Quantum reference frame transformations as symmetries and the paradox of the third particle
Marius Krumm1,2, Philipp A. Hohn¨ 3,4, and Markus P. Muller¨ 1,2,5
1Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria 2Vienna Center for Quantum Science and Technology (VCQ), Faculty of Physics, University of Vienna, Vienna, Austria 3Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa 904 0495, Japan 4Department of Physics and Astronomy, University College London, London, United Kingdom 5Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo ON N2L 2Y5, Canada
In a quantum world, reference frames are ulti- dox of the third particle’. We argue that it can mately quantum systems too — but what does it be reduced to the question of how to relation- mean to “jump into the perspective of a quantum ally embed fewer into more particles, and give a particle”? In this work, we show that quantum thorough physical and algebraic analysis of this reference frame (QRF) transformations appear question. This leads us to a generalization of the naturally as symmetries of simple physical sys- partial trace (‘relational trace’) which arguably tems. This allows us to rederive and generalize resolves the paradox, and it uncovers important known QRF transformations within an alterna- structures of constraint quantization within a tive, operationally transparent framework, and simple quantum information setting, such as re- to shed new light on their structure and inter- lational observables which are key in this reso- pretation. We give an explicit description of the lution. While we restrict our attention to finite observables that are measurable by agents con- Abelian groups for transparency and mathemat- strained by such quantum symmetries, and ap- ical rigor, the intuitive physical appeal of our re- ply our results to a puzzle known as the ‘para- sults makes us expect that they remain valid in more general situations.
3 2 group CnZn
4 ... arXiv:2011.01951v2 [quant-ph] 21 Aug 2021 Figure 1: The simplest example of this article’s setup: a discretization of wave functions in one spatial dimension under translation symmetry. The configuration space is the cyclic group Zn, and the one-particle Hilbert space is 2 n ⊗N H = ` (Zn) ' C . We have N distinguishable particles in a joint quantum state |ψi ∈ H , and we study QRF transformations that switch between the “perspectives of the particles”.
Marius Krumm: [email protected] Philipp A. H¨ohn: [email protected] Markus P. Muller:¨ [email protected]
Accepted in Quantum 2021-06-18, click title to verify. Published under CC-BY 4.0. 1 Contents
1 Introduction 2
2 Quantum information vs. structural approach to reference frames4 2.1 Describing physics with or without external relatum ...... 4 2.2 Communication scenarios illustrating the two approaches ...... 6
3 From symmetries to QRF transformations and invariant observables9 3.1 G-systems and their symmetries ...... 10 3.2 Invariant observables and Hilbert space decomposition ...... 12 3.3 Group averaging states ...... 14 3.4 Alignable states as states with a canonical representation ...... 16 3.5 QRF transformations as symmetry group elements ...... 16 3.6 Characterization of alignable states ...... 17 3.7 Alignable and relational observables ...... 18 3.8 Communication scenario of the structural approach revisited ...... 20
4 The paradox of the third particle and the relational trace 20 4.1 The non-uniqueness of invariant embeddings ...... 21 4.2 A class of invariant traces ...... 24 4.3 Definition of the relational trace ...... 25 4.4 Relational resolution of the paradox ...... 28 4.5 Comparison to the resolution by Angelo et al...... 28
5 Conclusions 29
Acknowledgments 32
References 32
1 Introduction ticle see the interferometer” [50]? A central topic in this approach is the QRF de- All physical quantities are described relative to pendence of observable properties like superposi- some frame of reference. But since all physical sys- tion, entanglement [38–40], classicality [39, 51, 52], tems are fundamentally quantum, reference frames or of quantum resources [53]. The corresponding must ultimately be quantum systems, too. This sim- QRF transformations admit an unambiguous defi- ple insight is of fundamental importance in a vari- nition of spin in relativistic settings by transform- ety of physical fields, including the foundations of ing to a particle’s rest frame [46, 47], they describe quantum physics [1–11], quantum information the- the comparison of quantum clock readings [42, 45], ory [12–19], quantum thermodynamics [20–28], and and they yield an alternative approach to indefinite quantum gravity [29–37]. causal structure [48,54]. Among other conceived ap- Recently, there has been a wave of interest in plications, they are furthermore conjectured to play a specific approach to quantum reference frames a crucial rule in the implementation of a “quan- (QRFs) that we can broadly classify as structural in tum equivalence principle” [55] as well as in space- nature, including e.g. Refs. [38–49]. This approach time singularity resolution [56] and the description extends the usual concept of reference frames by as- of early universe power spectra [57, 58] in quantum sociating them with quantum systems, and by de- gravity and cosmology. scribing the physical situation of interest from the Despite the broad appeal, several fundamental “internal perspective” of that quantum system. For and conceptual questions remain open. For exam- example, if an interferometer has a particle travel- ple, how should we make concrete sense of the ling in a superposition of paths, how “does the par- idea of “jumping into the reference frame of a parti-
Accepted in Quantum 2021-06-18, click title to verify. Published under CC-BY 4.0. 2 cle”? How are QRF transformations different from broader conceptual framework. any other unitary change of basis in Hilbert space?
What kind of physical symmetry claim is associ-
mation concepts into this context. For example,
In this article, we shed considerable light on all
Accepted in Quantum 2021-06-18, click title to verify. Published under CC-BY 4.0. 3 groups of particles hold “the same relation” to each either because it does not exist in the first place, one other within two distinct configurations, such that does not have access to it, or it is deliberately ig- the corresponding branches should interfere (see nored. This setting is schematically depicted in Fig- Figure6 on page 23). Mathematically, this corre- ure3. Examples include: sponds to the question of how to embed the alge- (i) Minkowski spacetime of special relativity, with bra of invariant N-particle observables into that of S the set of all possible states of matter (say, N + M particles. We show that no unique answer of classical point particles), and the Poincare´ to this question exists for the full set of invariant ob- group Gsym as the group of symmetry transfor- servables in Ainv: the answer always depends on mations. the physical choice of how to determine the particle group interrelations operationally. (ii) Electromagnetism in some bounded region of However, we show that a unique and natural em- spacetime. This is a gauge theory with Gsym the bedding does exist for the subset of relational ob- group of local U(1)-transformations as its sym- servables in Aphys. The trick is to use a coherent su- metry group. perposition of all operationally conceivable particle group relations, and it turns out this construction (iii) A spin in quantum mechanics with Hilbert preserves the algebraic structure of the N-particle space H and projective representation g 7→ Ug observables. We use this to define a relational no- of the rotation group G = SO(3). Here, S is the tion of the partial trace which arguably resolves the set of density matrices ρ, and Gsym consists of † paradox, and we compare this resolution to the one all maps of the form ρ 7→ UgρUg . proposed by Angelo et al. [50] before concluding in These three examples illustrate an important sub- Section5. tlety: to claim that ρ and Gρ are physically indistin- guishable, one needs to speak about ρ and Gρ as dif- 2 Quantum information vs. structural ferent objects in the first place. In other words, one ρ Gρ approach to reference frames has to somehow define and as distinct states. But in order to do so, one would need something external to the system S to refer to. Let us begin with the main element that both the quantum information as well as the structural ap-
Accepted in Quantum 2021-06-18, click title to verify. Published under CC-BY 4.0. 4 a system “in isolation” in the sense of Assumption 1 subsystems, as we will explain shortly. It suffices does not imply that the system S is literally a physi- to focus on invariant properties of S which have a cally isolated system. It simply means that we have meaning relative to an arbitrary choice of external chosen to describe the system without the external frame in order to successfully carry out communi- relatum relative to which the action of the symme- cation tasks in the absence of a shared frame. It try group is defined. Moreover, the setting does not is also worth emphasizing that there does not exist imply that the agent who treats ρ and Gρ as indis- a sharp distinction between the two approaches in tinguishable is herself part of the system S, but only the body of literature on QRFs. Since the structural that the agent considers S without the external rela- approach shares external relatum independent state tum. descriptions with the QI approach, there exist “hy- Here we argue that the essential difference be- brid” works which arguably incorporate elements tween the two approaches to quantum references from both. For example, Refs. [7–10,13,18,19,50] use frames can succinctly be stated as follows: standard quantum information techniques to define external relatum independent states, but also use the latter to explore to some degree the question of The quantum information (QI) approach as in how a quantum state is described relative to a sub- e.g. Refs. [12–17] emphasizes the fact that quantum system. However, these works do not study the re- states are often only defined relative to an external lations between the different such descriptions and relatum (as in Figure3), and that this relatum may thus, in particular, do not study the QRF transfor- ultimately be a quantum system, too. This leads mations. to questions like: how can quantum information- The structural approach to QRFs is sometimes il- theoretic protocols be performed in the absence of lustrated in ways that seem at first sight to be in a shared reference frame [12]? How well can quan- conflict to the characterization above. For example, tum states stand in as resources of asymmetry if Figure 1 in Ref. [38] suggests to think of QRFs as there is no shared frame [14, 15]? What are funda- physically attached to an observer and its labora- mental quantum limits for communicating or align- tory (defined by its own quantum state), similarly ing reference frames [12]? Addressing questions as as reference frames in Special Relativity are often these often involves encoding information in quan- thought of as being attached to an observer (defined tum states in an external relatum independent man- by its state of motion). QRF transformations would ner and, as such, requires external relatum indepen- then relate the descriptions of “quantum” observers dent descriptions of states. who are relative to each other in superposition in a The structural approach as in e.g. Refs. [38–40] is Wigner’s-friend-type fashion. not primarily concerned with operational protocols. However, we will show below that the structural While it shares the aim of external relatum indepen- framework of QRFs can be derived and analyzed dent descriptions of states with the QI approach, it exactly under an alternative and simpler interpre- goes further: it disregards the external relatum al- tation. As we will elaborate and generalize below, together, and instead asks whether and how phys- choosing a QRF amounts to aligning one’s descrip- ical subsystems of S can be promoted to an internal tion of the physics with respect to some choice of inter- reference frame. This emphasizes the fact that the nal quantum subsystem, such as the position of one distinction between quantum systems and their ref- of the particles. Two different observers can choose erence frames is not fundamental, but merely con- two different subsystems (say, particles) that are rel- ventional. It leads to questions like: what is the ative to each other in superposition, even if the ob- description of the quantum state relative to one of servers themselves are fully classical. Their descrip- its particles? Can we find a Hilbert space basis in tions will then be related by QRF transformations. which the description of the physics is simplified, The observer who assigns the quantum state may e.g., in which superpositions of subsystems of in- thus retain the status of a classical entity external terest may be removed? More generally, what are to the quantum system (at least in laboratory situa- the “QRF transformations” that relate the descrip- tions), as illustrated in Figure3. While more conser- tions relative to different choices of internal refer- vative, this new interpretation is operationally more ence frame? immediate, and it is sufficient to reconstruct and ex- tend the full machinery of QRF transformations, as In the QI approach, it is usually not necessary to we will see. take the extra step to internal frame choices and to The characterization above is also in line with ask how a system is described relative to one of its another version of the structural approach: the
Accepted in Quantum 2021-06-18, click title to verify. Published under CC-BY 4.0. 5 so-called perspective-neutral approach [39, 41–45] tum to be meaningful and cannot be communicated which, motivated by quantum gravity, is formu- purely classically between two agents who do not lated in the language of constrained Hamiltonian share a frame. systems [59,60]. The starting point of this approach is a deep physical and operational motivation: take 2.2.1 The quantum information approach: communi- the idea seriously that there are no reference frames, such cating quantum systems as rods or clocks, that are external to the universe. To implement this idea, one starts with a “kinematical Consider the scenario in Figure4. Alice holds a Hilbert space” that defines all the involved quan- quantum system S that she has prepared in some tum degrees of freedom and some gauge symmetry, state ρ ∈ S(H), and S(H) denotes the density ma- but is interpreted as purely auxiliary. The absence of trices on the corresponding Hilbert space H. We as- external references is then implemented by restrict- sume that there is a (for now, for simplicity) com- ing to the gauge-invariant subset of states where the pact group G of symmetries and a projective rep- description becomes purely relational. resentation G 3 g 7→ Ug such that G acts on S via † The actual mathematical machinery applied in Ug(ρ) = UgρUg . In this case, the symmetry group this approach still fits the description above: the is Gsym = {Ug | g ∈ G}. If we assume that Alice’s kinematical Hilbert space can be viewed as being quantum system S has the properties of Assump- described relative to a fictional external relatum. The tion 1, then the very definition of ρ is relative to her insight that there is nothing external to the uni- local frame of reference. verse motivates to ask — purely formally at first — whether some of the internal degrees of freedom of
the theory can be promoted to a frame of reference,
Accepted in Quantum 2021-06-18, click title to verify. Published under CC-BY 4.0. 6 cus on invariant physical properties that are mean- 2.2.2 The structural approach: agreeing on a ingful in any external laboratory frame. This does redundancy-free internal description of quantum states not require describing S relative to one of its sub- systems. The structural approach does not stop at an external relatum independent state description. It also asks Nevertheless, in the QI approach, the quantum for a description of a quantum state relative to an nature of reference frames is sometimes taken into internal frame that is part of the system of interest. account, for example, by “quantizing” them to over- A transparent way to understand the structural come superselection rules that arise in the absence approach operationally is as follows. Alice and Bob of a shared classical frame [12]. This “quantization” in their respective labs would like to agree on a con- of a frame means adding a reference quantum sys- crete description of the quantum state of a system tem R to the system of interest S in order to define without external relatum, i.e. in particular without relative quantities between R and S, such as rela- shared reference frame. They have the option of de- tive phases [12] or relative distances [13, 19], that scribing S in terms of the equivalence classes [ρ] of are invariant under Gsym and thereby meaningful quantum states. However, there is an evident redun- relative to any external laboratory frame. In a com- dancy in the description of each equivalence class munication scenario between two parties Alice and in terms of concrete quantum states: each member Bob who do not share a classical frame, the refer- of the equivalence class is a legitimate (and non- ence system R will typically be communicated to- unique) description of it. In order to break this re- gether with S. While this also constitutes an inter- dundancy and succeed in this task, they can take nalization of a frame in the sense thats the reference advantage of the fact that any equivalence class [ρ] system R is now a quantum system too, it is still of states admits certain “canonical choices” for its external to S. Furthermore, since the relative quan- description which are associated with different in- tities between R and S are meaningful relative to ternal reference frame choices. The transformations any external laboratory frame with respect to which relating these different canonical choices amount to a measurement will be carried out, it is not neces- “QRF transformations” and they will be elements of sary to take an extra step and ask how S is described the symmetry group Gsym defining the equivalence “from the perspective” of R in order for Alice and classes. Bob to succeed in their communication task.
In summary, in the QI approach, the quantum
gyroscope). Thus, “QRF transformations” relating
For example, one could imagine the following communication scenario depicted in Figure5 to il- lustrate the role of “canonical choices”: 1This includes Ref. [18], where transformations between dif- ferent “quantized” reference systems R1 and R2 are studied. However, in the spirit of the QI approach, the derived trans- • Referee Refaella informs Alice and Bob in formations proceed between different invariant states (i.e. es- their separate laboratories that she will pre- sentially G-twirls of ρ ⊗ ρ , i = 1, 2) and are thus not trans- S Ri pare quantum states of a particular system S formations between descriptions of the quantum state of S relative to different choices of subsystem, as we will see them (subject to Assumption 1) relative to her (freely later. In particular, the descriptions of the quantum state of aligned) frame, but that she will only commu- S relative to different subsystems will be different descriptions nicate the description of the respective equiva- of one and the same invariant state. lence classes [ρ] to Alice and Bob separately.
Accepted in Quantum 2021-06-18, click title to verify. Published under CC-BY 4.0. 7 • Alice’s and Bob’s task is to separately return a quantum reference frames: the system’s state breaks concrete (redundancy-free) description of each the fundamental symmetry, and admits, at least on quantum state to Refaella and they will win the the level of classical descriptions, a canonical choice game provided their descriptions always agree of representation. (either for all states of S, or for a particular class Example1 illustrates a general consequence of C of states). the symmetry structure: for any particular choice of QRF, the set of state descriptions relative to that • Alice and Bob are only permitted to communi- QRF corresponds in general only to a subset or sub- cate prior to the beginning of the game to agree space of states. In this example, any such choice only on a strategy. allows to describe a subset of states that corresponds to a classical bit: namely, the convex hull of the density Let us consider two examples for how this can be matrices diag(1, 0) and diag(1/2, 1/2). The follow- accomplished. These examples illustrate that there ing example demonstrates how a full subspace of will generally exist multiple “canonical choices” for states can be encoded. describing [ρ] in terms of concrete quantum states, however, that Alice and Bob can always agree in Example 2. Consider two spin -1/2 particles with ro- their communication beforehand which such choice tational symmetry. That is, the symmetry group is † † to pick. This will also give a hint on the rela- Gsym = {ρ 7→ U ⊗ UρU ⊗ U | U ∈ SU(2)}, acting 2 2 tion to quantum reference frames as described in on states in S(C ⊗C ). Let us make a somewhat ar- Refs. [38–40], and we will elaborate on this further bitrary, but nonetheless illustrative choice of a class in the following sections. C of states for which the above communication game can be played. These will be the pure states Example 1. Consider a single quantum spin-1/2 par- θ θ 2 C = cos |φ i + eiϕ sin |φi ⊗ |φi , (1) ticle, with state space S(C ). Let us assume that the 2 − 2 symmetry group is the full projective unitary group, where 0 ≤ θ ≤ π, 0 ≤ ϕ < 2π, |φ i is the singlet i.e. G = {ρ 7→ UρU † | U †U = 1}, which is iso- − sym state, and |φi ∈ 2 an arbitrary normalized state. morphic to the rotation group SO(3). C The set of states C is the disjoint union of the sets Let ρ be an arbitrary state that Refaella is for some C for which the two angles are fixed and |φi is reason interested in preparing. The equivalence class θ,ϕ still an arbitrary qubit state. Since U ⊗ U|φ i = [ρ] consists of all states with the same eigenvalues − |φ i, the C are orbits of the symmetry group, i.e. λ , λ as ρ. Describing [ρ] is equivalent to listing − θ,ϕ 1 2 equivalence classes of states. the eigenvalues λ , λ and this is what Refaella may 1 2 If Refaella gives Alice and Bob a description of communicate to Alice and Bob. Clearly, there are such an equivalence class [|ψi] = C , they can many ways to represent this information in terms of θ,ϕ agree on returning the standard description |ψ0i = a concrete quantum state ρ. cos θ |φ i + eiϕ sin θ |0i ⊗ |0i, for example. This pre- The strategy that Alice and Bob can agree on in 2 − 2 scription has the added benefit of preserving super- order to win the game, but prior to it starting, is position across different equivalence classes. Namely, trivial: they can agree to always choose a basis (i.e. if for i = 1, 2, we have |ψ i = α |φ i + β |φi ⊗ |φi a specific reference frame alignment) such that ρ i i − i such that |α |= 6 |α |, then |ψ i and |ψ i are in differ- described relative to it is a diagonal matrix. This 1 2 1 2 ent equivalence classes, and so are (in general) their leaves two “canonical choices” of representation: or- superpositions. But the states that Alice and Bob re- dering the eigenvalues such that λ ≥ λ they could 1 2 turn respect superpositions: if |ψi = κ|ψ i + λ|ψ i, decide to always return either ρ = diag(λ , λ ) or 1 2 1 2 then the returned states satisfy |ψ0i = κ|ψ0 i + λ|ψ0 i. ρ = diag(λ , λ ) to Refaella. The transformation 1 2 2 1 That is, this choice of QRF admits the description relating the two descriptions is the unitary “QRF ! of a subspace, a qubit, inside the joint state space. 0 1 transformation” U = . Other choices of QRF do so as well. These would 1 0 correspond to canonical descriptions where |0i ⊗ |0i This trivial example relies on the simple fact that is replaced by some arbitrary |φ0i ⊗ |φ0i, and they every quantum state has a canonical description: the are related by “QRF transformations” U ⊗ U. matrix representation in its own eigenbasis (up to There are also seemingly natural choices of QRF a choice of order of eigenvalues). In some sense, that, however, are deficient in that the set of admis- every quantum state defines a finite set of natural sible descriptions relative to them cannot encom- representations of itself. It is in this sense that the pass a state space, as the following example illus- structural approach interprets quantum systems as trates.
Accepted in Quantum 2021-06-18, click title to verify. Published under CC-BY 4.0. 8 Example 3. Consider again two spin-1/2 parti- states above corresponds precisely to a perspective- cles, but now under slightly different circumstances. neutral quantum state. As we will see, this means There is a canonical choice of factorization of the that the relational states of the structural approach Hilbert space: by looking at the system in isolation, are coherently (not incoherently as in the QI ap- observers can determine the decomposition into two proach) group-averaged states. distinguishable particles. If we assume that this is the only structure that can be determined by such observers, then we have the symmetry group 3 From symmetries to QRF transforma- † † † † tions and invariant observables Gsym = {ρ 7→ (U ⊗V )ρ(U ⊗V ) | U U = V V = 1}.
Under these circumstances, a canonical choice of Quantum reference frames as described in Refs. [38– frame is such that any pure state |ψi becomes iden- 40] have first been considered for the case of wave tical to its own Schmidt representation, |ψi = functions on the real line. We have a Hilbert space 1 2 √ of square-integrable functions, H = L (R), and a X α |iii, where α ≥ α . i 0 1 physical claim that there is no absolute notion of ori- i=0 While Alice and Bob could easily agree on such a gin. In other words, the “physics” does not change convention, the ensuing canonical description would under translations (we will soon formulate what not preserve complex superpositions and, in partic- this means in detail). If we have N particles on the 2 ⊗N ular, not lead to a subspace of states as its image, real line, the total Hilbert space is L (R) . owing to the real nature and ordering of the Schmidt- As noted in Ref. [40], the real numbers R play a coefficients. double role in this case: on the one hand, they la- bel the configuration space on which the wave func- A priori, a choice of QRF in the structural ap- tions are supported; on the other hand, they also proach can therefore be quite arbitrary. However, label the possible translations, i.e. the fundamental as the examples above motivate, a “good” choice symmetry group (R, +). of QRF will correspond to one that admits the de- In this section, we will analyze this particular scription of a set of states relative to it which carries situation in a simplified setting: one in which the sufficient convex or linear structure to encode clas- group is finite and Abelian. In the simplest case, we sical or quantum information. Preferably, that set of discretize the real line and make it periodic, as in states should correspond to a subspace of maximal Figure1. Formally, for some n ∈ N, we consider the size within C. cyclic group In the remainder of this article, we will focus on a more interesting realization of such a scenario Zn := {0, 1, 2, . . . , n − 1} (2) which reproduces the notion of QRFs in the struc- tural picture. We will define particular systems S with addition modulo n as its group operation. To that we call “G-systems”, and we will see that these this, we associate a single-particle Hilbert space carry an interesting group of symmetries Gsym. If we ask what kind of canonical choices of (redundancy- 2 H = ` (Zn) = span{|0i, |1i,..., |n − 1i} (3) free) description G-systems admit, such that Alice and Bob can succeed in the communication scenario and a total Hilbert space H⊗N for N distinguish- of Figure5, we will find that these correspond to able particles. We will denote the particles with la- choosing one of the subsystems of S as a reference bels A, B, C, . . ., and later in this paper with inte- system and to describing the remaining degrees of gers 1, 2, 3,.... Within this formalism, we can real- freedom relative to it. In this manner, we will re- ize the main ideas of quantum references frames as cover and generalize the “quantum states relative to in Refs. [38–40]. For the case N = 2, consider the a particle” of Refs. [38–40]. In particular, the trans- quantum state formations among the canonical choices of descrip- tion of S are elements of the symmetry group Gsym 1 |ψiAB = |0iA ⊗ √ (|1i + |2i) . (4) and exactly the QRF transformations of Ref. [40], 2 B which are also equivalent to the ones in [38, 39] (re- stricted to a discrete setting). In Ref. [72] we will We are interested in a situation where “only the re- further explicitly demonstrate the equivalence with lation between the particles” matters, but not their the perspective-neutral approach to QRFs [39] and total position. That is, in some sense, “applying el- elucidate that any equivalence class [ρ] of quantum ements of Zn to a quantum state doesn’t change the
Accepted in Quantum 2021-06-18, click title to verify. Published under CC-BY 4.0. 9 physics”. Intuitively, this means, for example, that Definition 4 (G-system). Fix some finite Abelian the quantum state group G, interpreted as a classical configuration space. That is, we regard the g ∈ G as perfectly 0 1 distinguishable orthonormal basis vectors |gi, span- |ψ iAB = |1iA ⊗ √ (|2i + |3i)B (5) 2 ning a Hilbert space H. Formally, this Hilbert space is H = `2(G), and it carries a distinguished basis should be an equivalent description of the system’s {|gi} , similarly as quantum mechanics on the real properties, since it is related to |ψi by a transla- g∈G line carries a distinguished position basis. tion. Motivated by Ref. [38], we can do some- Consider N distinguishable particles on such a thing more interesting. First, in the terminology of classical configuration space, where N ∈ . That Refs. [38–40], the form of |ψi can be interpreted as N is, the total Hilbert space is H⊗N , and it carries a saying that “particle B, as seen by A, is in the state √1 (|1i + |2i) natural orthonormal basis 2 ”. Second, we can then use the pre- B ⊗N scription of Refs. [38–40] to “jump into ’s reference H = span{|g1, . . . , gN i | gi ∈ G}. (7) frame”, and consider the state The physical system S described by this Hilbert space 00 1 |ψ i = √ (|n − 2i + |n − 1i)A ⊗ |0iB (6) will carry a group of symmetries Gsym as introduced 2 in Assumption 1 and Figure3. Clearly, the basic and conclude that “particle A, as seen by B, is in the Hilbert space structure of S, i.e. the notion of linear- state √1 (|n − 1i + |n − 2i)”. After all, this still ex- ity and the inner product, must not depend on the 2 orientation of the external reference frame. Hence, presses the fact that with amplitudes √1 , B is either 2 the symmetry group will be of the form one or two positions to the right of A. † We will now show that we can understand these Gsym = {U • U | U ∈ Usym}, (8) transformations as natural symmetry transforma- tions in a simple class of physical systems which we for Usym some group of unitaries. Furthermore, we call “G-systems”. Choosing one of the particles as a assume that the classical configuration space, i.e. the reference frame (as sketched above) will correspond set of basis vectors, {|g1, . . . , gN i | gi ∈ G}, is an to a choice of canonical representation of a state as internal structure of S that is defined without the ex- in the structural approach outlined above. This will ternal reference frame. We now postulate that the give the idea of “jumping into a particle’s perspec- classical configurations carry G-symmetry. In par- tive” a thorough operational interpretation. ticular, any given configuration |gi := |g , g , . . . , g i (9) 3.1 G-systems and their symmetries 1 2 N and its “translated” version We begin by considering a specific physical system which is motivated by translation-invariant quan- ⊗N Ug |gi = |ggi := |gg1, gg2, . . . , ggni (10) tum physics on the real line with Hilbert space 2 L (R). Here we consider a finite, discrete group- are internally indistinguishable. On the other hand, theoretic analogue, again using the group as both we postulate that the relation between the particles the configuration space and set of transformations. is accessible to observers without the external frame. Some aspects of QRF transformations in this case To formalize this, consider some tuple h ∈ GN−1 of were also considered in Ref. [40]. In contrast to group elements, i.e. h = (h1, . . . , hN−1). Any state Ref. [40], we restrict our attention to finite Abelian of the form groups G for simplicity. Due to the structure the- orem [73], every such G can be interpreted as the |g, h1g, h2g, . . . , hN−1gi =: |g, hgi (11) group of translations of a discrete torus of some di- has the same pairwise relations between its particles, mension. In the simplest case where G = Zn, this no matter what the state |gi of the first particle is. torus is the circle2, and we are in the setting of Fig- We now define G as the largest possible symme- ure1. sym try group that is compatible with these postulates. To 2This representation is not unique. For example, we can this end, Usym must be the group of unitary transfor- interpret Z6 as the translation group of six points on a circle, mations with the following properties: but the structure theorem tells us that Z6 ' Z2 × Z3. Thus, we can also interpret this group as the translations of a two- 1. U maps classical configurations to classical con- dimensional (2 × 3)-torus. 0 0 figurations, i.e. U |g1, ..., gni = |g1, ..., gni.
Accepted in Quantum 2021-06-18, click title to verify. Published under CC-BY 4.0. 10 2. On classical configurations, U preserves relative and the corresponding orthogonal projectors Πh. 0 0 ⊗N L positions, i.e. U |g, hgi = |g , hg i. Note that H = h∈GN−1 Hh, and the {Πh}h∈GN−1 define a projective measurement. 3. If two classical configurations are g-translations of each other, then U preserves this fact, i.e. Lemma 5. The symmetry group of a G-system is ⊗N ⊗N |gi = Ug |ji ⇒ U|gi = Ug U|ji . (12) M ⊗N Usym = U = U g ∈ G , (14) gh h h∈GN−1 A few words of justification are in place. While two choices of external reference frame may yield where U ⊗N denotes the global translation by g , but gh h a different description of any configuration, they restricted to the subspace Hh. must agree on the set of all possible configurations that S can be in, for otherwise their descriptions of S That is, the symmetries in Usym act as relation- 3 conditional global translations: every classical config- cannot be placed in full relation with one another. ⊗N uration is globally translated via some Ug , but the The set of basis vectors {|gi}g∈GN must thus be in- h dependent of the external frame and hence should amount of translation gh may depend on the rela- tion h between the particles. We will soon identify remain invariant under Gsym. It is also clear that the symmetry group must preserve the linear and prob- the QRF transformations of Refs. [38–40] with ele- abilistic structure of quantum theory and thereby ments of this group. Thus, the above highlights that leave the inner product on H⊗N invariant.4 Af- these transformations make sense in a purely clas- ter all, by Assumption 1, symmetry related quan- sical context; indeed, the corresponding classical tum states should be indistinguishable even prob- frame transformations were also studied in [39, 40] h and shown to be conditional on the interparticle re- abilistically. Furthermore, the label the ’relative 5 positions’ among the N particles. These are inter- lations. For example, they can also be applied if nal properties of S and so independent of any ex- one deals with statistical mixtures of particle posi- g tions instead of superpositions. Nonetheless, their ternal relatum. Finally, configurations that are - ⊗N translations of each other are by assumption inter- unitary extension to all of H leads to interesting nally indistinguishable. The symmetry group must quantum effects like the frame-dependence of en- preserve this indistinguishability. tanglement [38–40]. This is similar to the behavior Note that it is possible to drop assumption 3., and of the CNOT gate in quantum information theory, to assume only 1. and 2. In this case, one obtains which is defined by its classical action on two bits, similar results to those presented here, but with but nonetheless can create entanglement. modified structures: the algebra of invariant oper- Proof. Due to conditions 1. and 2. of Definition4, ators then becomes what we call Aalg in Lemma 26, the U ∈ Usym leave every Hh invariant. Thus, U and the symmetry group becomes the group of con- L decomposes into a direct sum U = h∈GN−1 Uh. Fix ditional permutations (not only conditional transla- N−1 some h ∈ G . Since Hh is invariant, there exists tions). Physically, this does not seem particularly some gh ∈ G such that U|e, hi = |gh, hghi. Now, well-motivated, and it leads to the loss of certain ⊗N for every g ∈ G, we have |g, hgi = Ug |e, hi. Thus, uniqueness results, including the uniqueness of U ∈ condition 3. of Definition4 implies that Usym in Theorem 18. ⊗N ⊗N The symmetry group of a G-system can now eas- U|g, hgi = Ug U|e, hi = Ug |gh, hghi ily be written down. To this end, define the sub- ⊗N ⊗N = Ugg |e, hi = Ug |g, hgi. (15) spaces h h This shows that U acts like U ⊗N on H . Hh := span{|g, hgi | g ∈ G} (13) h gh h
3This assumes that the external frame choices in the ambi- When working with pure state vectors, we some- ent laboratory that an agent may have access to are complete times want to allow global phases. Thus, we use the in the sense that all quantum properties of S can be described notation relative to them.
4 ⊗N ∗ iθ In constraint quantization, H corresponds to the kine- Usym := Usym × U(1) = {e U | U ∈ Usym, θ ∈ R}. matical Hilbert space and so the preservation refers here to the kinematical inner product. While one is usually only in- 5More precisely, in the perspective-neutral approach these terested in the physical inner product (i.e. the inner product classical reference frame transformations correspond to condi- on the space of solutions to the constraints), it nevertheless tional gauge transformations, i.e. the gauge flow distance de- holds that also the kinematical inner product is left invariant pends on the subsystem relations, see Appendix B of Ref. [39] by the group generated by the (self-adjoint) constraints. and also Refs. [41–43].
Accepted in Quantum 2021-06-18, click title to verify. Published under CC-BY 4.0. 11 Above, we have decided to denote the state of the invariant under all symmetry transformations A 7→ † † particles relative to the first particle, but this also de- UAU . A self-adjoint element A = A ∈ Ainv is fines the relations between all other pairs of parti- called an invariant observable. cles: the equation |gi = |g, hgi ∈ Hh means that Since all observable properties of our system are gi = hi−1g1 for i ≥ 2, but this implies that gi = −1 assumed to be invariant under Gsym, it follows that (hi−1h )gj for all i, j if we set h0 := e, the unit el- j−1 the observables in Definition7 comprise the set of all ement of the group. Thus, the Hh decompose the observables that can be physically measured by an global Hilbert space into sectors of equal pairwise observer who does not have access to the external relations. reference frame. It is clear that global G-translations are elements Clearly, all the Π are invariant observables, i.e. of the symmetry group, but they do not exhaust it: h Πh ∈ Ainv. However, due to the fact that we have Example 6. Given any G-system, the global trans- declared a classical basis to be a distinguished struc- ⊗N lations Ug are symmetry transformations. Since ture of the G-system, there are many more invari- they represent the global action of G on the N-particle ant observables. To determine the algebra Ainv, re- Hilbert space, this can be written as call the decomposition of U ∈ Usym from Lemma5. We can regard Usym as a representation of several G ⊂ G . (16) sym copies of the group G, and thus further refine this However, there are other symmetries that are not decomposition via basic representation theory of fi- global translations. For example, for N = 2 parti- nite Abelian groups [73]. cles, the unitary U which acts on all basis vectors A major role is played by the characters of G. These are the homomorphisms χ : G → S1, i.e. the maps |g1, g2i as −1 2 from G to the complex unit vectors S1 := {z ∈ U|g1, g2i := |g2, g1 g2i (17) C, | |z| = 1} with χ(gh) = χ(g)χ(h). In other words, is a symmetry transformation, i.e. U ∈ Usym. −1 the characters are the one-dimensional irreducible Namely, |g1, g2i ∈ Hh for h = g1 g2, and U im- G ⊗2 representations (irreps) of , and these turn out to plements the global translation Ug(h) on Hh, where exhaust all irreps. The set of all characters of G is g(h) = h. denoted Gˆ. On the other hand, the transformation Denote the order of the group by n := |G|, then gn = e for all g ∈ G [73]. Thus, every χ(g) must be V |g , g i := |g−1, g−1i (18) 1 2 2 1 among the n-th roots of unity: χ(g)n = 1. Moreover, is not a symmetry transformation: it satisfies condi- there are exactly n characters, i.e. |G|ˆ = n. tions 1. and 2. of Definition4, but violates condition Furthermore, note that dim Hh = n. We claim that 3. these subspaces can be decomposed as follows: M We will later see that QRF transformations corre- Hh = Hh;χ (19) spond to elements in Gsym \G. χ∈Gˆ
H 3.2 Invariant observables and Hilbert space de- with h;χ the one-dimensional subspace spanned by the vector composition 1 X Which observables can we internally measure in a |h; χi := χ(g−1)|g, hgi. (20) p|G| G-system, i.e. without access to the external relatum g∈G that was used to define the state space and the sym- Indeed, due to Ref. [73, Proof of Corollary III.2.3], metry group? These must be the observables that we have the well-known orthogonality relations are invariant under all symmetry transformations P 0 χ(g)χ (g) = nδχ,χ0 . Using this, direct cal- and which thus correspond to speakable informa- g∈G culation shows that the |h; χi are orthonormalized tion: states, and Definition 7 (Invariant observable). We define the in- U ⊗N |h; χi = χ(g)|h; χi for all g ∈ G. (21) variant subalgebra Ainv as g ⊗N Example 8. As a simple example, consider the cyclic Ainv = {A ∈ L(H ) | [U, A] = 0 for all U ∈ Usym}, group G = Zn = {0, 1, . . . . , n − 1} with addition where L(H) denotes the set of linear operators on modulo n, see Figure 4. This group can be inter- Hilbert space H. These are the operators A that are preted as a finite analogue of a part of the real line
Accepted in Quantum 2021-06-18, click title to verify. Published under CC-BY 4.0. 12 with periodic boundary conditions, by distributing Lemma 9. The subspace of relational states Hphys is finitely many possible positions along a ring. Its irre- isomorphic to H⊗(N−1). ducible representations and the respective characters i 2π kg To determine the invariant subalgebra, consider are given by χk(g) := e n with k ∈ {0, 1, . . . , n − any A ∈ L(H⊗N ) and develop it into the |h; χi- 1} [78]. Indeed, one can directly verify that the P 0 0 A = 0 0 a 0 0 |h; χihh ; χ | χ form one-dimensional representations of , and eigenbasis: h,h ,χ,χ h,h ,χ,χ . Us- k Zn U ∈ they are inequivalent. We explicitly obtain ing Eqs. (14) and (21), conjugation with some Usym yields n−1 1 X −i 2π kg † X 0 −1 0 0 |h; χ i = √ e n |g, g + hi , (22) UAU = χ(g )χ (g 0 ) a 0 0 |h; χihh ; χ |. k n h h h,h ,χ,χ g=0 h,h0,χ,χ0 where g +h means that g is added to each component This is equal to A for all U if and only if for of h, modulo n. Similarly, one can directly verify all h, h0, χ, χ0, one of the following is true: either 0 that ah,h0,χ,χ0 = 0 or χ(gh) = χ (gh0 ) for all possible ⊗N i 2π kg choices of gh, gh0 . The latter condition is automat- U |h; χ i = e n |h; χ i . (23) g k k ically satisfied if χ = χ0 = 1. Thus, all operators A
One can see that the |h; χki are obtained via a kind that are fully supported on the relational subspace of discrete Fourier transform [77] from the classical Hphys will be elements of Ainv. Let us denote the set configurations, and therefore they are reminiscent of of such operators by Aphys, then we have just shown momentum eigenstates. that Aphys ⊂ Ainv. For reasons that will become Since elements of Usym translate all particles by clear later, we will call the observables in Aphys re- the same amount, and momentum is the generator of lational observables. translations, one may identify |h; χki with the eigen- Now consider the other cases in which at least one 0 0 0 states of total momentum. However, since we are of χ or χ differs from 1. Clearly, if h = h and χ = χ 0 not explicitly interested in dynamics in this paper, then the character condition χ(gh) = χ (gh0 ) is triv- we will postpone any elaboration on this analogy to ially satisfied, and ah,h,χ,χ does not need to be zero. 0 0 our upcoming work, Ref. [72]. Consider the case h = h and χ 6= χ . Choosing any 0 gh with χ(gh) 6= χ (gh) shows that we must have From Eqs. (14)–(21) it is clear that the subspace 0 ah,h,χ,χ0 = 0. Finally, if h 6= h and at least one of χ spanned by the eigenstates with trivial character 0 or χ (say, χ) differs from 1, choose gh0 = e and gh χ = 1 is the subspace of Usym-invariant states, |ψi = such that χ(gh) 6= 1. This violates the character con- U|ψi for all U ∈ Usym. We denote it by dition and implies ah,h0,χ,χ0 = 0. In summary, we M n N−1o have proven the following: Hphys := Hh;1 = span |h; 1i | h ∈ G . h∈GN−1 Lemma 10. The invariant algebra consists exactly of (24) the block matrices of the form We have equipped the total invariant subspace with the label “phys” because it is the finite group ver- A = A ⊕ M M a |h; χihh; χ| , sion of the so-called physical Hilbert space of con- inv phys h;χ h∈GN−1 χ6=1 straint quantization. When the symmetry group is (25) generated by (self-adjoint) constraints, the physical where Aphys ∈ Aphys is supported on the relational Hilbert space corresponds to the set of solutions to subspace Hphys defined in Eq. (24), and the ah;χ are the quantum constraints and is thereby precisely the complex numbers. Hilbert space on which the group acts trivially. It is usually called ‘physical’ because quantum states A few words are in place regarding the physical of a gauge system are required to satisfy the con- interpretation of these observables. Due to Eq. (20), straints imposed by gauge symmetry. Nevertheless, χ labels the irreps of the global translations on state we will see that we can give quantum states that are space. As already mentioned in Example8, they can not invariant under the symmetry group a useful thus be interpreted as a discrete analog of (an expo- physical interpretation, and we will clarify their re- nentiated version of) total momentum. We can hence lation with the ‘physical’ states in Hphys in Ref. [72]. interpret the operator |h; χihh; χ| as describing a Being spanned by the states |h; 1i which encode the projective measurement that asks whether the rela- particle relations in an invariant manner, we shall tion between the particles is h, and whether the total mo- henceforth also refer to Hphys as the subspace of re- mentum corresponds to χ. Since this operator is con- N−1 lational states. Its dimension is |G| , and thus: tained in Ainv, this measurement can be performed
Accepted in Quantum 2021-06-18, click title to verify. Published under CC-BY 4.0. 13 by an observer without access to the external ref- Lemma 11. The two coherent group averaging oper- 0 erence frame. In the special case if χ = 1, i.e. on ations coincide, Πphys = Πphys, and Πphys is the or- the relational subspace Hphys which corresponds to thogonal projector onto the relational subspace Hphys.
“total momentum zero”, such an observer can also † perform measurements that correspond to superpo- Proof. Direct calculation shows that Πphys = Πphys 0† 0 0 02 sitions of different particle relations h. However, for and Πphys = Πphys, and that Πphys = Πphys and 2 0 “non-zero total momentum” (χ 6= 1), we obtain an Πphys = Πphys. Thus, Πphys and Πphys are orthogo- emergent superselection rule that forbids such su- nal projectors. Since H⊗N is spanned by the |g, hgi perpositions and the corresponding measurements. N−1 0 for g ∈ G and h ∈ G , the image im(Πphys) of 0 The reader familiar with constraint quantization Πphys is spanned by will notice that the invariant observables Aphys on 1 X |h; 1i the subspace Hphys are the finite group analog of Π0 |g, hgi = U ⊗N |g, hgi = .(27) phys |G| g0 p so-called Dirac observables [29, 30, 59]. Given some g0∈G |G| continuous group that is generated by an algebra 0 Since these states span Hphys, this proves that Πphys of constraints, Dirac observables are operators that is the orthogonal projector onto the physical sub- commute with the constraint operators (up to terms space. By construction, every |ψi ∈ im(Πphys) is in- proportional to the constraints themselves). As variant under every U ∈ Usym, and thus in particular such, they are invariant under the group generated ⊗N under every U ∈ Usym. Thus, im(Πphys) ⊆ Hphys. by the constraints and observables on solutions to g On the other hand, decomposing U ∈ Usym as in (14), the constraints, i.e. on the so-called physical Hilbert we get space. 1 There is, however, a subtlety in this analogy: usu- Π |h; 1i = X U ⊗N |h; 1i = |h; 1i phys gh ally the (continuous) group generated by the con- |Usym| U∈Usym straints would be the analog of the ‘classical’ group (28) G given here which is a strict subgroup of Gsym. since |h; 1i is invariant under global translations. Thus, it is natural to ask whether the Gsym-invariant 0 Thus, im(Πphys) ⊇ Hphys, and so Πphys = Πphys. subspace Hphys is a strict subset of the subspace of G-invariant states. Accordingly, one may wonder In conclusion, any basis state in Hh projects to the whether the entire invariant algebra Ainv defined in same invariant subnormalized state Πphys |g, hgi = 0 0 1 G Πphys |g , hg i = √ |h; 1i under coherent group terms of invariance under the larger group sym in |G| Definition7 is a strict subset of the algebra that is averaging, and it does not matter whether one av- G invariant under the smaller group . It is this latter erages with respect to the larger group Gsym or the algebra which thus gives rise to the actual analog of smaller G. Dirac observables for the finite groups considered However, we will now see that the set of invari- here. We will address these questions in the next ant observables, i.e. the observables resulting from subsection. incoherent group averaging (G-twirling), differs for the two choices, but only outside of the relational 3.3 Group averaging states subspace Hphys. These operations are defined by
1 X † Although we work with a representation of the Πinv(ρ) := UρU , (29) |Usym| larger group Gsym, Eqs. (14)–(24) indicate that the U∈Usym total Hilbert space decomposes naturally in terms 1 Π0 (ρ) := X U ⊗N ρ(U ⊗N )†. (30) of the representation of the smaller group G; e.g., the inv |G| g g physical Hilbert space is also precisely the subspace g∈G invariant under G. We will now clarify this obser- It is well-known, and easy to check by direct cal- 2 vation by considering the corresponding (coherent) culation, that these maps are projectors, i.e. Πinv = 02 0 group averaging operations, Πinv and Π inv = Πinv, and that they are orthogonal with respect to the Hilbert-Schmidt inner product, 1 X 0 1 X ⊗N ⊗N Πphys := U, Πphys := Ug , i.e. for all A, B ∈ L(H ), |Usym| |G| U∈Usym g∈G † † (26) tr A Πinv(B) = tr Πinv(A) B . (31) which are standard in constraint quantization [30, ⊗N 61, 62], and for which the following holds. If A ∈ L(H ) satisfies [U, A] = 0 for all U ∈ Usym, then Πinv(A) = A. Conversely, if B ∈ im(Πinv), then
Accepted in Quantum 2021-06-18, click title to verify. Published under CC-BY 4.0. 14 † UBU = B, i.e. [U, B] = 0, for all U ∈ Usym. Thus, demand that global classical translations are sym- Πinv projects into the invariant algebra Ainv. Simi- metries, then these superpositions remain allowed. 0 larly, Πinv projects into Let us return to the analogy with contraint quantization discussed in the previous subsection. A0 = {A ∈ L(H⊗N ) | [U ⊗N ,A] = 0 for all g ∈ G}. inv g Lemma 11 and Theorem 12 show that an observ- 0 Clearly, Ainv ⊆ Ainv, but are these algebras equal? able Aphys, i.e. the analog of a Dirac observable in The following lemma collects the above insights, our context, does not depend on whether it is con- and answers this question in the negative. structed relative to Gsym or its subgroup G. Later, we will also see that Aphys is the finite group ana- Theorem 12. Πinv is the orthogonal projector onto log of the algebra generated by so-called relational the invariant subalgebra Ainv. It can also be written in the form Dirac observables. These are invariant observables that encode relations between the subsystems, and X Πinv(ρ) = ΠphysρΠphys+ hh; χ|ρ|h; χi|h; χihh; χ|. they are common use in canonical quantum grav- h,χ6=1 ity [29–37,42–45]. This explains why we have called 0 the observables in Aphys “relational observables”. Similarly, Πinv is the orthogonal projector onto the strictly larger subalgebra They will become crucial in the resolution of the paradox of the third particle in Section4, and they will turn out to be tomographically complete for the A0 = M A = A ⊕ M A , (32) inv χ phys χ QRF states which we introduce in the next subsec- ˆ χ6=1 χ∈G tion. where every Aχ is an arbitrary operator supported on Recall the notion of equivalence classes [ρ] from N−1 the subspace Hχ := span{|h; χi | h ∈ G } (note Section2. We are now ready to introduce this notion that H1 = Hphys, so Aphys = A1). formally for G-systems: Proof. To see the claimed form of Π , note that inv Definition 13. We call two quantum states ρ, σ ∈ the combination of projections is a Hilbert-Schmidt- S(H⊗N ) symmetry-equivalent, and write ρ ' σ, if orthogonal projection with image A . It remains to inv there exists some symmetry U ∈ U such that be shown that A0 has the claimed form. Note that sym inv σ = UρU †. We call them observationally equivalent, g 7→ U ⊗N is a representation of the finite Abelian g and write ρ ∼ σ, if tr(Aρ) = tr(Aσ) for all invariant group G. It thus decomposes into one-dimensional observables (and thus all operators) A ∈ A . irreps, and the equivalence classes of irreps are la- inv belled by the characters χ. Thus, the form of A0 inv Clearly, if ρ ' σ then ρ ∼ σ, but the converse is follows again from Schur’s lemma. not in general true. The equivalence class [ρ] from This theorem has interesting implications for the Section2 can now be defined as [ρ] = {σ | σ ' ρ}. 0 physical properties of G-systems S. Recall our ini- In the case of pure state vectors |ψi, |ψ i, we must 0 tial scenario as depicted in Figure3. Suppose that allow global phases and write ψ ' ψ if and only if ∗ 0 we only demand symmetry of S with respect to ordi- there is some U ∈ Usym such that |ψ i = U|ψi. ⊗N nary, unconditional global translations Ug , and ask Observational equivalence can be characterized which observables can be measured by an observer in terms of the projection into the invariant subal- without access to the external relatum. The answer gebra: 0 is: all observables in Ainv. On the other hand, if we demand symmetry with respect to all conditional Lemma 14. Two states ρ and σ are observationally global translations in Usym — and we will soon see equivalent, i.e. ρ ∼ σ, if and only if that the QRF transformations of Refs. [38–40] are among those — then this turns out to be a more Πinv(ρ) = Πinv(σ). (33) stringent requirement. In this case, fewer observ- ables are measurable, namely only those in Ainv. Proof. This follows from the chain of equivalences In this sense, QRF transformations have fewer frame-independent observables than classical trans- ρ ∼ σ formations: if all QRF transformations are symme- ⇔ hA, ρiHS = hA, σiHS ∀A ∈ Ainv ⊗N tries, then superpositions of different particle rela- ⇔ hΠinv(B), ρiHS = hΠinv(B), σiHS ∀B ∈ L(H ) tions h are forbidden by an emergent superselec- ⊗N ⇔ hB, Πinv(ρ)iHS = hB, Πinv(σ)iHS ∀B ∈ L(H ) tion rule whenever the “total momentum is non- ⇔ Π (ρ) = Π (σ). zero”, i.e. χ 6= 1. On the other hand, if we only inv inv
Accepted in Quantum 2021-06-18, click title to verify. Published under CC-BY 4.0. 15 3.4 Alignable states as states with a canonical rep- Lemma 16. For every i ∈ {1,...,N} and h ∈ GN−1, resentation there is a unique basis vector |gi ∈ Hh with gi = g, −1 −1 namely |hi−1g, hhi−1gi, with h0 := e the unit ele- With these technical insights at hand, we are ready ment of G. to return to the discussion of Section2. In the structural approach to QRFs, we ask whether a Proof. If |gi ∈ Hh then |gi = |l, hli for some l ∈ G. given state has a natural representation, depend- Hence gi = hi−1l which is equal to g if and only if −1 ing only on internal data, such that the communi- l = hi−1g. cation task of Figure5 can be successfully accom- |ψ0i plished. In the following, let us focus on pure states This allows us to show that the state in Defini- |ψi ∈ H⊗N for simplicity. Our task is to find an- tion 15 is unique, and thus defines indeed a natural other state |ψ0i ∈ H⊗N that is symmetry-equivalent representation of the symmetry-equivalence class of |ψi to |ψi and that is in some sense distinguished, i.e. : yields a “canonical choice” for describing the set of Lemma 17. If |ψi ∈ H⊗N is i-alignable, then the symmetry-equivalent states, cf. Section 2.2.2. state |ϕi¯i in Definition 15 is unique up to a global In general, there may be many different possible phase. ways to define such a “canonical choice”. Let us pick one possible choice. Suppose that we fix one Proof. Suppose that both ψ0 and ψ˜0 are states that of the particles, say, particle i, where 1 ≤ i ≤ N. satisfy the conditions of Definition 15. In particular, Can we set the external reference frame such that this means that ψ ' ψ0 and ψ ' ψ˜0, and so there ∗ ˜0 0 this particle ends up at the “origin” — the unit ele- exists some U ∈ Usym such that |ψ i = U|ψ i, and iθ 0 ment of the group? In other words, can we align our U = e V for V ∈ Usym. In |ψ i = |eii ⊗ |ϕi¯i and ˜0 state “relative to particle i”? This is certainly pos- |ψ i = |eii⊗|ϕ˜i¯i, we decompose ϕ andϕ ˜ into product P sible in classical mechanics of N point particles in basis vectors: |ϕi = g∈GN−1 αg|g1, . . . , gN−1i, and one dimension, given translation-invariance. Clas- similarly for |ϕ˜i with amplitudesα ˜g. This implies sically, it would indeed define us a unique represen- that tation. We will now see that a similar construction X U αg|g1, . . . , gi−1, e, gi, . . . , gN−1i can be done for G-systems, and that it leads to the g∈GN−1 notion of QRF of Refs. [38, 40]. X = α˜g|g1, . . . , gi−1, e, gi, . . . , gN−1i. (35) Definition 15. Let i ∈ {1, 2,...,N}. A pure state g∈GN−1 |ψi ∈ H⊗N is called i-alignable if there exists some Now, according to Lemma 16, both of the decompo- state |ψ0i ∈ H⊗N with ψ ' ψ0 such that P sitions g∈GN−1 ... contain at most one basis vec- tor from every subspace H with non-zero ampli- 0 0 h |ψ i ≡ |ψ i1,...,N = |eii ⊗ |ϕi1,...,i−1,i+1,...,N . (34) −1 −1 tude, namely |hi−1, hhi−1i. But since U (and thus V ) leaves the subspaces H invariant, the last equa- In the following, we will also use the notation |ϕi h ¯i tion implies that V |h−1 , hh−1 i = |h−1 , hh−1 i for for the vector |ϕi . i−1 i−1 i−1 i−1 1,...,i−1,i+1,...,N every such vector that appears with non-zero ampli- −iθ tude αg 6= 0 (and thusα ˜g 6= 0). Henceα ˜g = e αg, The state |ϕi¯ in Definition 15 is exactly what is in- i and soϕ ˜ = e−iθϕ. terpreted in Refs. [38–40] as “the state of the remain- ing N − 1 particles as seen by particle i”. Similarly, we will thus interpret |eii ⊗|ϕi¯i as the description of 3.5 QRF transformations as symmetry group ele- the N particle system relative to the QRF ‘perspec- ments tive’ defined by particle i (which defines the origin). In the structural approach in Refs. [38–40], we can Not all pure states are i-alignable. For exam- “jump” from one particle’s reference frame into any ple, the relational state |h; 1i is an element of the ∗ other’s. How is this idea expressed in our formal- subspace Hphys, hence every U ∈ Usym satisfies iθ ism? To see this, let us first show the following. U|h; 1i = e |h; 1i for some θ ∈ R. Thus this state cannot be i-alignable for any i. While devoid of Theorem 18 (QRF state transformations). If there alignable states, we will see later that Hphys con- is some i ∈ {1, 2,...,N} such that |ψi is i-alignable, tains the complete relational information about all then |ψi is j-alignable for every j ∈ {1, 2,...,N}. alignable states. We will then simply call |ψi alignable. Moreover, for To analyze this notion further, the following every i, j ∈ {1,...,N}, there is a unique symmetry lemma will be useful. transformation U ∈ Usym such that U (|eii ⊗ |ϕi¯i) =
Accepted in Quantum 2021-06-18, click title to verify. Published under CC-BY 4.0. 16 |eij ⊗|ϕi¯j for all |ϕi¯i. Furthermore, if i 6= j then U is Example 19. Suppose we have N = 4 particles, and a proper conditional global translation, i.e. U • U † ∈ an alignable state |ψi such that Gsym \G. Every such U induces a unique unitary |ψi ' |ei ⊗ |g , g , g i. (38) (“QRF transformation”) Vi→j such that Vi→j|ϕi¯i = 2 1 3 4 |ϕi¯j. This transformation can be written Thus, the state relative to the second particle is X −1 ⊗(N−2) Vi→j = Fi,j |g ihg|j ⊗ Ug−1 , (36) |g1, g3, g4i. To determine the state relative to the g∈G third particle, compute where flips (swaps) particles i and j. This is the Fi,j X −1 ⊗2 discrete version of the form given in Refs. [38, 40]. V2→3|g1, g3, g4i = F2,3 |g ihg|3 ⊗ Ug−1 |g1, g3, g4i g∈G Proof. Fix i, j ∈ {1,...,N}. For every h ∈ GN−1, = |g−1ihg | ⊗ U ⊗2 |g , g , g i −1 F2,3 3 3 g−1 1 3 4 let gh := hj−1hi−1 (setting, as before, h0 := e). Then 3 −1 −1 −1 the global translation by gh satisfies = F2,3| g3 g1, g3 , g3 g4i | {z } |{z} | {z } U ⊗N |h−1 , hh−1 i = |h−1 , hh−1 i. (37) 1 3 4 gh i−1 i−1 j−1 j−1 = | g−1g , g−1, g−1g i, (39) L ⊗N 3 1 3 3 4 Set U := N−1 U , then U ∈ U . According h∈G gh sym | {z } |{z} | {z } to Lemma 16, for every h, U maps the unique basis 1 2 4 vector |gi ∈ Hh with gi = e to the unique basis where the integers at the bottom denote the particle 0 0 vector |g i ∈ Hh with gj = e, and it is clear that U labels. is the only symmetry transformation that does this.
Thus, U maps all states of the form |eii ⊗ |ϕi¯i to In Ref. [72], we will demonstrate equivalence of states of the form |eij ⊗ |ϕ˜i¯j. Furthermore, if i 6= j the above QRF transformations with the “quantum then there exist h, j such that gh 6= gj. Thus, any coordinate changes” of the perspective-neutral ap- such U is an h-dependent transformation and thus proach [39, 41–45]. cannot be a global translation. As Theorem 18 has shown, the symmetry group Fix an arbitrary orthonormal basis {|ϕi¯i}ϕ of Usym contains the QRF transformations which ⊗(N−1) H , then U (|eii ⊗ |ϕi¯i) = |eij ⊗ |ϕi¯j yields an- switch from state descriptions relative to particle i other orthonormal basis. Thus, we can view this as to descriptions relative to particle j. But these do ⊗(N−1) a unitary Vi→j from H into another copy of not exhaust the symmetry group. The following H⊗(N−1). Since its action on basis vectors is fixed, lemma gives another physically motivated exam- there can be no more than one such map. To deter- ple. mine that it has the form as claimed, simply look at its action on the basis vectors |g1, . . . , gN−1i. Example 20 (Center of mass). Consider again the cyclic group Zn with addition modulo n, as shown So indeed, for every alignable state, and any par- in Figure1. Let m1, . . . , mN be non-negative real j ∈ {1,...,N} N−1 ticle , there is a unique representation numbers and m := m1 + ... + mN . For h ∈ G , of that state “relative to the jth particle”. Further- define the group elements more, the QRF transformation from i’s to j’s ‘per- spective’ at the level of the full Hilbert space H⊗N 1 g(h) := − (m h + ... + m h ) , (40) corresponds to a symmetry transformation which m 2 1 N N−1 lies in Gsym, but not in G (if i 6= j). This observa- L ⊗N tion highlights the physical significance of the sym- and set U := h∈GN−1 Ug(h). If we interpret the mi metry group Gsym. While we have seen that the set as the masses of the particles, with the origin as the of invariant states Hphys is independent of whether position of particle 1 (i.e. h0 = 0), then this symme- one constructs it through coherently averaging over try transformation U describes a change of quantum Gsym or its ‘classical translation subgroup’ G, the coordinates such that the (integer part of) the “center symmetry group Gsym is key for understanding the of mass” becomes the origin. meaning of the QRF transformations (which trans- condi- form non-invariant descriptions): they are 3.6 Characterization of alignable states tional symmetry transformations that depend on the interparticle relation h. We have already seen that not all global states are To clarify the notation used in the definition of the alignable. The following lemma characterizes those QRF transformation Vi→j, we give a simple exam- that are. ple.
Accepted in Quantum 2021-06-18, click title to verify. Published under CC-BY 4.0. 17 Lemma 21. A state |ψi ∈ H⊗N is alignable if and Proof. Using (27), we obtain only if it can be written in the form 1 X X Πphys|ψi = p αh|h; 1i. (43) |ψi = αh|gh, h ghi (41) |G| h∈GN−1 h∈GN−1 Direct calculation shows that |hψ|h; χi|2 = |α |2/|G|. for some gh ∈ G and αh ∈ C. Moreover, the h The result then follows by using the form of Π as αh characterize the alignable state up to symmetry- inv equivalence. That is, for two alignable states |ψi and given in Theorem 12. Both notions of equivalence |ψ0i, we have ψ ' ψ0 if and only if their coefficients boil down to the fact that the states have the same iθ 0 amplitudes αh up to a global phase. Finally, for ρ satisfy αh = e αh for some θ ∈ R. any alignable state, the above form of Πinv(ρ) implies Proof. If |ψi is alignable, then it is in particular 1- alignable. That is, there exists U ∈ U ∗ such that tr(ρA) (44) sym X = tr(ρphysA) + hh; 1|ρphys|h; 1itr(Πh;χ6=1A) iθ X ⊗N |ψi = U |ei ⊗ |ϕi = e α U |e, hi, N−1 1 1¯ h gh h∈G h∈GN−1 for all A ∈ Ainv, where ρphys := ΠphysρΠphys. That where the αh are the coefficients of |ϕi1¯ in terms is, the expectation values of all invariant observables, of the product group basis, and the U ⊗N translate and thus the notion of observational equivalence, de- gh the basis vectors of the subspaces Hh. Hence we get pends only on the state’s projection into Aphys. the claimed form for |ψi. The converse direction of the proof of the first part of this lemma is analo- In other words, the algebra Aphys is tomograph- ically complete for the invariant information in gous. We also see that the αh only depend on the symmetry-equivalence class of the state |ψi (up to a alignable states. In this sense, it can be said that global phase); and, in the case of equality of those the external relatum independent states of the struc- tural approach are what we called the relational coefficients, two states must have the same |ϕi1¯ (up to a global phase), hence they must be symmetry- states, namely the ones in Hphys. That is, the ex- equivalent. ternal relatum independent states of the structural approach are the coherently group-averaged states, The above form shows that any maximal sub- while their counterparts in the QI approach are space of H⊗N which is contained in the set of the incoherently group-averaged states (cf. Subsec- alignable states has dimension |G|N−1, i.e. is isomor- tion 2.2.1).6 phic to H⊗(N−1). This shows that the QRFs as de- fined above are indeed “good” QRFs as explained 3.7 Alignable and relational observables in Subsection 2.2.2. Due to Lemma9, it also shows that the relational subspace Hphys has the right di- Given the notion of alignable states and the duality mension for its states to contain “all the particles’ between states and observables, it is natural to also internal QRF perspectives at once”. This observa- define alignable observables in the obvious way. tion is corroborated by the following useful Lemma Definition 23 (Alignable observables). An operator and will be further discussed in Ref. [72]. A ∈ L(H⊗N ) is called i-alignable for i ∈ {1,...,N} Lemma 22. Given any alignable state |ψi (which is if there exists U ∈ Usym such that hence of the form (41)), its projection onto the in- † variant subalgebra is UAU = |eihe|i ⊗ A¯i (45) ⊗(N−1) Πinv(|ψihψ|) (42) for some A¯i ∈ L(H ). If A is an observable, it α α |α |2 is called an i-alignable observable. = X h j |h; 1ihj; 1| + X h Π , |G| |G| h;χ6=1 h,j∈GN−1 h∈GN−1 This leads to the following extension of Theo- P rem 18 from QRF state to observable transforma- where Πh;χ6=1 := χ6=1 |h; χihh; χ|. Thus, for any two alignable states |ψi, |ψ0i, we have ψ ∼ ψ0 if tions. The proof is analogous and thus omitted. 0 and only if ψ ' ψ : such states are symmetry- Theorem 24 (QRF observable transformations). If equivalent if and only if they are observationally A ∈ L(H⊗N ) is i-alignable, then it is also j-alignable equivalent. Furthermore, they are equivalent if and for every j ∈ {1,...,N}, and so we will call A simply 0 0 only if hψ|A|ψi = hψ |A|ψ i for all A ∈ Aphys; that is, all invariant information of alignable states is fully 6We thank A. R. H. Smith for suggesting us to emphasize this technical distinction. contained in their projection Πphys|ψi.
Accepted in Quantum 2021-06-18, click title to verify. Published under CC-BY 4.0. 18 alignable. In particular, there is a unique U ∈ Usym alignable observable A. Hence, in particular we such that have
† FA¯,i = FA¯,j for all i, j ∈ {1,...,N}, (49) U (|eihe|i ⊗ A¯i) U = |eihe|j ⊗ A¯j, (46) i j and so the relational observable in Eq. (48) does not where A = V A V . Here, U is the unique ¯j i→j ¯i j→i depend on the choice of particle i. symmetry transformation from Theorem 18 such that This systematic equivalence of relational observ- U (|ei ⊗ |ϕi ) = |ei ⊗ |ϕi for all |ϕi and V is i ¯i j ¯j ¯i i→j ables is once more made possible by studying the the unitary “QRF transformation” induced by it. larger symmetry group Gsym rather than its sub- G These are the discrete versions of the observable group as usual in the literature. Indeed, Theo- transformations in Refs. [38, 39]. rems 18 and 24 demonstrate that any two particle P alignments of an observable are related by a unique Finally, note that if A1¯ = h,j∈GN−1 ah,j|hihj|, then symmetry group element which generically lies in Gsym \G. X ah,j Πˆ |eihe| ⊗ A¯ = |h; 1ihj; 1|, Lemma 22 (and its obvious generalization to phys 1 1 |G| h,j∈GN−1 alignable observables) yields an interesting insight (47) that will become relevant in Section4: if we look where Πˆ phys henceforth denotes the superoperator at the image of all alignable states and observables that acts as Πˆ phys(A) = Πphys A Πphys, as obvious under Πinv, then we do not obtain the full invariant from Lemma 22. Up to a factor of |G|, this is identical algebra Ainv. Instead, we always obtain an operator to the original representation of A1¯, but in another in the smaller subalgebra basis. This proves the following theorem, extending X a result from [44, 45]: Aalg := Aphys + ahΠh;χ6=1 , (50) h∈GN−1 Theorem 25. Consider the map where Aphys ∈ Aphys, and ah ∈ C are arbitrary com- A 7→ F := |G| · Πˆ |eihe| ⊗ A , (48) plex numbers. According to the definitions in Sub- ¯i A¯i,i phys i ¯i section 3.3, this gives us the subalgebra inclusions where A ∈ L(H⊗(N−1)) is any operator. This de- 0 ⊗N ¯i Aphys ⊂ Aalg ⊂ Ainv ⊂ Ainv ⊂ L(H ). (51) fines an isomorphism between the operators A¯ and i Hence, if we denote the orthogonal projection into Aphys, preserving products, linear combinations, and adjoints. Aalg by Πalg, we have Πalg ◦Πinv = Πinv ◦Πalg = Πalg. More specifically, the following holds.
This gives us two independent motivations to fo- Lemma 26. Aalg is the smallest subalgebra of A ⊗N cus on phys: for alignable states, the projection into L(H ) that contains Πinv(|ψihψ|) for all alignable this subalgebra contains all invariant information; states |ψi. and it does so in a way that preserves the natural Proof. It is clear that the rank-one projectors |ψihψ| structure of the alignable observables. for |ψi = |ei ⊗ |ϕi linearly span all of |eihe| ⊗ Furthermore, returning to the comparison with 1 1¯ 1 L(H⊗(N−1)), thus we are looking for the subalgebra constraint quantization, it follows from Refs. [44,45] A that is generated by the image of these operators that FA ,i is (the finite group analog of) the relational ¯i under Π . Lemma 22 shows that Π (|ψihψ|) is Dirac observable which encodes in an invariant man- inv inv contained in Aalg for every alignable state |ψi, hence ner the question “what is the value of A¯i given that P A ⊆ A . Conversely, if A = N−1 a |hihj|, particle i sits at the origin?” Such relational observ- alg h,j∈G h,j then ables are a standard tool in canonical quantum grav- ity, e.g. see Refs. [29–37, 42, 43]. Theorem 25 is the Πinv(|eihe| ⊗ A) (52) reason why we refer to A as the algebra gener- X ah,j X ah,h phys = |h; 1ihj; 1| + Π . ated by relational observables. |G| |G| h;χ6=1 h,j∈GN−1 h∈GN−1 Due to Aphys ⊂ Ainv, we have Setting A := |hihj| for h 6= j shows that ˆ ˆ |h; 1ihj; 1| ∈ A. But then, we also have |h; 1ihh; 1| = Πphys (Πinv(|eihe|i ⊗ A¯i)) = Πphys (|eihe|i ⊗ A¯i) . |h; 1ihj; 1|)(|j; 1ihh; 1|) ∈ A, and so all operators Since Πinv is the incoherent G-twirl over Usym, it is Aphys fully supported on Hphys are in A. Finally, clear that the image Πinv(|eihe|i ⊗ A¯i) ∈ Ainv only considering the image of A = |h; 1ihh; 1| shows that depends on the symmetry equivalence class of the Πh;χ6=1 ∈ A, and so A ⊇ Aalg.
Accepted in Quantum 2021-06-18, click title to verify. Published under CC-BY 4.0. 19 3.8 Communication scenario of the structural ap- ever we write a + b, we actually mean a + b mod n. proach revisited As described earlier, this is a discrete model of the translation group acting on the real line. We start We are now in a position to revisit the communi- with two particles that have been prepared (by an cation scenario elucidating the operational essence external observer with access to the reference frame) of the structural approach to QRFs in section 2.2.2 in the state (see also Figure5). It is clear how Alice and Bob 1 iθ can win the game proposed by Refaella in the case |ψi = √ | − ai1|bi2 + e |ai1| − bi2 , (53) 2 of N particles on the configuration space G in the absence of a shared external relatum, given that where a, b ∈ {0, 1, 2, . . . , n−1}, and θ ∈ R. This state 7 the class of states C that they are interested in is is symmetry-equivalent to the set of alignable states. For example, before the 1 iθ |ψi ' |0i1 ⊗ √ |a + bi2 + e | − a − bi2 . (54) game begins, they can agree to always use particle 2 1 as the internal reference system relative to which In our terminology, this means that the state is the remaining particles are described. This yields alignable. In Ref. [50], this form is used as a mo- the “canonical choice” to represent any equivalence tivation to declare: “Therefore, we conclude that par- class of (pure) states in the form |ei1 ⊗ |ϕi¯ and Al- 1 ticle 1 sees particle 2 in a pure state. Importantly this ice’s and Bob’s return to Refaella will always agree. implies that particle 1 can get access to the phase θ by in- Alice and Bob have, of course, the option to teracting with particle 2 alone, i.e. without access to the choose any of the N particles as a reference sys- external reference frame.” In our conceptual frame- tem, each likewise defining a “canonical choice”. It work, we would rather describe the situation as fol- is clear that all these different possible conventions lows: consider an external observer who has access by Alice and Bob are precisely related by the QRF to particles 1 and 2, but has no access to the external transformations of Lemma 18 and that each such reference frame. There are some observables that transformation is an element of the symmetry group this observer can measure for which the phase θ is U . sym relevant. This is because the state that is effectively seen by 4 The paradox of the third particle and this observer is the projection of ψ into the invariant subalgebra, which we can determine via Lemma 22. the relational trace √ The coefficients of this√ state are αh = 1/ 2 for h = iθ a + b and αj = e / 2 for j = −a − b, thus Ref. [50] presents a “paradox of the third particle” 1 e−iθ in the context of QRFs. We will now formulate this Πinv(|ψihψ|) = |h; 1ihh; 1| + |h; 1ih j; 1| apparent paradox in our formalism, and see that the 2n 2n eiθ 1 structure of observables, as elaborated in Section3, + |j; 1ihh; 1| + |j; 1ihj; 1| helps to clarify its physical background and to re- 2n 2n 1 1 solve it in terms of relational observables. As we + Πh;χ6=1 + Πj;χ6=1 (55) will see, the core of the problem is how to embed 2n 2n the two-particle observables into the set of three- and this state depends on θ in a nontrivial way. particle observables, and the key will be to do so in Now a third particle is introduced. From the a relational manner. This bears some resemblance external perspective, it is prepared in a pure state to the issue of boundaries and edge modes in gauge |ci independently of the other two particles, where theory and gravity [64–71], which is related to the c ∈ {0, 1, . . . , n − 1}. From that perspective, the question of how to embed the gauge-invariant ob- global state thus reads servables of neighbouring subregions in spacetime 1 iθ |Ψi = √ | − ai1|bi2 + e |ai1| − bi2 |ci3. (56) into the set of gauge-invariant observables associ- 2 ated with the union (‘gluing’) of these subregions. This state is still alignable. Relative to particle 1, it The setup of Ref. [50] consists of three particles in becomes one dimension, i.e. on the real line R. But since only 0 |0i1 iθ a finite number of positions is relevant for the para- |Ψ i = √ |a + bi2|a + ci3 + e | − a − bi2| − a + ci3 2 dox, we can discretize space and its translations. (57) Thus, we consider a G-system with G = Zn, the cyclic group of order n, as in Fig.1. The group oper- 7In fact, in the sequel we will assume that a 6= 0, for oth- ation is addition modulo n; in the following, when- erwise the states in Eqs. (53) and (54) coincide.
Accepted in Quantum 2021-06-18, click title to verify. Published under CC-BY 4.0. 20 with Ψ0 ' Ψ. This is a state for which particles 2 or may not be unital). and 3 are formally entangled. Now suppose that Now consider any quantum state ρ123 on the three particle 3 is very far away, such that our external particles. We would like to determine the state ρ12 observer (or, as the authors of Ref. [50] would say, that results if one has only access to particles 1 and such that particle 1) has no access to particle 3. If 2. By this, we mean the state that gives the same one now formally takes the partial trace over parti- expectation values as ρ123 on all local observables cle 3, one obtains a reduced state of particles 1 and A12. Thus, we demand 2 that is independent of the phase θ. This seems to con- tr(ρ A ) = tr(ρ Φ(A )) for all A . (59) tradict our earlier claim — now it looks as if an ex- 12 12 123 12 12 ternal observer without access to the external refer- If we write this in terms of the Hilbert-Schmidt in- ence frame or to particle 3 cannot see any observable ner product, then consequences of the phase θ. hρ ,A i = hρ , Φ(A )i = hΦ†(ρ ),A i . We arrive at an apparent paradox: computing 12 12 HS 123 12 HS 123 12 HS 0 † † the partial trace via Ψ or via Ψ gives different pre- That is, we must have ρ12 = Φ (ρ123), with Φ the dictions, even though both states are symmetry- Hilbert-Schmidt adjoint of Φ. But given Eq. (58), it † equivalent. Moreover, the result from tracing out is easy to see that Φ = Tr3, and this recovers the the third particle via Ψ0 seems absurd, given the partial trace. seemingly innocuous role that the third particle In the context of QRFs and G-systems, we have a plays in state Ψ. Can the phase θ be accessed by different structure of observables: what is measur- an observer without access to the external relatum able without access to the external reference frame and with restricted access to only particles 1 and 2? corresponds to the invariant observables. We there- It seems like there should be an objective answer to fore need an analog of the above construction for this question which does not depend on whether it the invariant subalgebra Ainv. In fact, since we are is asked in the context of state Ψ or Ψ0. only interested in alignable states, Lemma 26 tells Clearly, since the usual partial trace yields differ- us that we can focus on the subalgebra Aalg. ing results for Ψ and Ψ0, it cannot represent the cor- rect rule to compute reduced states in the setting of 4.1 The non-uniqueness of invariant embeddings QRFs. To shed light on the reason for why this is the case, let us reconsider how the standard partial In light of the above considerations, let us try to trace can be motivated. Consider three distinguish- construct an embedding of the relevant 2-particle (N) able particles as one usually does in quantum infor- observables (or more generally of Aalg ) into the 3- mation theory — in our setting, this implies that the (N+M) particle observables (more generally into Aalg , state of the particles is defined relative to an acces- where the superscript denotes the number of parti- sible reference frame. Denote by A12 some observ- cles). To obtain some crucial physical intuition, we able that is measurable if one has access to particles will now define one such embedding (called Φ˜ (1)) in 1 and 2 only (and to the reference frame). Then one an intuitive manner, before we turn to a more sys- can equivalently describe this as an observable on tematic treatment below. the three particles, such that the third particle is ig- We start our construction with the orthogonal nored. Formally, this can be done via a map (N) P projector Πh := χ∈Gˆ |h; χihh; χ|. Its embedding A12 7→ Φ(A12) = A12 ⊗ 13. (58) (N+M) must be on orthogonal projector in Aalg ; mea- That is, the 2-particle observables are naturally em- suring this projector amounts to asking whether the bedded into the 3-particle observables via some first N particles have pairwise relations described map Φ, which takes the tensor product with the by h. Clearly, the answer must be “yes” whenever identity observable. This map preserves all relevant the M + N particles are in some joint relation (h, g) M structure (as it must): it takes linear combinations to for arbitrary g ∈ G , and the answer must be “no” 0 0 linear combinations, products to products, the ad- for all other relations (h , g) whenever h 6= h. This joint to the adjoint, and the identity to the identity. suggests to choose the embedding as Formally, this is summarized by saying that Φ is ˜ (1) (N) X (N+M) ∗ Φ (Πh ) = Πh,g . (60) a unital -homomorphism. It defines what we mean g∈GM when we talk about “observables pertaining only to particles 1 and 2” within the set of all 3-particle The reason for the superscript ‘(1)’ will become observables. In the following, we will refer to ∗- clear shortly. Now consider the orthogonal pro- (N) P homomorphisms simply as embeddings (which may jector Πphys := h∈GN−1 |h; 1ihh; 1|. A state |ψi
Accepted in Quantum 2021-06-18, click title to verify. Published under CC-BY 4.0. 21 ˜ (N) (N+M) is in the image of this projector if and only if it Now let Φ be any embedding of Aalg into Aalg ⊗N N−1 is translation-invariant, i.e. Ug |ψi = |ψi for all which satisfies Eq. (64). Choose h, j ∈ G arbi- g ∈ G N + M . Suppose that a state of particles trary, and let A1¯ be the operator with ah,j = |G| is translation-invariant; then we would like to be and all other coefficients zero. Then Eq. (65) be- able to say that the state of the first N particles is (N+M) (M) comes |h; 1ihj; 1|, and Πalg |eihe|1 ⊗ A1¯ ⊗ 1 translation-invariant, too. This motivates us to de- becomes the right-hand side of Eq. (62). Now, still Φ˜ (1)(Π(N) ) = Π(N+M) mand phys phys : in other words, we for h 6= j, embed the translation-invariant observables of N particles into the translation-invariant observables (|h; 1ihj; 1|)† (|h; 1ihj; 1|) = |j; 1ihj; 1|. (66) of N + M particles. (N) (N) ˜ (1) We have |h; 1ihh; 1| = Πh Πphys. Since Φ is Since Φ˜ preserves adjoints and products, this proves supposed to be an embedding and thus multiplica- Eq. (62) also in the case h = j. Finally, choosing A1¯ tive, this implies such that ah,h = |G| and all other coefficients zero proves Eq. (63), which implies that Φ˜ is the linear Φ˜ (1)(|h; 1ihh; 1|) = X |h, g; 1ihh; g; 1|. (61) extension of Eqs. (62) and (63). g∈GM
(N) To exhaust all of A , we still need to embed the What this lemma demonstrates is that our con- alg ˜ (1) operators |h; 1ihj; 1| for h 6= j. Given Eq. (61), it struction of Φ can be interpreted in an alternative seems formally natural to define way — at least on “alignable” observables. What our embedding map does to those observables is as (1) X Φ˜ (|h; 1ihj; 1|) := |h, g; 1ihj, g; 1|. (62) follows: write them in the form |eihe|1 ⊗ A1¯, and embed g∈GM them into the total Hilbert space according to its defining tensor product structure. Then demand that Φ˜ (1) maps Natural as this definition may seem, we will soon the invariant part of |eihe|1 ⊗ A1¯ to the invariant part see that it relies on quite subtle physical assump- (M) of its embedding |eihe|1 ⊗ A1¯ ⊗ 1 . This defines a tions. For now, let us work with this definition (N) (N+M) particular, natural embedding of A into A . and explore its consequences. First, linearity and alg alg But this suggests directly that our candidate re- Eqs. (60) and (61) imply lational trace is defective: its definition is implicitly ˜ (1) X based on the choice of particle 1 as our reference. Indeed, Φ Πh,χ6=1 = Πh,g;χ6=1. (63) g∈GM the following lemma shows that there is a large class of invariant embeddings. In particular, choosing an- These demands yield an embedding that can equiv- other particle as the reference particle will in general alently be defined as follows: lead to inequivalent embeddings. The proof is given Lemma 27. There is a unique unital embedding Φ˜ (1) by a straightforward calculation and thus omitted. of A(N) into A(N+M) that satisfies alg alg Lemma 28. Let U ∈ Usym be any symmetry transfor- mation (for example, a QRF transformation). Then ˜ (1) (N) (N+M) (M) Φ Πalg (ˆe1 ⊗ A1¯) = Πalg eˆ1 ⊗ A1¯ ⊗ 1 ˜ U (N) there is a unique unital embedding Φ of Aalg into (64) (N+M) ⊗(N−1) Aalg which satisfies for all A1¯ ∈ L(H ), where eˆ1 := |eihe|1 and 1¯ := {2, 3,...,N}. It is given by the linear extension ˜ U (N) (N+M) † (M) of Eqs. (62) and (63). Φ Πalg (ˆe1 ⊗ A1¯) = Πalg U(ˆe1 ⊗ A1¯)U ⊗ 1 . (67) P ⊗N Proof. Write A¯ = h,j∈GN−1 ah,j|hihj| and use L 1 Writing U = h∈GN−1 Ug(h), it acts as Lemma 22 to obtain a (N) X h,j Φ˜ U (|h; 1ihj; 1|) Πalg (|eihe|1 ⊗ A1¯) = |h; 1ihj; 1| |G| X h,j∈GN−1 = |h, g(h)−1g; 1ihj, g(j)−1g; 1|, (68) a + X h,h Π . (65) g∈GM |G| h;χ6=1 N−1 ˜ U X h∈G Φ Πh;χ6=1 = Πh,g;χ6=1. (69) A similar representation can be obtained for the g∈GN right-hand side of Eq. (64). Thus, it is clear that U ˜ U (N) the linear extension of Eqs. (62) and (63) satisfies Via Φ := Φ ◦ Πalg , this extends to a completely Eq. (64). In particular, the unit is preserved. positive unital map ΦU : L(H⊗N ) → L(H⊗(N+M)).
Accepted in Quantum 2021-06-18, click title to verify. Published under CC-BY 4.0. 22 (N) (N) (N) 4
In particular, if we choose U as the QRF transforma- tion of Theorem 18 that “changes from the perspec- Figure 6: The state |ψi in (71) describes a superposition (indicated in yellow) of two particles (black dots) being either tive of particle 1 to particle i”, we obtain a natural one or two places apart. If there is a third particle (white dot) Φ˜ (i) i invariant embedding “relative to particle ”. It such that the resulting state is still identical to the pure state satisfies |ψi on the first two particles, then that particle should carry no information as to “which branch” is actualized. That ˜ (i) (N) (N+M) (M) Φ Πalg (ˆei ⊗ A¯i) = Πalg eˆi ⊗ A¯i ⊗ 1 , is, its relation to the first two particles should be the same (70) in both branches. But whether this is the case depends on our convention of how we define its relation to the first two and acts on the basis elements of Aphys as particles. This results in different embedding maps ΦU . ˜ (i) X −1 Φ (|h; 1ihj; 1|) = |h, g; 1ihj, hi−1ji−1g; 1|. g∈GM
For a better understanding of the physical reason then the local state of the first two particles will re- of this non-uniqueness of embedding, let us recon- main coherent if and only if the third particle inside sider our intuitive construction of Φ˜ (1) above. First, the configuration (h, g) has the same relation to the note that all the Φ˜ U satisfy Eq. (61), which had a first two particles as the third particle in the config- 0 clear physical motivation. However, the Φ(i) violate uration (j, g ). But the crucial insight is that this will Eq. (62), which we had motivated purely by formal depend on what we mean by “relation to the first two analogy. To shed light on Eq. (62) and its general- particles”. ization, Eq. (68), suppose for concreteness that we For instance, suppose that g = 4. If we choose the are interested in embedding N = 2 particles into convention to say that the relation is identical if the N + M = 3 particles, and our group G is the cyclic relation to the first particle is the same, then this will be 0 group Zn with addition modulo n. Consider the or- the case if g = g = 4. But if we demand instead that thogonal projector |ψihψ|, where the relation to the second particle is the same, then we need g0 = 5. Our definition of Φ(1) is implicitly rely- 1 |ψi = √ (|1; 1i + |2; 1i) . (71) ing on the former convention, while Φ(2) would rely 2 on the latter. The reason why Eq. (62) (for Φ = Φ(1)) (2) looks so simple is that we have implicitly labelled This is an element of Hphys, the 2-particle subspace the relations h ∈ GN−1 as relative to the first particle in of invariant states, and it describes a superposition all of this work, recall Eq. (11). This is no loss of gen- of two particles either being one or two places apart, erality, and it had no implications whatsoever for see Figure6. Note that there is no origin that would Section3, but here it becomes relevant. The factor locate the particles absolutely; all we have is their −1 of hi−1ji−1 in Eq. (68) adapts the convention. Note relations. that it does not alter the pairwise relations among Now suppose we would like to embed the cor- the first N particles, or the pairwise relations among responding observable |ψihψ| into the three-particle the last M particles, but only the relation between observables. The essence of the problem lies in em- the two groups of particles. bedding |h; 1ihj; 1|, where h = 1 and j = 2. The lo- Is there a way to escape the non-uniqueness of cal state of the two particles will remain coherent if embeddings via some formal construction that is the third particle carries no information on whether manifestly relational, but does not depend on a configuration h or j is actualized — that is, if the choice of reference within the N-particle subsystem properties of the third particle are the same in both relative to which the relation of the new M par- branches of the superposition. But the only proper- ticles is defined? The right-hand side of Eq. (70) ties of the third particle are its relations to the other shows that the maps Φ˜ (i) embed the invariant oper- two particles. Thus, if we complete |ψi to a state on Π(N)(ˆe ⊗ A ) three particles ator alg i ¯i by embedding the original, non- invariant operator eˆi ⊗A¯i into the total Hilbert space, 1 0 followed by the projection into the global subalge- |Ψi = √ |h, g; 1i + |j, g ; 1i , (72) ˜ (i) 2 bra Aalg. While the resulting map Φ is invariant,
Accepted in Quantum 2021-06-18, click title to verify. Published under CC-BY 4.0. 23 its definition is therefore not. Can we perhaps make A similar argument applies if we try to embed (N) (N+M) eˆ ⊗A¯ the definition invariant by embedding not i i di- Ainv into Ainv : the analog of the above construc- rectly, but its invariant part? The following lemma (N+M) (N+M) tion, with Πalg replaced by Πinv , does not answers this question in the negative: yield a valid embedding. ˜ (N) (N+M) Lemma 29. Define the map Φ: Aalg → Aalg as
˜ (N) (N+M) (N) (M) Φ(A ) := Πalg A ⊗ 1 . (73)
Then this map is not a valid embedding. Namely, it 4.2 A class of invariant traces is not in general multiplicative, i.e. there exist A, B ∈ A(N) with Φ(˜ AB) 6= Φ(˜ A)Φ(˜ B). alg For every embedding ΦU of Lemma 28, we obtain a Proof. A tedious but straightforward calculation corresponding “invariant trace”: yields 1 Φ(˜ |h; 1ihj; 1|) = X X |h, g; 1ihj, gg; 1| |G| ⊗N g∈G g∈GM L Lemma 30. For every U = h∈GN−1 Ug(h) ∈ Usym, 1 U + δ X Π . (74) define a corresponding “invariant trace” Trinv := h,j h,g;χ6=1 † |G| U g∈GM Φ . It is trace-preserving, and it maps invari- But then, for h 6= j, we obtain ant operators to invariant operators. In particular, TrinvU A(N+M) = A(N), and can be explicitly ˜ ˜ ˜ (M) alg alg Φ(|h; 1ihj; 1|)Φ(|j; 1ihh; 1|) 6= Φ(|h; 1ihh; 1|). written in the following form:
tr(Π ρ) TrinvU ρ = X |h; 1ihj; 1| X hh, g(h)−1g; 1|ρ|j, g(j)−1g; 1i + X Π X h,g;χ6=1 . (M) h;χ6=1 |G| − 1 h,j∈GN−1 g∈GM h∈GN−1 g∈GM
We omit the straightforward proof. straightforward calculation gives Since the different invariant traces formalize dif- 1 1 Trinv1|ΨihΨ| = |h; 1ihh; 1| + |j; 1ihj; 1| ferent ways to embed the two-particle observables 3 2n 2n into the three-particle observables, it is clear that 1 1 + Π + Π the answer to the question raised at the beginning 2n h;χ6=1 2n j;χ6=1 of this section will depend on the embedding. In (2) 1 1 = Πinv |eihe|1 ⊗ |hihh| + |jihj| , other words, the question of whether the phase θ 2 2 2 is accessible on the first two particles depends on Up to observational equivalence, we hence get a the operational definition of how to access the first mixed state two particles within the total three-particle Hilbert 1 1 Trinv1|ΨihΨ| ∼ |eihe| ⊗ |hihh| + |jihj| . space. 3 1 2 2 To illustrate this fact, let us apply two different 2 invariant traces to the paradox. In contrast to the In particular, the phase θ has disappeared. This is 1 usual partial trace, every invariant trace yields iden- not surprising: ultimately, Trinv amounts to tak- tical results when applied to the equivalent states Ψ ing the usual partial trace in the representation of 0 and Ψ0 in Eqs. (56) and (57) (hence the name “invari- the state relative to particle 1, i.e. of the state |Ψ i of ant”). That is, Eq. (57). On the other hand, consider the symmetry trans- U U 0 0 Trinv3 |ΨihΨ| = Trinv3 |Ψ ihΨ | (75) formation U from Example 20 which transforms to the center of mass. Like in Ref. [50], let us choose for all U ∈ Usym. However, different U yield dif- m and m such that m a = m b. Using Lemma 30, ferent results. For example, consider the invariant 1 2 1 2 It is clear that trace Trinv1 which is the adjoint of the embedding (1) U Φ that we have constructed in Subsection 4.1.A Trinv3 Πh;χ6=1 = Πh1;χ6=1 (76)
Accepted in Quantum 2021-06-18, click title to verify. Published under CC-BY 4.0. 24 and similarly for h replaced by j. Furthermore,
m m TrinvU (|l; 1ihp; 1|) = X |h; 1ihj; 1| X hh, g + 2 h ; 1|l; 1ihp; 1|j, g + 2 j ; 1i 3 m m h,j∈Zn g∈Zn = |l1; 1ihp1; 1|δ m2 m2 (77) l2−b m l1c,p2−b m p1c
2 U for all l, p ∈ Zn. Since Trinv3 |ΨihΨ| = the two particle groups. It is therefore independent U Trinv3 (Πalg(|ΨihΨ|)), we can expand Πalg(|ΨihΨ|) of the various physically distinct conventions dis- into basis elements and apply the above equations. cussed in the previous subsections. On the rela- As a result, this yields exactly Eq. (55) (since a is an tional subspace Hphys, the paradox is therefore un- integer, it turns out that we can ignore all b·c). That ambiguously resolved. It is sufficient to focus on is, this subspace because Lemma 22 tells us that the re- lational observables are tomographically complete TrinvU |ΨihΨ| = Π |ψihψ| ∼ |ψihψ|. (78) 3 inv for the invariant information in all alignable states. Thus, H contains all the information we are in- That is, up to observational equivalence, we ob- phys terested in. tain the original pure alignable state |ψi of Eq. (53). Moreover, in this observational equivalence class, ˜ (N) (N+M) Lemma 31. The map Φphys : Aphys → Aphys , de- |ψi is unique up to symmetry equivalence: namely, fined by |ψ0i if there is another pure alignable state with U 0 0 0 ˜ (N) ˆ (N+M) (N) (M) Trinv3 (|ΨihΨ|) ∼ |ψ ihψ |, then ψ ∼ ψ . But due Φphys(Aphys) := Πphys Aphys ⊗ 1 (80) to Lemma 22, this implies that ψ ' ψ0. Thus, in par- is an embedding. It simplifies to ticular, there is (up to a global phase) a unique rep- resentation of this state relative to the i-th particle, ˜ (N) (N) (M) Φphys(Aphys) = Aphys ⊗ Πphys, (81) |ψi ' |eii ⊗ |ϕi¯i. According to Eq. (54), it is but it is not unital. 1 iθ |ϕi¯ = √ |a + bi2 + e | − a − bi2 (79) 1 2 Recall that this construction does not yield an em- (N) relative to the first particle. Thus, under the “cen- bedding of all invariant observables Ainv , or even U (N) ter of mass” relational trace Trinv , the phase θ sur- of Aalg , as we have seen in Lemma 29. Thus, it is vives, in contrast to the result for Trinv1. remarkable that it works for the subalgebra of rela- tional observables. 4.3 Definition of the relational trace (N) (N) Proof. Noting that every Aphys ∈ Aphys satisfies (N) ˆ (N) (N) ˜ Recalling the invariant algebra inclusions Aphys ⊂ Aphys = Πphys(Aphys), we can recast Φphys in the Aalg ⊂ Ainv, we have thus far focused on construct- form ing an invariant trace for Aalg, since by Lemma 26, (N) (N+M) (N) (M) (N) (M) Φ˜ phys(A ) = Π Π ⊗ 1 A ⊗ 1 it is the smallest algebra containing the invariant phys phys phys phys (N) (M) (N+M) part of all alignable states and observables. How- × Πphys ⊗ 1 Πphys . ever, in the previous subsection, we have seen that By considering the action on an arbitrary basis state there does not exist a unique invariant trace on it, |g1, . . . , gN+M i, it is easy to verify that and the same conclusion applies to Ainv. We will now show that, by contrast, there does exists a nat- (N) (M) (N+M) (N) (M) Πphys ⊗ 1 Πphys = Πphys ⊗ Πphys. (82) (N) ural embedding of the algebra Aphys of relational (N+M) Hence, using idempotence of the projector, N-particle operators into the algebra A of re- phys ˜ (N) (N) (M) lational (N + M)-particle operators. This will also Φphys(Aphys) = Aphys ⊗ Πphys, (83) lead to a natural definition of an invariant trace in and the image commutes with Π(N+M). Check- terms of relational states. phys ing the embedding properties is now trivial. Fur- This trace, which we hence call the relational trace, thermore, note that the unit element of A(N) is has a natural and simple definition that is mani- phys (N) (N+M) (N+M) festly invariant under relative translations between Πphys, while the unit element of Aphys is Πphys .
Accepted in Quantum 2021-06-18, click title to verify. Published under CC-BY 4.0. 25 ˜ (N) (N) (M) However, Φphys(Πphys) = Πphys ⊗ Πphys. To see Specifically, note that (N) (N+M) that Φ˜ phys(Π ) does not act as the identity on (N) (N) ˆ phys tr Φphys(A ) ρ = tr Φphys(A ) Πphys (ρ) (N+M) (N) (N) Hphys , observe that tr Πphys = dim Hphys = (N) = tr Φphys(A ) ρphys . (85) N−1 (N) (M) N+M−2 |G| , hence tr Πphys ⊗ Πphys = |G| , but (N+M) N+M−1 This is precisely the expectation value of the rela- tr Πphys = |G| . (N) (N+M) tional observable Φphys(A ) ∈ Aphys in the re- (N+M) Returning to the discussion of how to define the lational state ρphys ∈ S(Hphys ), evaluated in the relations of the additional M particles to the first (N+M) manifestly invariant inner product on S(Hphys ). group of N particles (cf. Fig.6), we now have a The relational trace is thus unambiguously defined (M) ˜ unique answer: the projector Πphys in Φphys takes in terms of the so-called physical inner product the coherent average over all possible such relations. of constraint quantization [30, 61, 62], i.e. the inner This can also be seen by inspecting the action of product on Hphys. We will analyze this in more de- ˜ Φphys on the basis elements of Aphys: tail in our upcoming work [72]. 1 Theorem 33. The relational trace takes the explicit Φ˜ (|h; 1ihj; 1|) = X |h, g; 1ihj, gg; 1|. phys |G| form g∈G,g∈GM h (N) (M) (N) (M) i Trel(M)ρ = Tr(M) Πphys ⊗ Πphys ρ Πphys ⊗ Πphys , In contrast to Eq. (72) and the embeddings of Sub- section 4.1, this embedding does not assign to every where Tr(M) is the standard partial trace over M-particle configuration g another one, g0, which particles N + 1,...,N + M. It maps rela- has “the same relation” to the first N particles in tional operators onto relational operators, i.e. (N+M) (N) branches h and j. Instead, it generates the uniform Trel(M) Aphys = Aphys, and is trace-preserving superposition of all the possibilities. (N⊗M) for states in S(Hphys ), but trace-decreasing out- This averaging is also the reason for the failure side of it. Furthermore, it preserves observational of the unitality property. However, as we will see equivalence, i.e. ρ ∼ σ implies Trel(M)ρ = Trel(M)σ. shortly, the absence of unitality is precisely the rea- son why the relational trace defined below maps re- Proof. The first statement follows from lational (N + M)-particle states into relational N- (N) h (N) (N) (N) (M) i tr Φphys(A ) ρ = tr ΠphysA Πphys ⊗ Πphys ρ particle states. It will thus be rather a feature than a h (N) (M) (N) (M) (N) (M) i failure. = tr A ⊗ 1 Πphys ⊗ Πphys ρΠphys ⊗ Πphys , While Φ˜ phys is not unital, note that the embed- which holds for any A(N) ∈ L(H⊗N ) and any ρ ∈ N Φ˜ (Π(N) ) = ding of the relational -particle unit phys phys L(H⊗(N+M)). (N) (M) Πphys ⊗ Πphys certainly does act as the identity on its Given the conjugation of its input with the pro- image, (N) (M) jector Πphys ⊗ Πphys, it is clear that Trel(M) is trace- (N⊗M) (N⊗M) n (N) (M) o ⊗(N+M) preserving for states in S(Hphys ), but not for Hphys := Πphys ⊗ Πphys |ψi |ψi ∈ H . states outside of it. It is also clear that Trel(M) maps (N + M) (N+M) (N) This subspace of the space of relational - operators from Aphys into operators in Aphys, since (N+M) particle states H will be essential below when (N) phys the projectors Πphys can be taken outside of the resolving the paradox of the third particle. trace over particles N + 1,...,N + M. To see The natural embedding induces a natural trace. that it is surjective, it is straightforward to check 1−M (N) (M) (N) Definition 32 (Relational trace). The relational trace that |G| Trel(M) Aphys ⊗ 1 = Aphys for all is defined to be the Hilbert-Schmidt adjoint Trel := A(N) ∈ A(N) . † phys phys Φphys of the extended embedding map Φphys : Finally, note that the image of Φphys is contained L(H⊗N ) → L(H⊗(N+M)) defined by Φ := Φ˜ ◦ (N+M) ˆ (N+M) phys phys in Aphys , thus Φphys = Πphys ◦ Φphys. Taking ˆ (N) Πphys. That is, the relational trace is the unique map the Hilbert-Schmidt adjoint of this equation yields ˆ (N+M) with the property Trel(M) ◦ Πphys = Trel(M). Now suppose we have ˆ (N+M) (N+M) (N) (N) ρ ∼ σ, then Lemma 14 and Πphys ◦ Πinv = tr Φphys(A ) ρ = tr A Trel(M)ρ (84) ˆ (N+M) ˆ (N+M) ˆ (N+M) Πphys imply Πphys (ρ) = Πphys (σ). Alto- (N) ⊗N ⊗(N+M) for all A ∈ L(H ) and all ρ ∈ L(H ) (in gether this implies that Trel(M) preserves observa- particular for all states). tional equivalence.
Accepted in Quantum 2021-06-18, click title to verify. Published under CC-BY 4.0. 26 We can write Proof. It follows from Theorem 33 and Eq. (82) that tr Trel ρ(N+M) = tr Π(N) ⊗ Π(M) ρ(N+M) ˆ (N) ˆ (N+M) (M) phys phys Trel(M) = Π ◦ Tr(M) ◦ Π . (86) phys phys (N+M) (N+M) ≤ tr Πphys ρ , (89)
(N) We can thus view the relational partial trace Trel(M) hence ρ is not supernormalized. Now, the proba- as an invariant extension of the standard partial bility that an initial global measurement of the pro- ˜ (N+M) trace Tr(M). The non-unitality of Φphys is reflected jector Πphys yields outcome “yes”, and then a sub- (N) in the final application of Πˆ . Without this pro- (N) phys sequent local measurement of Ephys yields “yes” too, jection, the image of Trel(M) would not in general be is (N) contained in A . For example, the uniform mix- (N) (N+M) phys Prob E , Π (N+M) phys phys ture ρ(N+M) := Π /|G|N+M−1 on the physical phys (N) (N+M) (N+M) (N+M) = tr Φphys(E )Πˆ (ρ ) subspace of N + M particles yields Tr(M)ρ = phys phys 1(N)/|G|N (N) ˆ (N+M) (N+M) , whose decomposition according to The- = tr EphysTrel(M) ◦ Πphys (ρ ) (N) orem 12 contains operators outside of A .8 Thus, phys = tr E(N) Trel ρ(N+M) , (90) non-unitality is the price to pay for remaining relational. phys (M)
This leads to Trel(M) being trace-decreasing, unless where we have used Definition 32 and Eq. (86). The the initial state is fully supported on the subspace rest of the claim follows from the definition of con- (N⊗M) ditional probability. Hphys . Should we be worried about this fact — shouldn’t marginals of normalized quantum states Recall that in Lemma 22, we have seen that the be normalized? Not in this case. In contrast to the (N+M) projection ρ := Πˆ (ρ) for alignable states standard partial trace, the relational trace is not sup- phys phys ρ is subnormalized, but is sufficient to determine posed to tell us what the reduced quantum state on the expectation values of all invariant observables. a subsystem is. Instead, it is constructed to tell us Thus, ρ should not be seen as the marginal of ρ precisely the following: phys on some subsystem, but as the “relational part” of ρ. The “relational weight” tr ρphys is in general less than one,9 and it can decrease when disregarding Theorem 34. Given some (N + M)-particle state 9 ⊗(N+M) ˆ (N+M) (N+M) ⊗(N+M) This assumes that ρ ∈ L(H ) in ρphys = Πphys (ρ) ρ ∈ S(H ), the following conditional is normalized, as appropriate in the context of our manuscript state of N particles is normalized or subnormalized: where there is an external observer who could measure also non-invariant observables in the presence of an external frame. (N+M) (N) Trel(M)ρ By contrast, in the perspective-neutral approach [39, 41–45] ρ := . (87) (and more generally in constraint quantization), one disregards (N+M) (N+M) tr ρ Πphys any external structure and works directly with normalized re- lational states. That is, one would define the normalization of (N) (N) the (N + M)-particle relational state as trρphys = 1. Indeed, Consider any relational projector 0 ≤ Ephys ≤ Πphys note that the full standard trace trρ = tr Π(N+M)ρ is which we interpret as a “relational event”. Then the phys phys state ρ(N) tells us the probabilities of this N-particle precisely the extension of the so-called physical inner prod- uct [30, 61, 62] to density matrices. However, note also from event, conditioned on the (N + M)-particle system the previous footnote that the standard partial trace Tr(M) is being relational: not in general appropriate for relational states ρphys. The ad- ditional projection with Πˆ (N) in the relational trace fixes this (N) (N) (N+M) phys tr E ρ(N) = Prob E | Π . (88) issue, but reduces the norm of states with support outside of phys phys phys (N⊗M) Hphys . This has a transparent physical interpretation which (N) is best seen through the norm reduction of invariant basis That is, the renormalized result ρ of the rela- N−1 states: Trel(M)(|hN, hM; 1ihjN, jM; 1|), where hN , jN ∈ G M 1 tional trace gives us the expectation values of all and hM , jM ∈ G , is equal to |G| |hN ; 1ihjN ; 1| if hM = gjM N-particle relational observables, conditioned on for some g ∈ G, and is zero otherwise. The variables hM , jM M M−1 the global (N + M)-particle state being fully rela- lying in G rather than G reflects the fact that they en- code not only the M − 1 internal relations of the M particles, tional. but also a definite relation between the two particle groups, which is a property of both groups together. The normaliza- tion reduction factor 1/|G| comes from the coherent averaging over the relations between the two particle groups (which is 8More generally, the image of H(N+M) \H(N⊗M) under phys phys partially a property of the N particles), and quantifies the (N) Tr(M) does not lie in the N-particle relational subspace Hphys. corresponding ignorance of the relational N-particle state ob-
Accepted in Quantum 2021-06-18, click title to verify. Published under CC-BY 4.0. 27 1 −1 −1 some of the particles. Using 3hg3|h; 1i = √ |g3h , g3h h1i yields |G| 2 2
h i Trel (|ΨihΨ|) = Π(2) Tr Π(3) (|ΨihΨ|)Π(3) Π(2) 4.4 Relational resolution of the paradox 3 phys 3 phys phys phys = Π(2) |ψihψ| Π(2) In order to apply these insights to the paradox of phys phys 1 e−iθ the third particle, let us first discuss some further = |h; 1ihh; 1| + |h; 1ih j; 1| aspects of the interplay of composition and rela- 2n2 2n2 M + N eiθ 1 tional trace, for the generic case of parti- + |j; 1ihh; 1| + |j; 1ihj; 1|. cles. Suppose ρ(N) is a state of the N particles be- 2n2 2n2 fore taking the additional M particles into account, Since |Ψi is alignable, Lemma 22 tells us that prepared by an observer with access to the external (N+M) reference frame. The corresponding relational state hΨ|Πphys |Ψi = 1/n. Thus, computing the condi- (N) ˆ (N) (N) tional state of Theorem 34, we obtain the projection is ρphys = Πphys(ρ ). Next, suppose the M addi- tional particles are prepared in a normalized state of the state in Eq. (55) into the relational subalgebra. ρ(M), and the composite state of all particles is of Hence, we recover exactly the relational state of the the product form ρ(N+M) = ρ(N) ⊗ ρ(M). We can first two particles which we had before adding the θ then construct the relational state corresponding to third. In particular, the phase is preserved: as ex- this composition. Here it is important to note that pected, it remains accessible on the first two parti- (N+M) ˆ (N+M) (N+M) ˆ (N+M) (N+M) cles. ρphys = Πphys ρ = Πphys ρ˜ , The algebra Aphys generated by relational observ- ρ˜(N+M) ' ρ(N+M) for any . That is, all members ables and the subspace Hphys of relational states (N+M) of the symmetry equivalence class of ρ (incl. is the arena of the perspective-neutral approach to states featuring entanglement between the two par- QRFs [39,41–45]. As such, there is no paradox of the ticle groups) yield the same relational (N + M)- third particle in this approach. Furthermore, it pro- particle state. However, due to Theorem 33, it is vides a compelling conceptual interpretation of this (N⊗M) only the projection into Hphys that matters for the resolution: the relational states are the perspective- relational trace. But Eq. (82) implies that neutral states, i.e. they correspond to a description of the composite particle system prior to choosing (N) (M) (N+M) (N) (M) an internal reference relative to which the state is Πphys ⊗ Πphys ρphys Πphys ⊗ Πphys (N) (M) (N) (M) (N) (M) described. The perspective-neutral states contain = Πphys ⊗ Πphys ρ ⊗ ρ Πphys ⊗ Πphys the entire information about all internal QRF per- (N) (M) spectives at once. The relational trace is performed = ρphys ⊗ ρphys, (91) at the perspective-neutral level, and consistency at
(N+M) (N) (M) that level implies consistency in all internal perspec- and so we have Trel(M)ρphys = ρphys · tr(M)ρphys. tives. We will further elaborate on this in Ref. [72]. Up to a constant factor, this is precisely the initial relational N-particle state that we had before taking the additional M particles into account. 4.5 Comparison to the resolution by Angelo et al. Let us see what this implies for the paradox of the Angelo et al. [50] also propose a resolution to the third particle. Thus, let us concretely compute the apparent paradox that they have raised in their pa- relational trace Trel3 of the state |Ψi of Eq. (56). Due per. Let us recapitulate their resolution in our ter- to invariance, the result will be identical if we apply 0 minology and compare the two approaches. First, it to the state |Ψ i of Eq. (57). As a first step, we find −2i(a+b)ˆp they introduce an operator T := e r2 which −iθ in our notation simply implements a translation of (3) (3) 1 e Πphys(|ΨihΨ|)Πphys = |h; 1ihh; 1| + |h; 1ihj; 1| the two particles, given by T |g1, g2i = |g1 − 2a, g2 + 2n 2n 2bi iθ . Computing the expectation value in the two- e 1 1 iθ + |j; 1ihh; 1| + |j; 1ihj; 1|. particle state of Eq. (53) yields hψ|T |ψi = 2 e . Thus, 2n 2n this operator (or rather its real and imaginary parts) admit the measurement of the phase θ. tained via Trel(M). By construction, the latter contains all information about the relational N-particle observables which Since we are in the framework of Example8, we are independent of the relation between the particle groups. can use the explicit form of the characters to see that However, the ‘ignorance factor’ 1/|G| has to be taken into ac- count. T |h; χki = χk(−2a)|h + 2a + 2b; χki. (92)
Accepted in Quantum 2021-06-18, click title to verify. Published under CC-BY 4.0. 28 In particular, the invariant part of this translation move. However, the paradox of the third particle is can be expressed in the form a completely kinematical thought experiment: it can be formulated without any reference to the particle T |h; 1i = |h + 2a + 2b; 1i. (93) masses or time evolution whatsoever, as explained at the beginning of Section4. Thus, for the para- That is, T increases the relative distance of the par- dox, there is in any case no reason to privilege the ticles by 2a + 2b. Indeed, if we define center of mass embedding over any other choice of invariant embedding, and hence this type of argu- T := Π (T ) = Π (T ) = Πˆ (T ), (94) inv inv alg phys mentation is insufficient to conclude that the phase θ is preserved. then Tinv ∈ Aphys satisfies Eq. (93). Note, however, In contrast, our resolution amounts to the con- that T contains strictly more information than Tinv: not only does it tell us that the relative distance of struction of a relational embedding for which no such the particles increases by 2a + 2b, but it also tells us choice has to be made in the first place. Nonethe- what happens to their absolute positions. less, Angelo et al’s insight is still important: embed- Angelo et al. write: “The crucial (and surprising) ding fewer into more particles will in general “do observation is that T actually shifts the relative coordi- something” to the additional particles, and care has nate of particle 3 as well as that of particle 2.” Strictly to be taken of how the embedding is accomplished. speaking, this is not a claim about T , but about the embedding T (3) of T into the three-particle observ- ables. Using the obvious embedding T (3) = T ⊗ 1, 5 Conclusions (3) we obtain T |g1, g2, g3i = |g1 − 2a, g2 + 2b, g3i, and The aim of this article is to elucidate the operational thus essence and interpretation of the recent structural (3) T |h1, h2; 1i = |h1 + 2a + 2b, h2 + 2a; 1i. (95) approach to QRFs [38–40], and to also clarify the meaning of its QRF transformations as symmetry That is, the relative coordinate of particle 3 is also transformations. These insights have then been ex- shifted by 2a. This reproduces Angelo et al.’s claim, ploited to illuminate the physics behind the appar- but it is important to understand where the shift of ent ’paradox of the third particle’ of Ref. [50] and h2 7→ h2 + 2a comes from. It is certainly not possi- to resolve it at a formal level through relational ob- ble to deduce this shift from Tinv alone. Instead, it servables. comes form the specific choice of implementing Tinv We began by providing a careful conceptual com- (a relative shift of 2a + 2b) via T (absolutely shift- parison of the quantum information [12–19] and ing particle 1 by −2a and particle 2 by 2b). And structural approaches to QRFs, illustrating the dif- the latter choice comes from Angelo et al.’s decision ference in their operational essence in terms of two of preserving the center of mass, which fixes the non- communication scenarios. While both approaches invariant action of T . focus on an external relatum independent descrip- In summary: Angelo et al.’s proposed resolution tion of physical observables and quantum states, of the paradox comes from deciding to embed the two- they do so in different manners and with different particle observables via the center-of-mass embedding goals. As we have seen, technically a distinction that we have described in Subsection 4.2. As shown can be drawn between the two in terms of how they there, this leads to a preservation of the phase θ. describe external relatum independent states fun- While the center-of-mass may be the most demo- damentally: they are the incoherently and coherently cratic choice among the particles, as also discussed group-averaged states in the QI and structural ap- in Subsection 4.2, there exist other well-motivated, proach, respectively. but physically inequivalent choices of embedding, A key ambition of the QI approach is to elucidate e.g. relative to one of the particles, which may or how to perform communication protocols between may not preserve θ. Basing a resolution of the para- different parties in the absence of a shared external dox in the context of frame covariance on a spe- laboratory frame. To this end, it suffices to focus cific choice of frame seems unsatisfactory. Further- on physical properties of the communicated quan- more, even if one wanted to argue that the center- tum system that are meaningful relative to an ar- of-mass is a privileged choice in mechanics, this bitrary choice of external frame. For example, this would have to rely on its dynamical properties — in can be achieved by restricting to speakable informa- fact, postulating certain particle masses in the first tion that is encoded in decoherence-free subspaces place is nothing but a claim about how the particles or by communicating an additional reference quan-
Accepted in Quantum 2021-06-18, click title to verify. Published under CC-BY 4.0. 29 tum system that serves as a token for the sender’s lecting any one of the N particles as a reference sys- reference frame. Either way, the QI approach main- tem to define the origin and to describing the re- tains the reference frame external to the system of maining N − 1 particles relative to it. These canon- interest. For successfully carrying out such opera- ical choices are the ’internal QRF perspectives’ on tional protocols it is also not necessary to take an the N-particle system of Refs. [38–40]. extra step and choose an internal reference frame The symmetry group Gsym contains the ’classi- within the system of interest, and to ask how the cal translation group’ G as a strict subgroup, but quantum system is described relative to one of its it also contains ‘relation-conditional translations’ subsystems. which turn out to include the QRF transforma- However, this additional step is precisely what tions of Ref. [40], which are equivalent to those of the structural approach aims for. Its primary goal Refs. [38, 39]. While it is evident from these works is not the implementation of protocols for com- that the QRF transformations are conditional trans- municating physical information. It has rather a lations, the present article clarifies that they are sym- more fundamental ambition: to dissolve the dis- metry transformations with a precise and transpar- tinction between quantum systems and reference ent physical interpretation. frames and thereby to extend the set of available Being translations conditional on the particle rela- reference frame choices to include subsystems of tions, the QRF transformations make sense in a clas- the physical system of interest. Its focal point are sical context when dealing with, for example, statis- thus not only external relatum independent state tical mixtures of particle positions rather than su- descriptions, but internal state descriptions. The op- perpositions, and indeed have classical analogs that erational essence of the structural approach can be have been exhibited in Refs. [39–43]. Nevertheless, illustrated in a scenario in which different agents just like the CNOT gate has a classical meaning, but agree on a (redundancy-free) description of physical can generate entanglement, the QRF transforma- quantum states without adhering to an external re- tions similarly lead to interesting quantum effects latum. They can always achieve this task by invok- such as a QRF dependence of, e.g. entanglement ing certain “canonical choices” in the representation and superpositions [38–40], classicality [39, 51, 52], of quantum states that exploit the internal structure spin [46, 47], certain quantum resources [53], tem- of the quantum system to be described. In particu- poral locality [44, 48], and of comparing quantum lar, these canonical choices of representation are re- clock readings [42, 45].10 lated by transformations that coincide with the QRF Given the two groups Gsym and G in the setup, transformations in the structural approach. one has a priori two distinct ways to construct in- To show this explicitly, we have then formal- variant states and observables. Interestingly, as we ized these conceptual observations in the context of have shown, the invariant (pure) states and thus the an N-particle quantum system (“G-system”) where subspace Hphys of relational states do not in fact de- the configuration space of each particle is a finite pend on whether one requires invariance under the Abelian group G. We chose this simple setting in action of Gsym or its subgroup G. By contrast, the order to avoid technicalities and to render all ap- set of invariant observables does depend on which pearing structures completely transparent. But we group one works with: the operator algebra in- emphasize that our observations are of more gen- variant under Gsym is a strict subset of the opera- eral validity. They apply directly to laboratory situ- tor algebra invariant under its subgroup G. How- ations in which agents simply disregard, or do not ever, the two invariant operator algebras coincide have access to, a relatum external to the system of again in their restriction to the space of relational interest, but in principle also to the case that no ex- states Hphys, which is the algebra Aphys generated by ternal physical relatum exists in the first place as, so-called relational observables [29–37, 42–45]. The e.g., in quantum cosmology (see Refs. [43, 76] for a space of relational states Hphys and the relational op- related discussion). erator algebra Aphys are key structures in constraint We determined the symmetry group Gsym as- quantization [30, 61, 62] and the platform of the sociated with a G-system, which preserves all perspective-neutral approach to QRFs [39, 41–45] its external-relatum-independent structure. We (part of the structural approach), which are thus in- showed that states from the corresponding symme- dependent of the distinction between Gsym and G. try equivalence class are alignable to a choice of ref- erence system through a Gsym transformation: each 10It would be interesting to study the recent proposals [79– equivalence class contains “canonical choices” of 81] for quantum time dilation effects in terms of the temporal state representations, and these correspond to se- QRF transformations as in Refs. [42–45, 48].
Accepted in Quantum 2021-06-18, click title to verify. Published under CC-BY 4.0. 30 The difference between Gsym and its subgroup G solved; in this sense, the perspective-neutral ap- is, however, crucial when aligning the non-invariant proach does not feature any paradox of additional description of quantum states to a particle at the particles. However, we have not discussed what level of the full N-particle Hilbert space. it would mean for an agent to operationally imple- These observations also permitted us to first clar- ment this resolution in the lab and, specifically, how ify the physics behind the ’paradox of the third par- they may operationally restrict to relational states ticle’ discussed in Ref. [50], and subsequently to re- and observables (although we believe this to be pos- solve it at a formal level. First, we have illumi- sible). In this light, our resolution of the paradox is nated why the usual partial trace is not suitable in formal. the context of QRFs, because it ignores the equiv- The paradox of the third particle and our reso- alence classes of states that are operationally indis- lution can be viewed as a finite-dimensional ana- tinguishable in the absence of an external relatum. log of the problem of boundaries and edge modes Next, we have explained that, in order to take the in gauge theory and gravity [64–71]. Boundaries in observational equivalence classes into account, one space or spacetime usually break gauge-invariance has to construct a partial trace in terms of the in- and constitute challenges for gauge-invariant ob- variant observables. However, even when attempt- servables. The latter are typically non-local (such ing to do so, we have seen that there does not ex- as Wilson loops) and can thus have support in ist a physically distinguished choice for such an two neighbouring regions separated by a bound- invariant partial trace outside the space of relational ary. Those gauge-invariant observables with sup- states. The reason is that an invariant partial trace port in both regions determine the physical relation demands a suitable embedding of the two-particle between the two and are accounted for in terms of invariant observables into the three-particle invari- so-called edge modes when one of the regions is ant observables. Yet such an embedding (while in- ignored. This is analogous to the joining of two variant under symmetry transformations) depends groups of N and M particles and asking for the on how one defines the relation of the third to the invariant relations between the two groups. The first two particles, and there are multiple physically relative distances between the two groups of parti- inequivalent ways (e.g., distance to the first parti- cles are the finite-dimensional analog of the gauge- cle, center of mass, etc.). The two-particle reduced invariant observables in gauge theories and grav- state then depends on one’s convention of how to ity that have support in two neighbouring regions. define the relation between the third and the first As we have seen, ignoring one group of particles by two particles, despite restricting attention to invari- simply taking the standard partial trace may indeed ant observables. lead to an invariance breaking in analogy to the field However, when restricting attention further to theory case, i.e. N-particle states that are not rela- the algebra Aphys generated by relational observ- tional. This is because the set of relational observ- ables and the space of relational states Hphys, we ables for the joint (N+M)-particle system is not only showed that there does exist a physically distin- the union of the sets of relational N- and M-particle guished embedding of the relational two-particle observables, again in analogy to two neighbouring observables and states into the relational three- subregions in spacetime. Our relational trace de- particle observables and states. Physically, this em- fines a purely relational, i.e. invariant way of ‘ignor- bedding corresponds to coherently averaging over all ing’ a group of particles, and it would be interesting possible relations between the third and the first to extend this tool to the study of edge modes in two particles, and thereby defines an entirely invari- gauge theories and gravity. ant embedding. This permitted us to define an un- Lastly, we emphasize that our novel interpreta- ambiguous relational partial trace that determines tion of the structural approach applies in particu- the expectation values of relational observables on lar also to temporal quantum reference frames, i.e. subsets of particles. In particular, this trace achieves quantum clocks. For instance, the example of the for relational states what a consistent partial trace cyclic group could model a set of quantum clocks should do: if a third particle is independently pre- each with a finite set of readings. The external frame pared, then the two-particle reduced state, obtained would then be some laboratory clock that one ex- from the relational three-particle state, coincides ternal observer has access to, but another may not. with the relational two-particle state prior to tak- Nevertheless, the two observers can agree on the ing the third particle into account. At the level description of the flow of time by focusing on a of relational observables and relational states, the purely internal choice of clock that leads to a rela- paradox of the third particle of Ref. [50] is thus re- tional notion of time entirely independent of any ex-
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