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Quantum Reference Frames for General Symmetry Groups

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Quantum Reference Frames for General Symmetry Groups Quantum reference frames for general symmetry groups Anne-Catherine de la Hamette1,2 and Thomas D. Galley2 1Institute for Theoretical Physics, ETH Zurich,¨ Wolfgang-Pauli-Str. 27, 8093 Zurich,¨ Switzerland 2Perimeter Institute for Theoretical Physics, 31 Caroline St. N, Waterloo, Ontario, N2L 2Y5, Canada 23 November 2020 A fully relational quantum theory necessar- 1 Introduction ily requires an account of changes of quan- tum reference frames, where quantum refer- In quantum mechanics, physical systems are implic- ence frames are quantum systems relative to itly described relative to some set of measurement de- which other systems are described. By in- vices. When writing down the quantum state of a troducing a relational formalism which iden- system of interest, say a spin-1/2 system in the state tifies coordinate systems with elements of a |↑zi, we mean that the state of the system is `up' rel- symmetry group G, we define a general op- ative to a specified direction zˆ in the laboratory. In erator for reversibly changing between quan- practice, this direction will be associated to a macro- tum reference frames associated to a group G. scopic physical system in the lab. If we assume that This generalises the known operator for trans- quantum mechanics is a universal theory and there- lations and boosts to arbitrary finite and lo- fore applicable at all scales, the systems we make ref- cally compact groups, including non-Abelian erence to to describe quantum systems should eventu- groups. We show under which conditions one ally be treated quantum mechanically as well. Refer- can uniquely assign coordinate choices to phys- ence systems that are themselves treated as quantum ical systems (to form reference frames) and systems are referred to as quantum reference frames. how to reversibly transform between them, Following the success of Einstein's theory of relativ- providing transformations between coordinate ity and its inherently relational nature, one may seek systems which are ‘in a superposition’ of other to adopt a relational approach to quantum theory as coordinate systems. We obtain the change of well. In such an approach, most physically mean- quantum reference frame from the principles ingful quantities are relational, i.e. they only take of relational physics and of coherent change of on well defined values once we agree on the reference reference frame. We prove a theorem stating system (or the observer) relative to which they are that the change of quantum reference frame described. In his papers [1,2], Rovelli suggested that consistent with these principles is unitary if quantum mechanics is a complete theory about the and only if the reference systems carry the left description of physical systems relative to other phys- and right regular representations of G. We ical systems. In his Relational Quantum Mechanics also define irreversible changes of reference (RQM) he rejected the idea of observer-independent frame for classical and quantum systems in the states of systems and values of observables. The im- case where the symmetry group G is a semi- portance of changes of reference frame in special and direct product G = N o P or a direct prod- general relativity suggests the development of an ac- uct G = N × P , providing multiple examples count of changes of quantum reference frame in RQM. of both reversible and irreversible changes of Such an account is given in the present work. quantum reference system along the way. Fi- Recently, there has been an increased interest in nally, we apply the relational formalism and analysing spatial and temporal quantum reference changes of reference frame developed in this frames and in establishing a formalism that allows work to the Wigner’s friend scenario, finding arXiv:2004.14292v3 [quant-ph] 23 Nov 2020 to switch between different perspectives [3{7]. The similar conclusions to those in relational quan- present work is partially based on these approaches tum mechanics using an explicit change of ref- which define changes of quantum reference frames for erence frame as opposed to indirect reasoning systems that transform under the translation group using measurement operators. (in space and time) and the rotation group in three di- mensional space. As opposed to other more standard approaches [8{11], this formalism stresses the lack of an external reference frame from the outset and de- fines states of subsystems relative to another subsys- Anne-Catherine de la Hamette: tem. In standard approaches to quantum reference [email protected] frames [9, 10], one often starts from a description rel- Thomas D. Galley: [email protected] ative to an external reference frame and removes any Accepted in Quantum 2020-11-16, click title to verify. Published under CC-BY 4.0. 1 dependence on this reference frame by applying a G 2 Relational approach to quantum twirl (a group averaging over all possible configura- theory tions of the external reference frame). In some cases, one can refactor the total Hilbert space into global In the construction of a relational formalism of quan- and relational subsystems and trace out the global tum mechanics, an essential task is to write quantum degrees of freedom [9, 10, 12, 13]. The main empha- states of systems relative to a specified reference sys- sis of these standard approaches is often to obtain A tem. We introduce the following notation: |ψiB indi- the physically meaningful (or reference frame inde- cates the state of system B relative to system A. In pendent) quantities, in a similar fashion to identifying contrast to the approach of [6], we assign a Hilbert noise free subsystems in error correction. In the work space to the system whose perspective is adopted and of [3{7, 14, 15] however emphasis is given on the rela- assign to it the trivial state, corresponding to the tional nature of the description (always starting from identity element of the group. Hence, by convention, a state that is given from the viewpoint of one of the system A is in a default `zero-state' relative to itself. subsystems) and the main object of study is the re- Once we introduce the notion of symmetry groups and lation between different accounts. We make a similar how they enter into the formalism, we will see that emphasis in the present work. We abstract the for- this default zero-state corresponds to the identity ele- malism of [6] and introduce an approach which makes ment of the group that describes the transformations heavy use of the inherently group theoretic nature of of the system. Thus, to be more precise, one can write quantum reference frames. This allows us to gener- A A alise the known results beyond the translation and ro- |0iA ⊗ |ψiB . (1) tation groups to arbitrary finite and locally compact groups (including non-Abelian groups). The upper index refers to the system relative to which the state is given while the lower index refers to the system that is being described (similarly to the per- In Section2 we outline the relational approach to spectival approach of [16]). This description does not quantum theory embraced in the present work as well make use of any external abstract reference frame nor as give a simple example of a change of reference frame does it assume the existence of absolute, observer- for classical bits and an example of a change of quan- dependent values of physical observables. We ob- tum reference frame for qubits. In Section3 we define serve that since system A can only ever assign itself the notion of a reference frame in terms of reference a single state there are no state self-assignment para- systems and coordinate systems, as well as give a full doxes [17, 18]. account of active and passive transformations as left A natural question to address on the relational ap- and right regular group actions. Combining these we proach to quantum theory is how to change reference define changes of reference frame under a group G systems. Namely if the state of B relative to A is ∼ A A A B for classical systems with configuration space X = G. |ψiB = |0iA ⊗ |ψiB, what is the state |ψiA of A rela- In Section4 we extend the classical change of refer- tive to B? This is the problem which will be addressed ence frame to quantum systems L2(G) following the in the present work. principle of coherent change of reference system; and Before introducing the general framework we will define a general unitary operator which implements be using, we give two simple examples of changes of this change of reference system. We prove a theorem reference frame for relational states. The first is classi- stating that only systems carrying a regular repre- cal and the second its quantum generalisation. These sentation of G can serve as reference frame, subject should hopefully provide the reader with an intuitive to the principle of coherent change of reference sys- picture of the general mechanisms at play. tem. Following this we extend the change of quantum Example 1 (Z change of classical reference frame). reference frame operator between L2(G) systems de- 2 Let us consider the case where systems can be in two scribing systems which do not carry the right regular states ↑ or ↓. Every system considers themselves to be representation of G. In Section5 we define irreversible in the state ↑ (for example an observer free floating changes of reference frames for groups G = N P o in empty space would always consider the up direc- and G = N × P via a truncation procedure. In Sec- tion to be aligned from their feet to their head). Con- tion6 we extend this change of reference frame to sider classical systems where the state relative to A is quantum reference frames using the principle of co- ↑A↑A↓A.
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