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These instruments indicate a preferred Cartesian refer- seen by Wigner and his friend. Although we ultimately ence frame for the laboratory, representing the labelling do not succeed in this task, we are able to find an exam- of the (x,y,z) co-ordinates in the lab. Since the Carte- ple that transcends the usual system-apparatus division sian reference frames of the laboratories are related by a in the absence of an external reference frame, indicating proper rotation, the associated symmetry group is SO(3). that future efforts along similar lines might bear fruit. For quantum particles, we typically seek a unitary repre- The outline of the paper is as follows. In Section II we sentation of the rotation group, hence = SU(2). consider a system composed of spins and ask what prop- G Most of the literature to date has restricted attention erties should be the same for all observers. We argue that to transformations that preserve the separation between all observers should be subject to the same constraints system and laboratory. That is to say, if one laboratory on their ability to distinguish between subsystems of the performs a measurement on the system, the state is up- total system. Taking the quantum fidelity between sub- dated according to the outcome of this measurement for systems as a measure of their distinguishability, we ob- all laboratories, not just the laboratory in question. This tain U(2) as the symmetry group that preserves them. In assumption excludes the possibility of treating the labo- Section III we examine the bearing this result has on the ratories themselves as quantum systems relative to one microscopic / macroscopic distinction and the Wigner’s another. For example, consider the following thought ex- friend thought experiment. Section IV contains our con- periment due to Wigner [9]: an experimenter called Fran clusions and outlook. (a friend of Wigner) performs a measurement of the sys- tem and obtains an outcome. It should then be pos- sible for another experimenter in a different laboratory II. THE INTERNAL PROPERTIES OF SPINS (Wigner) to treat Fran and her laboratory as a quantum system, and the measurement as a physical interaction Our task is to define the internal properties of general that entangles the state of Fran to the state of the sys- pure states Ψ of N spins, namely, the attributes of the tem. This physical interaction is a transformation that system that| doi not depend on the observer’s laboratory. applies not just to the system alone, but also to the whole In what follows, we assume that all laboratories can agree laboratory of Fran, thereby including this laboratory and on a labeling of the N spins, so that if one observer spec- Fran herself as part of the ‘system’. Thus, Fran’s labora- ifies a subsystem of spins, say “spins numbered 7 12”, tory is in a superposition relative to Wigner’s laboratory. all the observers agree on which subsystem this refers− to. So far, there have been few attempts to apply the tools Measurements of spin performed within a laboratory are of quantum reference frames to physical settings of this relative to the (arbitrary) orientation of that laboratory’s more general type. spatial axes. The procedure for measuring ‘spin up’ in a We note the following exceptions. Palmer et. al. [10] given direction is assumed to be the same for all spins, have considered the task of changing reference frames hence labeling the ‘up’ direction for one spin fixes the in a setting where the reference frame is an explicitly label for all spins in the same lab. Finally, we assume all quantum object; however they do not extend their anal- observers can measure the number of spins in any sub- ysis to ‘Wigner’s friend’-type scenarios. Angelo et. al. system without difficulty. Other than that, the devices [11, 12], and Pereira and Angelo [13], following work by in the laboratories are not calibrated with each other. Aharonov et. al. [14], consider the problem of treating The problem can now be understood as follows. If quantum systems as observers and changing frames be- a system of N spins is prepared according to a fixed tween them. However, that work is concerned with linear preparation procedure, it will nevertheless have a differ- position and momentum degrees of freedom, whereas we ent state according to different laboratories. We want to consider angular momentum and spin (since the rotation know: how much can the states seen by two laboratories group is compact, we can apply the standard tools of differ? Clearly, any differences can only be attributed to quantum reference frames). Finally, H¨ohn et. al. [15–17] a change of laboratory frame, since the preparation of have managed to reconstruct quantum mechanics and its the system is invariant. We have assumed that the labo- key symmetry groups for qubits using an operational ap- ratories are related by collective transformations of their proach that inspired the approach used in the present apparatuses that preserve the number of spins and the work; however they also do not address the Wigner’s spin ordering, but otherwise the symmetry group that friend scenario. relates the laboratories is unknown. In this work, we make an incremental step towards Our first clue comes from asking how well the labora- extending the formalism of quantum reference frames tories are able to distinguish different preparation proce- to more general scenarios of the kind applicable to the dures. Imagine that a machine prepares N-spin systems Wigner’s friend thought experiment. Our main idea is according one of two preparation procedures, A or B. to try and derive the symmetry group of reference frame In the first phase of the experiment, it prepares many transformations from first principles, rather than pos- systems using procedure A and sends multiple copies to tulating it from a classical limit, in the hopes that the each laboratory for analysis. In the second phase, it does resulting physical symmetry will permit us to draw an the same but only using procedure B. As a result, each equivalence between the apparently irreconcilable states laboratory is able to perform tomography on the systems 3 they received and determine the state obtained from each copy of the state is provided, then the appropriate mea- preparation procedure relative to their own instruments. sure of distinguishability is the quantum error probability. In general, the pair of states ρA,ρB seen by one lab will More generally, after randomly choosing whether to pre- ′ ′ not be the same as the pair ρA,ρB in another lab. pare A or B, the machine could send a fixed number of In the third phase of the experiment, the machine copies of the system to each lab. In this case, there is sends only a single copy of the system to each laboratory, a whole zoo of distinguishability measures that could be and in each case it chooses randomly which preparation used (see eg. Ref [18]). We will adopt the quantum fi- procedure to use: with probability p it applies procedure delity as the measure of interest in the remainder of this A, and with probability (1 p) it applies B. Each lab is work, but we conjecture that the results obtained here then asked to guess whether− the system they received was hold also for other measures of distinguishability. The prepared using procedure A or B. They can use only the quantum fidelity can be expressed as: knowledge they have gained in the previous two phases, plus the value of p. F (ρA,ρB) := min X pTr [ρAFk] Tr [ρBFk] , (1) {Fk} Since the pair of states being distinguished will in gen- k eral differ among laboratories, it is a priori possible that one laboratory might consistently perform better at the where the minimisation is over all possible POVMs Fk { } task than another laboratory (for instance, if the pair on the Hilbert space of the systems. of states seen by one laboratory are more alike than the Why choose the fidelity? For spins, the fidelity can states as seen by another laboratory). Hopefully, this be related to the relative spatial angles between the spin possibility strikes the reader as rather unnatural. After vectors. Specifically, if we restrict attention to a tensor all, the laboratories are all trying to distinguish systems product of N pure spin states, then the state space is iso- produced by the same pair of preparation procedures, morphic to a set of unit vectors in 3-dimensional space and these procedures are frame-independent notions, so (Bloch vectors). In that case, a natural candidate for the we intuitively expect that no laboratory has any special internal properties of the system are the relative angles advantage over another. Hence, the likelihood of suc- between the Bloch vectors. The angle θij between the ith ceeding should be independent of which laboratory (i.e. and jth vectors is directly related to the overlap φj φi observer) is asked to perform the task. between their respective states, which is just the|h fidelity| i| Without loss of generality, we can imagine that in this case. For more general (entangled) states, the fi- the machine implements the two preparations by first delities can be related to angles between Bloch vectors of preparing a single system according to a fixed procedure the purifications of the spin states (Uhlmann’s theorem, and then choosing one of two N-spin subsystems, Ref. [19]). The connection between the fidelities and the designated A or B, as the output. For example, let relative angles between spatial vectors is one of the main N = 3. Starting with an initially prepared state of inspirations for using the fidelity as the measure of distin- M N spins, the machine could choose either spins guishability. However, it is merely a heuristic motivation, 5, 11≥, 14 (subsystem A) or spins 17, 42, 99 (subsystem as the Bloch vectors of higher-dimensional systems obvi- B), discard the rest, and re-label the chosen spins 1, 2, 3. ously do not exist in 3-dimensional space. Ultimately, we In this way, the choice of preparation procedure reduces have chosen the fidelities mostly because they are math- to a choice of subsystems sampled from a larger system ematically convenient and lead to interesting results. with a fixed preparation. With this perspective, the An N spin system can be partitioned into subsys- considerations of the previous paragraph amount to the tems of different numbers of spins. Obviously, if the statement that all laboratories should be equally good subsystems have different sizes, they are perfectly at distinguishing a given pair of subsystems of a total distinguishable by measuring the number of spins, so system prepared by a fixed procedure. This forms our the only nontrivial cases arise when the subsystems first postulate: have equal numbers of spins. This is important, because the fidelity is defined only for subsystems of equal size. Postulate 1: Let A and B label two (not necessarily Using the fidelity, we can now give a mathematical disjoint) equal-sized subsystems of a quantum system version of Postulate 1 for spins: prepared by a fixed procedure. We assume that all observers have access to identically prepared copies of Postulate 1 for spins: Given an arbitrary pure this total system and its subsystems. Then the ability of state Ψ consisting of N spins, the internal properties | i any observer to distinguish A from B (when presented are the set of fidelities F (ρA,ρB) between all subsystems with a random sample of either one) must be the same ρA,ρB with equal numbers of spins. for all observers. We next show that these fidelities are preserved by (Note that although we motivated this postulate from (and only by) actions of the unitary group U(2): the example of spins, its scope is quite general). For a given pair of states ρA and ρB, quantum mechanics sup- Theorem 1: The set of fidelity-preserving transfor- plies bounds on their distinguishability. If just a single mations acting on the N-spin Hilbert space is the image 4 of a unitary representation of the group U(2), namely: Although Theorem 1 states that fidelities are guaran- teed to be preserved by all and only those unitaries with π : V U(2) V ⊗N U(2N ) . (2) the form U = V ⊗N , this does not imply that two states ∈ 7→ ∈ having the same fidelities must necessarily be related by Proof: The action of a unitary U on Ψ induces the a unitary map of this form. An example of this intriguing following transformations on the reduced| states:i situation will be explored in the next section.

† ρA Tr=A U Ψ Ψ U := (ρA) , → 6  | ih |  E † ′ ρB Tr=B U Ψ Ψ U := (ρB) . (3) → 6  | ih |  E III. PHYSICAL INTERPRETATION It can be shown using Uhlmann’s theorem [19] that the fidelity will be preserved only if the CPT maps are the same, i.e. ′ = . We then make use of a theorem E E In analogy with special relativity, let us assume that due to Molnar [20], which states that the fidelities are transformations that preserve the internal properties preserved if and only if (ρ)= V ρV † for some unitary V E relate physically equivalent situations. That is, the acting on the Hilbert space of the subsystems (this is a internal properties constitute the observer-independent generalisation of a well-known theorem of Wigner [21]). core of quantum mechanics: It follows that U decomposes into U = UAUBUR where R labels the systems not in A or B, and UA = UB = V . Applying this to all pairs of single-spin subsystems Postulate 2: In the absence of a preferred external leads to the conclusion that U preserves the fidelities iff reference frame, two quantum states of the same number U = V ⊗N with V U(2). Under the map V V ⊗N , of spins represent the same physical situation iff they the image of U(2)∈ in U(2N ) is therefore precisely7→ the have the same internal properties (i.e. the same fidelities set of fidelity preserving maps acting on the space of N between their subsystems). spins.  This leads to the following counter-intuitive result: a While U(2) is the appropriate group for states in microscopic superposition as seen by one lab can appear Hilbert space, one is typically concerned only with to be a macroscopic superposition as seen by a different ‘physical states’ in the form of projectors Ψ Ψ , which lab. Let := η1, η2, ...ηM with η 0, 1 be the live in the projective Hilbert space. In that| ih case,| the computationalB basis{| for a seti} of M spins.∈ { Consider} the following corollary applies: following two states, expressed in the basis : B Corollary: Under the representation mentioned in φ = 00...0 (α 0 + β 1 ) Theorem 1, fidelity-preserving maps that act on the | i | i | i | i projective Hilbert space constitute the image of the := 0 1 Φ , | i | i group SO(3). φ′ = (α 00...0 + β 11...1 ) 0 | i ′| i | i | i := Φ 0 M . (4) Proof: Theorem 1 implies that the fidelity-preserving | i| i unitaries acting on the projective space have the form W ⊗N , where W is any element of the 1-spin projective The first state φ might represent, for example, a large | i unitary group PU(2). This defines the representation magnet consisting of M 1 spins, with the Mth spin in of PU(2) by fidelity-preserving unitaries acting on the a superposition relative− to the magnet. The basis is projective space, and since PU(2) is isomorphic to a natural or preferred choice of labeling in a laboratoryB SO(3), the statement of the Corollary follows.  where the physical states corresponding to elements of are easy to prepare, and where superpositions of theseB The symmetry group of reference frame transforma- elements are easy to prepare whenever the correspond- tions is usually taken for granted in the literature, but we ing elements of are similar to each other. An example have derived it using a novel assumption (Postulate 1). is the state φ ,B whose superposed states differ at only Note that since the fidelities are not in general indepen- a single spin| ini the basis ; we call φ a microscopic dent, they should be reducible to a smaller set of indepen- superposition relative to suchB a lab. Conversely,| i if the dent internal properties, which we have not attempted to two superposed states are very different when expressed identify here. It is straightforward to do so using tools in the natural basis of the lab, the state represents a from the literature on quantum reference frames; for ex- macroscopic superposition. Thus φ′ is macroscopic rel- ample, Eq. 3.20 in [7] presents the decomposition into ative to the lab for which is the| naturali basis. If we collective and relative (i.e. internal) degrees of freedom have a preferred basis inB which to prepare both these for the group SU(2), and one sees that the internal de- states, we will find theyB appear physically different, the grees of freedom span the degenerate subspaces of states state φ′ being significantly harder to prepare than the with definite total angular momentum. state |φ i. We now observe that these two states have the | i 5 same internal quantities, namely: precise. In order to do so, it would be necessary to treat Fran as a physical system (perhaps herself composed of F (ρA,ρB) = 1, if A, B exclude spin M; spin-half particles) initially entangled to spin M. The 2 F (ρA,ρB) = α if M appears in one of A, B; task would then be to discover a state having the same | |4 fidelities as the original, but in which the spin M has a F (ρA,ρB) = α if M appears at different | | definite value, in accordance with what we expect Fran sites in both A, B; to see. The problem is that the equivalence of φ′ and | i F (ρA,ρB) = 1 if M appears at the same φ seems to be rather a special case: simple generaliza- | i site in both A, B . (5) tions, like adding more spins to the initially superposed state, or even adding an extra magnet pointing in a dif- Postulate 2 then asserts that, in the absence of a pre- ferent direction, lead to states whose fidelity-equivalent ferred external reference frame, these states should be counterparts don’t seem to possess the desired proper- physically equivalent. To see how this can occur, con- ties. For the present, it is left as an open problem to sider a different basis ′ related to by a transforma- classify the sets of physically equivalent states induced tion that exchanges 0B 1 in all elementsB except for by the fidelities via Postulate 2, but our efforts so far 00...0 and 11...1 . For↔ instance, for M = 3 the super- seem to indicate that these equivalence classes are not | i | i position ( 000 + 001 )/√2 in the basis corresponds to rich enough to capture the duality between Wigner and | i | i B ( 000 + 110 )/√2 in the basis ′. Notice that φ and Fran’s conflicting experiences. φ| ′ switchi | rolesi when expressed inB terms of basis |′.i This Despite this disappointment, the example does show means| i that if they are both prepared in a lab forB which that if we take internal properties as fundamental, and ′ is the preferred basis, φ′ will be microscopic (hence their symmetry transformations as merely derived, then easierB to prepare) and φ |willi appear macroscopic. This the line between apparatus and measured system can be example clearly shows| thati if no preferred basis is speci- transgressed. fied (i.e. there is no external lab to tell us which basis is most natural) the states φ′ and φ may be regarded as representing the same physical| i situation.| i Comment: there is a certain intuition according to IV. CONCLUSIONS which one might expect an electron in superposition with respect to a large magnet to ‘see’ the magnet as being in a We have shown that it is possible to derive U(2) as the superposition. This procedure of ‘stepping into the elec- group that preserves the internal degrees of freedom of tron’s shoes’ and talking about ‘what the electron sees’ spins by postulating (cf. Postulate 1) that the internal seems to fit with the transformation from φ to φ′ . One degrees of freedom are the fidelities between subsystems must be careful in making such claims, however,| i | sincei as of spins. Furthermore, we proposed that states having we have shown above, this physical equivalence of frames the same internal quantities should represent the same can only be maintained in the absence of a preferred ex- ‘physical situation’, differing only in terms of the descrip- ternal basis. In practice, there always exists a preferred tion relative to some laboratory (Postulate 2). This led basis for the external laboratory, which is singled out by to the observation that, in absence of a preferred external decoherence of the system and measuring apparatus by a laboratory, microscopic and macroscopic superpositions common environment. This is why, for example, placing become equivalent. a single particle in a spatial superposition over a long dis- The work presented here was inspired by the idea that, tance is considered ‘macroscopic’: the environment tends in certain circumstances, a single quantum particle can to decohere systems in the position basis. An interesting be considered an observer. Indeed, the approach shows task for future work would be to include the environment promise for reconciling the viewpoints of Wigner and his in the preceding analysis, to see explicitly how it breaks friend in the thought experiment, by utilizing the notion the symmetry between the states φ′ and φ . of an equivalence class of states all having the same inter- Setting aside decoherence, one| mighti still| i expect this nal properties. Unfortunately, we were unable to achieve surprising equivalence of states to have some bearing on this goal using the fidelities as the relevant degrees of the problem of Wigner’s friend. After all, it presents a freedom. dual perspective that seems to correspond to the conflict- The author thanks Caslavˇ Brukner, Philipp H¨ohn, ing experiences of the two parties: in one point of view Alexander Smith and an anonymous referee for helpful (Wigner), the Mth spin is in a superposition, while in the comments. This work has been supported by the Euro- other point of view (Fran) it’s spin has a definite value. pean Commission Project RAQUEL, the John Temple- The fact that the two points of view are reconciled by ton Foundation, FQXi, and the Austrian Science Fund Postulate 2 seems promising. Alas, further investigation (FWF) through CoQuS, SFB FoQuS, and the Individual leads us to conclude that the analogy cannot be made Project 2462. 6

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