum reference frames, the internal (relative) properties of a system are defined as those which are preserved under an arbitrary change of reference frame. For a system of quantum spins, these are all properties preserved by proper spatial rotations of the laboratory. However, this approach does not account for the hypothetical possibility of the laboratory becoming entangled to the system, as described by a second laboratory (the ‘Wigner’s friend’ scenario), in which case the relationship between the two laboratories is not a rotation, but is fundamentally quantum. To overcome this limitation, we re-define the reference frame transformations to be those that preserve the fidelities between subsystems. This enables us to derive U(2) as the correct symmetry group for transformations of a system of N spin-half particles. Next, we propose that systems having the same internal properties should be regarded as physically equivalent in the absence of an external frame. Remarkably, this implies that a single spin in a superposition relative to a spin magnet is equivalent to a macroscopic superposition of the magnet relative to the spin. We discuss the implications of this result for the Wigner’s friend paradox. I. INTRODUCTION spired by Rovelli’s relational interpretation, according to which different observers may give different accounts of the same physical situation. It follows that any disagree- What is the meaning of a quantum state Ψ ? It can ments between them must be about external quantities. ultimately only have meaning in the context| i of a lab- However, since the observers themselves must obey quan- oratory in which it is possible to prepare and measure tum mechanics, their accounts must also be consistent quantum systems under controlled conditions. The state with each other to some degree. The internal proper- represents a comparison between the state of the ob- ties are precisely those facts that all observers can agree served system and the states of the measuring devices. upon. Our aim is to pin down what these properties If we assume the physical state of the laboratory is fixed, are, and to decide exactly what it means for two states then any change in the state of the system must be inter- to represent the ‘same physical situation’ in a relational preted as a physical change in the system (and not the quantum world. laboratory). More generally, the laboratory can itself be treated as a physical system subject to change, and hence It is clear that the distinction between external and changes in the state of the system only represent changes internal properties cannot be fundamental, so long as we in the relationship between system and laboratory. This give no fundamental significance to the attendant dis- is called a relational interpretation of the state. tinction between the laboratory and the system. Never- theless, the distinction is relevant to practical considera- Relational interpretations are not new; they include tions. For example, consider two laboratories attempting Everett’s ‘relative state interpretation’ [1], Rovelli’s rela- to exchange quantum information, but whose measuring tional interpretation [2], and Quantum Bayesianism [3]. devices are not mutually aligned. Since the receiver is ig- These approaches all agree that the quantum state de- norant of the sender’s reference frame, he must sum over scribes relative information, but disagree on whose in- all possibilities, in effect applying a random transforma- formation it is, and what the information is about. In tion to the external degrees of freedom. This destroys all cases, however, the measurable properties of a sys- any information encoded in these degrees of freedom, re- tem can be split into two categories: those that change ducing the parties’ ability to communicate. This is the when the system is measured in a different laboratory, arXiv:1601.07320v2 [quant-ph] 6 Oct 2016 basis for the literature on quantum reference frames [6– and those that are measured to be the same in all lab- 8], which provides an extensive toolbox of methods for oratories. We call these properties external or internal, performing quantum communication, cryptography, tele- respectively [22]. portation and other information-theoretic tasks in situa- At one extreme, a subjective Bayesian (see eg. Ref. tions where the laboratories do not share a given refer- [4]) might say that the quantum state represents only the ence frame. beliefs of a rational agent. In this case all properties are The work on quantum reference frames proceeds from external, since every state is a valid description of the the assumption that the separation between internal and beliefs of some agent, and the laws of physics describe external degrees of freedom is known. Usually, it is given how these beliefs should be updated in light of measure- in the form of a symmetry group that is postulated to ment outcomes. At the other extreme, a psi-ontologist preserve only the internal degreesG of freedom. The sym- (see eg. Ref. [5]) would claim that two different states metry group is justified by appealing to some natural represent two different objective realities, hence all prop- symmetry of the physical setting in question. For ex- erties of the system are internal and do not depend on ample, if the systems to be communicated are spin-half the observer. particles, it seems logical that the measuring instruments In this work, we wish to explore a middle ground, in- in the laboratories are Stern-Gerlach magnets or similar. 2 These instruments indicate a preferred Cartesian refer- seen by Wigner and his friend. Although we ultimately ence frame for the laboratory, representing the labelling do not succeed in this task, we are able to find an exam- of the (x,y,z) co-ordinates in the lab. Since the Carte- ple that transcends the usual system-apparatus division sian reference frames of the laboratories are related by a in the absence of an external reference frame, indicating proper rotation, the associated symmetry group is SO(3). that future efforts along similar lines might bear fruit. For quantum particles, we typically seek a unitary repre- The outline of the paper is as follows. In Section II we sentation of the rotation group, hence = SU(2). consider a system composed of spins and ask what prop- G Most of the literature to date has restricted attention erties should be the same for all observers. We argue that to transformations that preserve the separation between all observers should be subject to the same constraints system and laboratory. That is to say, if one laboratory on their ability to distinguish between subsystems of the performs a measurement on the system, the state is up- total system. Taking the quantum fidelity between sub- dated according to the outcome of this measurement for systems as a measure of their distinguishability, we ob- all laboratories, not just the laboratory in question. This tain U(2) as the symmetry group that preserves them. In assumption excludes the possibility of treating the labo- Section III we examine the bearing this result has on the ratories themselves as quantum systems relative to one microscopic / macroscopic distinction and the Wigner’s another. For example, consider the following thought ex- friend thought experiment. Section IV contains our con- periment due to Wigner [9]: an experimenter called Fran clusions and outlook. (a friend of Wigner) performs a measurement of the sys- tem and obtains an outcome. It should then be pos- sible for another experimenter in a different laboratory II. THE INTERNAL PROPERTIES OF SPINS (Wigner) to treat Fran and her laboratory as a quantum system, and the measurement as a physical interaction Our task is to define the internal properties of general that entangles the state of Fran to the state of the sys- pure states Ψ of N spins, namely, the attributes of the tem. This physical interaction is a transformation that system that| doi not depend on the observer’s laboratory. applies not just to the system alone, but also to the whole In what follows, we assume that all laboratories can agree laboratory of Fran, thereby including this laboratory and on a labeling of the N spins, so that if one observer spec- Fran herself as part of the ‘system’. Thus, Fran’s labora- ifies a subsystem of spins, say “spins numbered 7 12”, tory is in a superposition relative to Wigner’s laboratory. all the observers agree on which subsystem this refers− to. So far, there have been few attempts to apply the tools Measurements of spin performed within a laboratory are of quantum reference frames to physical settings of this relative to the (arbitrary) orientation of that laboratory’s more general type. spatial axes. The procedure for measuring ‘spin up’ in a We note the following exceptions. Palmer et. al. [10] given direction is assumed to be the same for all spins, have considered the task of changing reference frames hence labeling the ‘up’ direction for one spin fixes the in a setting where the reference frame is an explicitly label for all spins in the same lab. Finally, we assume all quantum object; however they do not extend their anal- observers can measure the number of spins in any sub- ysis to ‘Wigner’s friend’-type scenarios. Angelo et. al. system without difficulty. Other than that, the devices [11, 12], and Pereira and Angelo [13], following work by in the laboratories are not calibrated with each other. Aharonov et. al. [14], consider the problem of treating The problem can now be understood as follows. If quantum systems as observers and changing frames be- a system of N spins is prepared according to a fixed tween them.
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