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Stephen 1 ueslcinrlsi h hoyo unu nomto p information quantum of re theory article the This in entanglement. rules of superselection of manipulation context the the and in develo tion, particularly been encodings, also resou relational have of using Methods kind supers etcetera. new quantified, local a pleted, and becomes l global information that in unspeakable or quantum result frames that investigations reference many restrictions lack frames, been that also parties have t for interesting There appropriate is efficiency. it which of exampl for limit an and o information is frames can phy unspeakable instances, reference of the former important distant to are the correlating indifferent clocks of string: synchronizing are problem latter bit The “unspe the th classical of riers. while way a freedom same kind to of the new stands degrees in a time information in in quantum regular ment interest to much stands been that has there Recently, 2007) April 4 (Dated: Kingdom United 4 3 ueslcinrule superselection ueslcinrules superselection .Etnin n plcto oohrsses18 16 systems other to application and Extensions 4. a 15 without entanglement bi-partite Quantifying distillation entanglement reference 3. and phase Activation shared a 2. without Entanglement 1. .Cneune o unu nomto rcsig13 processing 11 information quantum frame for Cartesian Consequences shared a 3. without Communication 2. .Cmuiainuigpooswtotashared a without photons using Communication 1. .Gnrlzto ocmoiesses24 23 22 21 systems composite to Generalization controversy coherence 4. optical reference The phase a 3. of reference Dequantization phase a 2. of Quantization 1. pisScin lcetLbrtr,Ipra olg Lond College Imperial Laboratory, Blackett Section, Optics colo hsc,TeUiest fSde,Sde,NwSou New Sydney, Sydney, of University The Physics, of School eateto ple ahmtc n hoeia hsc,U Physics, Theoretical and Mathematics Applied of Department nttt o ahmtclSine,Ipra olg Londo College Imperial Sciences, Mathematical for Institute hrdrfrnefae17 frame reference shared hs reference phase 1 er Rudolph, Terry ,3 2, n oetW Spekkens W. 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I. INTRODUCTION – WHY CONSIDER REFERENCE It is critical to note that when one has a system encod- FRAMES IN QUANTUM INFORMATION? ing directional information, such as a spin-1/2 particle in a pure state, the direction is not defined with respect Classical information theory is typically concerned to any purported absolute Newtonian space, but rather with fungible information, that is, information for which with respect to another system, for instance, a set of the means of encoding is not important. Shannon’s cod- gyroscopes in the laboratory. Similarly, a system that ing theorems, for instance, are indifferent to whether the contains phase information, such as a two-level atom in two values “0” and “1” of a classical bit correspond to a coherent superposition of ground and excited states, is two values of magnetization on a tape, two voltages on not defined relative to any purported absolute time, but a transmission line, or two positions of a bead on an rather relative to a clock. We refer to the systems with abacus. Most information-processing tasks of interest to respect to which unspeakable/nonfungible information is computer scientists and information theorists are of this defined, clocks, gyroscopes, metre sticks and so forth, as sort, whether they be communication tasks such as data reference frames. The tasks we have highlighted thus far compression, cryptographic tasks such as key distribu- can all be described as the alignment of reference frames. tion, or computational tasks such as factoring. Nonethe- Nonfungible information is nonfungible precisely because less, there are many tasks that cannot be achieved with it can only be defined with respect to a particular type fungible information but that are also aptly described of reference frame. as “information processing” tasks. Examples include the Even a quantum information theorist who is uninter- synchronization of distant clocks, the alignment of dis- ested in tasks such as clock synchronization and Carte- tant Cartesian frames, and the determination of one’s sian frame alignment must necessarily consider physical global position. Imagine for instance that Alice and Bob systems which make use of reference frames. The reason are in separate spaceships with no shared Cartesian frame is that although fungible information can be encoded into (in particular, no access to the fixed stars). There is any degree of freedom, and thus defined with respect to clearly no way for Alice to describe a direction in space any reference frame, it is still the case that some degree to Bob abstractly, that is, using nothing more than a of freedom must be chosen, and consequently some refer- string of classical bits. Rather, she must send to Bob a ence frame is required. For instance, if a two-level atomic system that can point in some direction, a token of one qubit is being used for some task, one still requires a clock of the axes of her own Cartesian frame. This token can- in the background in order to implement arbitrary prepa- not be spherically symmetric; it must have a degree of rations and measurements on this qubit even if the task freedom that can encode directional information. On the is to perform abstract quantum information-processing other hand, if she wishes to synchronize her clock with rather than as a means of distributing phase informa- Bob’s by sending him a token system, she will need to tion. In this example, one can change the relative phase make use of a system that has a natural oscillation. The between the ground and excited states of a two-level atom information that is communicated in these sorts of tasks by a specified amount by turning on a static electric field is said to be nonfungible. These two sorts of information, for a specific time interval, but this requires a suitably fungible and nonfungible, have also been referred to as precise clock as well as alignment of the field with the speakable and unspeakable (Peres and Scudo, 2002b). atomic dipole moment. The relatively young field of quantum information It follows that to lack a reference frame for a particu- theory has been primarily concerned with developing lar degree of freedom has an impact on the success with a quantum theory of speakable information. Investi- which one can perform certain quantum information pro- gators have sought to determine the degree of success cessing tasks. On several occasions there has been consid- with which various abstract information-processing tasks erable controversy over the performance of certain tasks can be achieved assuming that the systems used to im- because this impact was ignored, or not treated properly. plement these tasks obey the laws of quantum theory. As we will see, the lack of a reference frame can be treated Nonetheless, there has also been progress in developing within the quantum formalism as a form of decoherence – a quantum theory of unspeakable information, outlining, quantum noise. As opposed to the typical source of deco- for instance, the success with which tasks such as clock herence, which is due to correlation with an environment synchronization and Cartesian frame alignment can be to which one does not have access, this decoherence can achieved in a quantum world. be viewed as resulting from correlation with a (possibly That one must look to physics to answer questions of hypothetical) reference frame to which one does not have interest to computer scientists is a fact that has not al- access. This is a powerful result, because if the lack of a ways been obvious. (Landauer (1993) summarized this reference frame can be viewed as a form of decoherence, point in the slogan, “Information is physical”.) That one the now-standard techniques of combating decoherence must look to physics to answers questions about the pro- in quantum information theory (in particular, the use of cessing of unspeakable information, on the other hand, decoherence-free subsystems) can be applied. comes as no surprise. Nonetheless, the quantum theory As it turns out, the restriction of lacking a reference of unspeakable information is only just beginning to be frame is mathematically equivalent to that of so-called explored. superselection rules – postulated rules forbidding the 3 preparation of quantum states that exhibit coherence be- Regardless of the degree of freedom in question, a ref- tween eigenstates of certain observables. Originally, su- erence frame is always associated with some physical sys- perselection rules were introduced to enforce additional tem. As such, it may be treated within the formalism of constraints to quantum theory beyond the well-studied quantum mechanics. In this case, we speak of quantum constraints of selection rules (conservation laws). They reference frames. Indeed, one can imagine an extreme were considered to be axiomatic restrictions, applying case wherein the only system in a party’s possession that to only certain degrees of freedom. For instance, a su- plays the role of a reference frame (or plays the role of a perselection rule for electric charge asserts the impossi- shared reference frame with another party) is of bounded bility of preparing a coherent superposition of different size. For instance, one can imagine a quantum clock con- charge eigenstates. As we shall see, however, for superse- sisting of an oscillator with a small maximum number of lection rules associated with compact symmetry groups, excitations, or a quantum gyroscope consisting of a hand- the presence of appropriate reference frames can actually ful of spin-1/2 systems. It is then natural to ask how allow for the preparation of such superposition states, well such a bounded-size reference frame approximates thereby obviating the superselection rules in practice. one that is of unbounded size. This shows that there is an intimate connection between The ability of a bounded reference frame to stand in for the restriction of lacking a reference frame and that of a an unbounded reference frame is analogous to the ability superselection rule. of an entangled state to stand in for the possibility of im- As Schumacher (2003) has emphasized, interesting re- plementing non-local operations. Recall that the telepor- strictions on experimental operations yield interesting in- tation protocol permits entanglement and classical com- formation theories. For instance, the fact that classical munication to substitute for a non-local operation. More channels and local operations are a cheap resource com- generally, when one lacks the ability to perform non-local pared to quantum channels leads us to study what can operations (such as when qubits are remotely separated), be achieved with local operations and classical communi- entanglement becomes a quantifiable resource. Similarly, cation (LOCC). The resulting information theory is the when one is subject to a superselection rule (i.e., when theory of entanglement. As another example, the relative one lacks a reference frame for some degree of freedom) ease with which one can implement Gaussian operations bounded reference frames become a quantifiable resource in quantum optics leads one to consider the information about which we can ask the same sorts of questions as theory that results from the restriction to these oper- we do for entanglement. For instance, we may ask the ations. By comparing and contrasting the information following questions: Which states are interconvertible? theories that result from various different restrictions we How many states of a standard form can be distilled from are led to a much broader perspective on all of them. In a given state and how many are required to form a given particular, analogies between the resulting theories allow state? How much of the resource is required for a given us to apply the insights gained in the context of one to task? How quickly is it used up? etc. solve problems arising in the context of another. In this Finally, because it is all too easy to forget about the sense, studying the restriction of a superselection rule – presence of reference frames, these are at the root of var- or equivalently, as we shall demonstrate, the restriction ious conceptual confusions. These include: the interpre- of lacking a reference frame – may yield lessons for the tation of quantum states exhibiting coherence between rest of quantum information theory. number states in a single mode (a subject of controversy In some cases, it is difficult to imagine lacking a refer- in quantum optics, Bose-Einstein condensation and su- ence frame. For example, Cartesian frames with precision perconductivity); the quantification of entanglement in on the order of fractions of a degree and clocks with preci- systems of bosons or fermions, or in situations when op- sion on the order of fractions of a second are sufficiently erations are restricted; the efficiency with which frames ubiquitous that their presence typically does not even may be aligned, clocks synchronized, etc.; and the signif- warrant mention. However, these same reference frames icance of superselection rules on the possibility of imple- become quite difficult to prepare and maintain if one re- menting various quantum information processing tasks. quires very high precision or very good stability. Further- In this article, we provide a review of the recent in- more, there are certain kinds of reference frames that are vestigations into these and related issues. In Sec. II, we difficult to prepare even if one requires only low precision introduce the formalism for treating the lack of a gen- and poor stability. For instance, a Bose-Einstein conden- eral reference frame in quantum theory, and show how sate of alkali atoms can act as a reference frame for the this is equivalent to a superselection rule. Sec. III con- phase that is conjugate to atom number, and the reli- siders quantum information processing without a shared able preparation of these has only been achieved in the reference frame. Sec. IV considers how to treat reference past decade. In addition, it is straightforward to imagine frames within the quantum formalism, which provides two parties with reference frames that are uncorrelated the starting point for a theory of distributing quantum (such as the example of the space-faring Alice and Bob reference frames – the topic of Sec. V. The effect of provided earlier). In this case we say that they lack a bounding the size of reference frames for quantum infor- shared reference frame. All of these facts demonstrate mation processing is considered in Sec. VI. Finally, we that reference frames must be considered as resources. provide an outlook to the future of this field in Sec. VII. 4

II. FORMALIZING REFERENCE FRAMES AND representations. SUPERSELECTION RULES Because group theory provides a powerful mathemati- cal tool for analyzing the role of reference frames in quan- A. Reference frames in quantum theory tum systems, we will make frequent use of group theoretic techniques throughout this review. We present a short Reference frames (RFs) are implicit in the definition of introduction to the relevant techniques in this section, quantum states. For example, in the position represen- but the reader may wish to consult a standard group tation of the wavefunction of a quantum particle, ψ(x), theory text, such as Fulton and Harris (1991) or Stern- x parameterizes the position of the particle relative to berg (1994), for further details. Also, for an introduction a spatial reference frame. More generally, the quantum to the standard mathematical techniques of quantum in- state of a system is a description of the system relative formation, we direct the reader to Nielsen and Chuang to a suitable reference frame. (2000). Consider a quantum system with Hilbert space , pre- We begin by exploring an illustrative example. H pared in a state ψ0 relative to a reference frame. We can now consider a transformation| i that changes this relation. Such a transformation can be active, changing the system B. Lacking a phase reference implies a photon-number such that it subsequently holds a different relation to the superselection rule reference frame, or passive, in which case the system is unchanged but is now described relative to a new refer- In this section, we investigate an explicit example of a ence frame. In both situations, the transformation can reference frame – a phase reference – and demonstrate be represented by a unitary operator, T (g), where g de- that if one lacks a phase reference then the resulting notes the transformation; the transformed system is then quantum theory is equivalent to one in which there is described by the state T (g) ψ . Note that these opera- a superselection rule for photon number. | 0i In quantum optical experiments, states of an optical tions can be composed, so that T (g′g) T (g′)T (g) is a ≡ mode are always referred to some phase reference. Con- transformation if both T (g′) and T (g) are, and this com- sider K optical modes as described by some party, Alice, position is associative (i.e., T (g′′g′)T (g)= T (g′′)T (g′g)). 1 relative to a phase reference in her possession – for exam- Also, there exists an inverse transformation T (g− ) to 1 ple, a high intensity laser. Let n1,...,nK be the Fock every transformation T (g), such that T (g− )T (g) = I, | (K) i 1 state basis for the Hilbert space describing these the identity. If this inverse is unique, the set of all trans- H modes, with ni the number of photons in the mode i, formations form a group G. We use g G to denote an ˆ abstract transformation within the group,∈ and say that and Ni the number operator for this mode. T is the unitary representation of this group on the quan- Consider another party, Charlie, who has a different tum system (or equivalently, on the Hilbert space ). phase reference. Let φ be the angle that relates Char- H lie’s phase reference to Alice’s. Alice can perform an In this review, we will often use two common exam- active transformation on her system of optical modes ples of a reference frame to illustrate the concepts and by allowing them to evolve under a Hamiltonian pro- ideas we cover. The first example is a phase reference, ˆ K ˆ for which the relevant group of transformations is U(1), portional to Ntot i=1 Ni, the total photon num- ber operator. Specifically,≡ the unitary transformation the group of real numbers modulo 2π under addition. A ˆ P representation of U(1) on a quantum system determines U(φ) = exp(iφNtot) will actively advance her system by how that system transforms under phase shifts. The an angle φ. Using the equivalence between the represen- second example that we use extensively in this review tations for active and passive transformations, we thus is a Cartesian frame specifying three orthogonal spatial conclude that states prepared by Alice are represented directions; the group of transformations of orientation by Charlie relative to his phase reference by performing relative to a Cartesian frame is the group of rotations a passive transformation of φ, using the representation U of U(1) on K modes given by U(φ) = exp(iφNˆ ). If ψ SO(3). An element Ω SO(3) can be given, say, by a tot | i set of three Euler angles.∈ The representation of SO(3) is the state relative to Alice’s phase reference, then this on a quantum system, then, determines how that system same state relative to Charlie’s phase reference is given transforms under rotations; for example, a spin-j particle by the transformed state transforms according to the unitary representation Rj (a ˆ U(φ) ψ = eiφNtot ψ . (2.1) Wigner rotation matrix). We will often extend the group | i | i of rotations SO(3) to the group SU(2) to allow for spinor For example, let Alice prepare the single-mode coher- ent state

∞ n α 2/2 α 1 α c n , c e−| | , (2.2) If the inverse is not unique, then the RF is instead associated | i≡ n| i n ≡ √ n=0 n! with a coset space. This situation occurs when the RF itself is X invariant under some transformations. We consider an example with α C; this state has a phase arg(α) relative to Al- of such an RF – a direction (as opposed to a full Cartesian frame) ∈ – in Sec. V. ice’s phase reference. Charlie would describe this same 5 state relative to his phase reference by a coherent state Using Eq. (2.6) yields with the same amplitude but with phase arg(α)+φ. This 2π passive transformation agrees with that of Eq. (2.1) be- dφ inφ in′φ ψ ψ = e Π ψ ψ Π ′ e− cause U | ih | 2π n| ih | n Z0 n,n′ ˆ   X eiφα = eiφN α , (2.3) 2π dφ i(n n′)φ | i | i = Π ψ ψ Π ′ e − n n 2π where Nˆ is the number operator on this single mode. ′ | ih | 0 n,n Z  As another example, let Alice prepare the two-mode X = Π ψ ψ Π . (2.8) state ( 01 + 10 )/√2. Because this state is an eigen- n| ih | n | i | i n state of Nˆtot, the transformation U(φ) induces only an X unobservable overall phase when acting on this state. Because this result applies to any state ψ , we can ex- Thus, Charlie also represents the state of the system as press the action of on an arbitrary density| i operator ρ ( 01 + 10 )/√2 relative to his phase reference. This as U two-mode| i | statei is an example of an invariant state; it is defined independently of any phase reference. [ρ]= Π ρΠ . (2.9) U n n It will be useful for us to decompose the Hilbert space n X (K) of K modes into subspaces that transform in a sim- Hple way under the action of the group U(1), as follows. The map removes all coherence between states of dif- fering totalU photon number on Alice’s systems. It follows Defining n to be the Hilbert space consisting of states of H in particular that [ρ] is invariant under phase shifts, K modes with precisely n total photons, i.e., eigenspaces U of Nˆ with eigenvalue n, we can express the Hilbert tot [ [ρ],U(φ)]=0 , φ . (2.10) space (K) as a direct sum U ∀ H ∞ Thus, if states are described relative to Charlie’s phase (K) = . (2.4) reference, Alice faces a restriction in what she can pre- H Hn n=0 pare. This restriction is characterized by the quantum M operation , which ensures that Charlie will describe any Any state ψn n transforms under phase shifts, i.e., U under the representation| i ∈ H U of U(1), as state prepared by Alice as block-diagonal in total photon number, or equivalently, as invariant under phase shifts. inφ U(φ) ψn = e ψn , ψn n . (2.5) We note in particular that the only pure states that Alice | i | i | i ∈ H can prepare are those which lie entirely within a single Define Πn to be the projector onto n. Then an arbitrary eigenspace n. state ψ (K) transforms as H H | i ∈ H Now consider the related question for operations: If a unitary operation V is performed by Alice relative to U(φ) ψ = einφΠ ψ . (2.6) | i n| i her phase reference, how is this operation described by n X Charlie relative to his phase reference? Let σ be the Now consider the situation where Charlie has no state of the system relative to Charlie’s phase reference. knowledge of the angle φ that relates his phase refer- To describe the action of V on this state if the angle φ ences to Alice’s, i.e., the laser serving as his phase ref- that relates Charlie’s phase reference to Alice’s is known, erence is not phase-locked to hers.2 Let Alice prepare a Charlie could transform this state into Alice’s frame, then quantum state ψ of K modes relative to her phase ref- apply the unitary V , then transform back to his frame; | i erence. Given that φ is completely unknown, one must the resulting state is average over its possible values to obtain the state rela- tive to Charlie. This averaging yields the mixed state U(φ)VU(φ)†σU(φ)V †U(φ)† , (2.11) 2π dφ relative to Charlie. Thus, the operation is described by ψ ψ U(φ) ψ ψ U(φ)† . (2.7) U | ih | ≡ 2π | ih | Charlie by the unitary V = U(φ)VU(φ) . If the phase Z0 φ †   φ is unknown, then Charlie would instead describe the operation by an incoherent mixture of unitaries of this form, i.e., by the map 2 It should be noted that if the phase between Alice and Char- lie’s references is changing in time in a known manner, then the 2π dφ transformation relating their descriptions is still of the form of ˜[σ] U(φ)VU(φ)†σU(φ)V †U(φ)† . (2.12) V ≡ 2π Eq. (2.1) but with φ a function of time. Given that this function Z0 is known, the parties can compensate for this effect. However, a lack of knowledge of how this phase is changing, for instance, an A notable special case is if the system was prepared by unknown drift, can eventually lead to Alice and Charlie having Alice, so that the state σ relative to Charlie’s RF is of no information about the relative phase between their references. the form σ = [ρ] as in Eq. (2.9). In this case, In such a situation, the timescale of the drift relative to the op- U erations they perform is critical; a slow drift may have negligible ˜[σ]= [V σV †] , (2.13) effect on a quick protocol. V U 6 so that ˜[σ] is also block-diagonal in total photon num- a unitary representation T , ensuring that they are com- ber. Thus,V if operations are described relative to Char- pletely reducible (Sternberg, 1994). Many of the tech- lie’s phase reference, then Alice experiences a restriction niques in this review can be applied to other groups with on what operations she can perform. some modification, but there are many technical difficul- We note that a restriction that requires states to be ties which are beyond the scope of this review. block-diagonal in the eigenspaces of some operator is Let g G be the group element relating Charlie’s refer- ∈ common in quantum theory: it is formally equivalent ence frame to Alice’s, i.e., the element in G that describes to a superselection rule (SSR) (Giulini, 1996). Many the passive transformation from Alice’s to Charlie’s ref- superselection rules in non-relativistic quantum theory, erence frame. Furthermore, suppose that g is completely such as the superselection rule for charge (Wick et al., unknown, i.e., that Alice’s reference frame and Charlie’s 1952), are characterized by an inability to prepare states are uncorrelated. It follows that if Alice prepares a state with coherence between eigenspaces of some “charge op- ρ on relative to her frame, the state of the system is H 3 erator” corresponding to different eigenvalues. Thus, we represented relative to Charlie’s frame by the state can refer to the restriction described above as a super- selection rule for photon number (Sanders et al., 2003). ρ˜ = dgT (g)ρT †(g) Alice cannot prepare, say, a coherent state α relative to G | i Z Charlie’s phase reference, but she can prepare a phase- [ρ] , (2.14) invariant state such as ( 01 + 10 )/√2. In addition, she ≡ G cannot perform the unitary| i displacement| i operation that with T (g) a unitary representation of g on , and dg the takes the vacuum 0 to a coherent state α , but she can group-invariant (Haar) measure.4 We callH the operation perform any unitary| i operation on the two-dimensional| i the “G-twirling” operation. If we choose to always subspace spanned by 01 and 10 . representG preparations by Alice relative to the reference | i | i We note that in the present context the SSR only re- frame of Charlie, then all states are of the formρ ˜ = [ρ]. G stricts preparations and operations by Alice (or any party Anyρ ˜ of this form satisfies who does not share Charlie’s phase reference). The SSR [˜ρ,T (g)]=0 , g G . (2.15) does not forbid states with coherence between different ∀ ∈ total photon-number eigenstates from existing within the theory, and in particular, Charlie (or any party who does and thus is said to be G-invariant. The proof follows from the fact that T (g)˜ρT (g)= dg T (gg )ρT (gg )=˜ρ. share Charlie’s phase reference) experiences no such re- † G ′ ′ † ′ striction on what states he can prepare. Thus, it makes Let ( ) denote the set of all bounded operators on . GivenB thatH ( ) forms a HilbertR space under the Hilbert-H sense within this context to consider what manipulations B H Alice can perform under the restriction of an SSR on gen- Schmidt inner product (σ, τ) = Tr(σ†τ), linear maps can be regarded as operators acting on ( ). These are eral (possibly coherent) states. For example, Alice can B H perform the relative phase shift which takes the state called superoperators to distinguish them from operators acting on . It is useful to define the superoperator (g) ( 0 + 1 )/√2 to ( 0 1 )/√2. Also, we note that Alice H T | i | i | i− | i by (g)[ρ]= T (g)ρT †(g), which is the unitary represen- is able to (incoherently) change the total photon number, T i.e., she can perform an operation that maps the vacuum tation of G on ( ). We may then express simply as = dg (g).B H G 0 to the single-photon state 1 . Thus, this restriction is G G T |noti equivalent to a conservation| i law for photon number. We now consider the representation of transformations. TheR most general transformation upon a quantum sys- tem, i.e., the most general quantum operation, is repre- sented by a completely positivity-preserving superopera- C. A general framework for reference frames and tor : ( ) ( ). (See Nielsen and Chuang (2000) superselection rules for theE definitionB H → B andH properties of these superoperators.) The question of interest to us is the following: if an op- In this section, we consider how to generalize the basic eration is represented by the superoperator relative to idea of the previous section – that lacking a reference Alice’s frame, how is this same operation representedE rel- frame leads to a superselection rule – beyond the case of ative to Charlie’s frame? Generalizing the justification a phase reference. We present some formal mathematical given for Eq. (2.12) in the case of a phase reference, we tools, in particular, tools from group theory and linear algebra, that we will use throughout this review paper. Suppose two parties, Alice and Charlie, are consid- ering a single quantum system described by a Hilbert 3 The invariant measure is chosen using the maximum entropy space . Let this system transform via a group G rel- principle: because Charlie has no prior knowledge about Alice’s ative toH some reference frame. Throughout this review, reference frame, he should assume a uniform measure over all we will consider both finite groups and continuous (Lie) possibilities. 4 If the group G is instead a finite group, this expression is groups. For the latter, we will restrict our attention to −1 † Gfinite[ρ] ≡ |G| Pg∈G T (g)ρT (g). In the following, we use Lie groups that (i) are compact, so that they possess a the Lie group notation exclusively; however, all results apply group-invariant (Haar) measure dg; and (ii) act on via equally well to finite groups. H 7 conclude that relative to Charlie’s frame the operation is the adjoint of a superoperator is defined relative to the represented by the superoperator ˜, where Hilbert-Schmidt inner product on the operator space, E Tr( †(σ)τ) = Tr(σ (τ)). E E ˜[ρ]= dgT (g) [T †(g)ρT (g)]T †(g) , (2.16) Recalling how operations transform under a change of E G E reference frame, if a measurement is represented by the Z set of superoperators relative to Alice’s frame, then or, equivalently, {Ek} it is represented by the set of superoperators ˜k relative ˜ {E } 1 to Charlie’s, where k is given by Eq. (2.17). Taking ˜ = dg (g) (g− ) , (2.17) E E T ◦E◦T the superoperator adjoint of Eq. (2.17), and using the ZG fact that Ek = k†[I], it follows that the POVM Ek where [ρ]= [ [ρ]]. Given that (g) is a representa- relative to Alice’sE frame is represented by the POVM{ } A◦B A B T tion of G on ( ), Eq. (2.17) has the form of Eq. (2.14) E˜k relative to Charlie’s frame where except with operatorsB H replaced by superoperators. We { } ˜ therefore refer to the map taking to as “super-G- E˜k = [Ek] . (2.20) twirling”. Any superoperator of theE formE of ˜ satisfies G E It follows that [ ˜, (g)]=0 , g G , (2.18) E T ∀ ∈ [E˜k,T (g)]=0 , g G , (2.21) where [ , ]= is the superoperator com- ∀ ∈ A B A◦B−B◦A mutator. Thus, ˜ is invariant under the action of G; it that is, the POVM E˜ is G-invariant. E k is a G-invariant operation. Thus, relative to{ Charlie’s} reference frame, the prepa- The superoperator ˜ acts on a G-invariant operator A˜ rations, operations and measurements that Alice can im- E as plement are represented by states, superoperators and POVMs of the form of (2.14), (2.17), and (2.20), respec- 1 ˜[A˜]= dg (g) (g− )[ [A˜]] tively. We now demonstrate that this restriction has the E T ◦E◦T G ZG same mathematical characterization as that of a supers- = dg (g) [A˜] election rule for a (possibly non-Abelian) group G. T ◦E◦G First, we note that the representation T of the group ZG = [A˜] , (2.19) G allows for a decomposition of the Hilbert space into G◦E◦G charge sectors q, labeled by an index q, as 1 H where we have used the fact that A˜ = [A˜] and (g− ) G T ◦ = . = q , (2.22) H H G G q Every completely positivity-preserving superoperator M admits an operator-sum decomposition of the form where each charge sector carries an inequivalent repre- [ρ] = A ρA† where the A are called Kraus op- k k k k sentation T of G. In the U(1) phase reference exam- erators.E Clearly, a sufficient condition for an operation q ple presented above, the charge sectors corresponded to to be a GP-invariant operation is for all of its Kraus opera-E eigenspaces of total photon number. Each sector can be tors A to be G-invariant operators. In general, however, k further decomposed into a tensor product, this is not a necessary condition. Note, in particular, that if V is a unitary operator that is G-invariant, then q = q q , (2.23) [ ] = V ( )V † is a G-invariant superoperator and the H M ⊗ N associatedV · · unitary transformation can be implemented of a subsystem q carrying an irreducible representa- without an RF. Nonetheless, there may exist G-invariant M tion (irrep) Tq of G and a subsystem q carrying a trivial superoperators arising from G-noninvariant unitary op- representation of G. (Recall that a representationN acts ir- erators. reducibly on a space if there are no invariant subspaces.) Finally, we consider the representation of measure- Note that this tensor product does not correspond to the ments. The most general measurement on a quantum standard tensor product obtained by combining multiple system is represented by a set of completely positivity- qubits: it is virtual (Zanardi, 2001). The spaces q and preserving superoperators , the sum of which is M {Ek} q are therefore virtual subsystems. The q and q are trace-preserving. The probability of outcome k for the Nsometimes referred to as gauge spaces andM multiplicityN measurement is pk = Tr( k[ρ]) and upon obtaining this spaces respectively.5 For the U(1) phase reference ex- outcome, ρ is updated toE [ρ]/p . The probability Ek k ample, the subsystems q are one-dimensional, and so of outcome k may also be specified by pk = Tr(Ekρ) the additional tensor productM structure within the irreps where the set E is a positive operator valued mea- { k} sure (POVM) (defined by the conditions Ek 0 and E = I). The POVM E that is associated≥ with k k { k} a measurement is obtained from the set of superoper- 5 P In high energy physics, the Mq are called colour spaces and the ators associated with it by E = †[I], where Nq are called flavour spaces. {Ek} k Ek 8 is not required; for a general superselection rule corre- that is not restricted in this way is operationally indis- sponding to a non-Abelian group G, however, they can tinguishable from a state that is, so one may as well as- be non-trivial. sume this restriction for the states also.) The standard Expressed in terms of this decomposition of the Hilbert notion of a superselection rule for an arbitrary (possi- space, the map takes a particularly simple form. Be- bly non-Abelian) group G is a restriction of the physi- cause of the broadG utility of this form, we present it as a cal states and observables to those that commute with theorem. every element of G (Giulini, 1996). This restriction on states is precisely what is asserted in Eq. (2.15), and Theorem. The action of in terms of the decomposition the restriction on observables is simply Eq. (2.21) ap- G plied to the special case of a projective measurement. = , (2.24) The restriction on transformations has traditionally only H Mq ⊗ Nq q been articulated for unitary transformations and asserts M that only G-invariant Hamiltonians are physical. This is given by is equivalent to asserting that the unitary itself be G- invariant, and such unitaries were identified above as the = ( ) q , (2.25) only ones that can be achieved when lacking an RF for G DMq ⊗ INq ◦P q the group G. Eq. (2.18) is a generalization of this restric- X tion to irreversible transformations. Thus, one can view where q is the superoperator associated with projection the restrictions of Eqs. (2.15), (2.18) and (2.21) as the P into the charge sector q, that is, q[ρ] = ΠqρΠq with formalization of the restrictions of a superselection rule P Πq the projection onto q = q q, denotes associated with the group G in the language of quantum H M ⊗ N DM the trace-preserving operation that takes every operator information theory. We shall say that the restriction due on the Hilbert space to a constant times the identity to the lack of a reference frame for G is equivalent to a M operator on that space, and denotes the identity map superselection rule associated with the group G. over operators in the space IN. N We note that although the term “superselection rule” was initially introduced to describe an axiomatic re- We provide a short proof of this theorem at the end of striction on quantum states, observables, and opera- this section. tions (Wick et al., 1952), it has been emphasized by Note that the operation has the general form of de- Aharonov and Susskind (1967) that whether or not co- coherence. Whereas decoherenceG typically describes cor- herent superpositions of a particular observable are pos- relation with an environment to which one does not have sible is a practical matter, depending on the availability access, in this case the decoherence describes correla- of a suitable reference system. We return to this issue in tion to a reference frame to which one does not have Sec. IV. access. Given that acts as identity on subsystems G Finally, although we have thus far mentioned only the q, these subsystems are called decoherence-free subsys- temsN (also known as noiseless subsystems) (Knill et al., two limiting possibilities for the correlations that might 2000; Zanardi and Rasetti, 1997). In stark contrast, hold between Alice and Charlie’s reference frames – com- acts as the completely depolarizing operation on the pletely correlated or completely uncorrelated – in gen- G eral one might wish to consider the intermediate sce- subsystems q; these are called decoherence-full subsys- tems (BartlettM et al., 2004a). nario wherein they are partially correlated. To model It follows, in particular, that a G-invariant operator this, one replaces the uniform Haar measure appearing A˜ = (A) must have the form in Eq. (2.14) with the non-uniform measure that charac- G terizes Charlie’s partial knowledge of the group element g in order to obtain a weighted G-twirling operation. Like A˜ = I A , (2.26) Mq ⊗ Nq G-twirling, this operation is noiseless on the multiplicity q M spaces, but unlike G-twirling, which is completely de- where the I are identity operators on the subsystems cohering on the gauge spaces, the weighted G-twirling Mq operation is only partially decohering on these spaces. q and the A are arbitrary operators on the subsys- M Nq tems q. WeN are now in a position to see how the restriction of Proof of Theorem 1. Our proof, which follows Nielsen lacking a reference frame for the group G is equivalent (2003), will make use of two central theorems of group to the standard notion of a superselection rule associated representation theory known as Schur’s Lemmas. We with this group. Superselection rules are most commonly state these lemmas here without proof. discussed in the context of Abelian groups where they can be described simply as a restriction of the physical states and observables to those that are block-diagonal with re- Lemma (Schur’s first). If T (g) is an irreducible repre- spect to the inequivalent representations of G (Giulini, sentation of the group G on the Hilbert space , then any H 1996). (Occasionally, this restriction is argued to hold operator A satisfying T (g)AT †(g)= A for all g G is a ∈ for the observables alone, but in this case every state multiple of the identity on . H 9

Lemma (Schur’s second). If T1(g) and T2(g) are in- III. QUANTUM INFORMATION WITHOUT A SHARED REFERENCE FRAME equivalent representations of G, then T1(g)AT2†(g) = A for all g G implies A =0. ∈ In implementing multi-partite cryptographic and com- We begin by decomposing the representation T (g) ap- munication tasks using quantum systems, it is gener- pearing in Eq. (2.14) into a sum of irreducible represen- ally presumed, at least implicitly, that all parties share tations, T (g) = T (g) where q labels inequivalent ⊕q,λ q,λ perfect reference frames for all relevant degrees of free- irreps and λ is a multiplicity index. It follows that dom. Moreover, one might think that in order to achieve some or all of these tasks, they must share such refer- [A]= dgTq,λ(g)ATq†′,λ′ (g) . (2.27) ence frames; for instance, one might think that if they G ′ ′ q,qM,λ,λ Z wish to achieve quantum communication using the Fock space of an optical mode, they must share a phase refer- Define Aq,q′ ,λ,λ′ = dgTq,λ(g)ATq†′,λ′ (g). Because of the invariance of the measure dg, it follows that ence, and if they wish to do so using spin-1/2 systems, R they must share a reference frame for spatial orientation. Tq,λ(g)Aq,q′ ,λ,λ′ Tq†′,λ′ (g)= Aq,q′ ,λ,λ′ , g G . (2.28) This impression is mistaken; quantum information pro- ∀ ∈ cessing tasks can be achieved without first establishing a Thus, by Schur’s second lemma, A ′ ′ = 0 for q = q′. q,q ,λ,λ 6 shared reference frame by using entangled states of mul- Eq. (2.27) can then be expressed as tiple systems, that is, relational encodings. A classical analogue is elucidating. If two parties do [A]= dgTq,λ(g)ATq,λ† ′ (g) . (2.29) not share a Cartesian frame, then they cannot commu- G ′ q,λ,λM Z nicate any classical information to one another through Let Π be the projection of onto the carrier space of encodings in the directional degree of freedom of a sys- q,λ H Tq,λ, and let Πq = λ Πq,λ. Then the above equation tem. For instance, if Alice encodes information into the can be expressed as orientation, relative to her frame, of a physical arrow or P gyroscope, Bob cannot access this information because he

[A]= dgT (g)Π AΠ T † ′ (g) , (2.30) can compare the system with his frame only. Nonethe- G q,λ q q q,λ q,λ,λ′ Z less, they can still communicate by encoding information X in the relative orientations of two or more such systems. and thus we can express as G We shall be concerned with the quantum analogue of such = , (2.31) relational encodings. G Gq ◦Pq q The essential idea is to use the result, presented in X Sec. II, that the effect of lacking a shared RF can be ex- where q[Aq] = ′ dgTq,λ(g)AqT † ′ (g) is a super- pressed as a form of decoherence. We then make use of G λ,λ q,λ operator on , and recall that [A] = Π AΠ . the techniques of decoherence-free subspaces and subsys- Hq P R Pq q q We now determine the form of q in terms of the tensor tems (Kempe et al., 2001; Knill et al., 2000; Zanardi and product structure = .G The projector Π can Rasetti, 1997) to find quantum states that are protected Hq Mq⊗Nq q,λ be expressed in terms of this tensor product as Πq,λ = from the noise. These techniques (and variants thereof)

Π q Πλ, where Π q is the projector onto q, and can be interpreted as yielding relational encodings. They ΠMis⊗ the rank-1 projectorM on that “picksM out” the are in fact ideally suited to the problem of overcoming the λ Nq representation λ of G. The rank-1 projectors Πλ form a lack of a shared RF because the existence of decoherence- basis for , so that Π is the identity on . Given free subspaces and subsystems relies on there being non- Nq λ λ Nq that Tq(g) acts nontrivially only on q, we can write trivial symmetries in the noise, something that may not P H Tq,λ(g)= Tq(g) Πλ. It follows that occur for a realistic noise model, but which is guaranteed ⊗ to occur in the present context. For instance, in order [A]= dg T (g) Π Π AΠ T †(g) Π ′ to redescribe, relative to one RF, a qubit state that is G q ⊗ λ q q q ⊗ λ q,λ,λ′ Z defined relative to a second, uncorrelated RF, one must X

apply to it an unknown unitary. To redescribe, relative = dg Tq(g) Πλ ΠqAΠq T †(g) Πλ′ to this RF, many qubits that were all prepared relative to ⊗ q ⊗ q Z λ λ′ the same RF, one must apply precisely the same unitary X X
X
to each. = ( ) q[A] , (2.32) GMq ⊗ INq ◦P q We begin in Sec. III.A by applying these techniques X and others to determine the efficiency with which clas- where the superoperator takes an operator B on q GMq M sical and quantum communication can be performed in to q [B]= dgTq(g)BTq†(g). By Schur’s first lemma, GM the presence of such noise. The implications for quan- q [B] is a multiple of identity on q. Therefore, be- tum key distribution are discussed in Sec. III.B. We also GM R M cause the map is trace-preserving, q = q , the entanglement G GM DM discuss the important issue of sharing be- trace-preserving map that takes every operator on q tween two parties who lack a shared RF; we demonstrate to a constant times the identity on . M Mq in Sec. III.C that a rich structure emerges in bipartite en- 10 tanglement of pure states when this restriction applies. port on the qubit Hilbert space spanned by 0 , 1 , Finally, in Sec. III.D, we investigate the cryptographic and any such state is represented by Bob as{| [iρ|]i} = U 1 power of private shared RFs, where it is assumed that p0 0 0 + p1 1 1 for pi = i ρ1 i , i.e., as an incoherent it is an eavesdropper Eve who fails to have a sample of mixture| ih | of the| ih zero-| and one-photonh | | i states. Any qubit Alice and Bob’s RF. state is completely depolarized according to Bob. Thus, quantum communication cannot be performed by using only a single mode with at most one photon. This nega- A. Communication without a shared reference frame tive result is one of many disadvantages to this encoding of a qubit into states spanned by 0 and 1 , known as | i | i 1. Communication using photons without a shared phase the “single-rail” encoding (Kok et al., 2006). Clearly, reference no quantum communication can be performed using any number of photons in a single mode, because Bob rep- Consider the following problem: Alice wants to com- resents all states prepared by Alice as being diagonal in municate some amount of classical or quantum informa- the photon number basis. tion to Bob using an optical channel, i.e., using quantum Now consider the case where Alice can make use of two states of some number of optical modes, when they do not modes in her communication to Bob. Noting that Bob share a phase reference. Using the formalism of Sec. II.B, will represent any preparation by Alice as block-diagonal a state ρ prepared by Alice is represented by Bob as the in the eigenspaces of total photon number, Alice should prepare states lying in just one of these eigenspaces if she (generally mixed) state [ρ]= n ΠnρΠn. This problem thus takes the form ofU a more standard one from quan- wishes to communicate quantum information. For exam- tum communication: how to communicateP quantum or ple, the one-photon eigenspace of two modes (labelled a classical information through a noisy channel described and b) is two-dimensional, and a general pure state on by a decoherence map . this eigenspace has the form The communication mayU be constrained in some addi- ψ = α 1 0 + β 0 1 , (3.1) tional way, such as by a limit on the number of usable | in=1 | ia| ib | ia| ib optical modes, or by an energy limit that bounds the for α, β C satisfying α 2 + β 2 = 1. Any such state maximum number of photons that can be transmitted, or ∈ | | | | satisfies [ ψ 1 ψ ] = ψ 1 ψ ; this two-dimensional sub- both. Because of these constraints, Alice and Bob wish space is aUdecoherence-free| i h | | i h subspace| of . Using states of to use a communication protocol that makes optimal use this form, Alice can communicate a singleU qubit to Bob of these resources. without requiring a shared phase reference. We note that Let’s first consider classical communication. The sim- this encoding is the commonly-used “dual-rail” encoding plest possible problem is the one wherein Alice is re- of optical quantum computing (Kok et al., 2006). Ev- stricted to sending at most one photon to Bob, using a idently, to communicate quantum information using at single optical mode. Clearly, using such a channel, Alice most N photons in M modes without a shared phase ref- can communicate a single classical bit to Bob by send- erence, Alice and Bob should make use of the eigenspace ing either a single photon 1 or no photon (the vacuum) 6 | i of total photon number N ′ (N ′ N) that has the largest 0 . This protocol does not rely on Alice and Bob shar- dimension. This eigenspace is the≤ one corresponding to ing| i a phase reference, because both the states 0 and 1 N ′ = N, and has dimension (N + K 1)!/N!(K 1)!. are invariant under the superoperator . Generalizing| i | i − − U Using multiple modes of the optical field raises addi- this result, if Alice can send at most N photons in a sin- tion issues regarding the use of reference frames, depend- gle mode, she can communicate N +1 classical messages ing on how these modes are identified, and this can lead (equivalently, log2(N + 1) classical bits) to Bob. With to a much richer structure. For example, in the dual-rail K > 1 modes, one has to consider all the possible ways encoding of Eq. (3.1), the modes a and b could represent of distributing N photons among K modes. The dimen- different spatial or temporal modes, in which case Alice sion of the resulting Hilbert space is (N + K)!/N!K!, and Bob would require a shared Cartesian frame or a and specifies the number of classical messages Alice can clock in order to identify these modes. Another common communicate using eigenstates of photon number. implementation for this encoding is for a and b to rep- What about quantum communication? Again, con- resent the two polarization modes of the single photon sider a situation wherein Alice is restricted to sending (for example, horizontal and vertical polarization) – a at most a single photon to Bob using a single optical so-called “polarization encoding” (Kok et al., 2006). For mode. Any state ρ1 Alice prepares must then have sup- Alice and Bob to share quantum information using such an encoding, although they do not need to share a phase reference, they do need to share a reference frame for

6 polarization, i.e., to agree on an axis for their polarizing Such an encoding is not feasible in practice, because all photon- materials that are used to prepare, manipulate, and mea- based communication schemes rely on obtaining a detector event (a “click”) for every message. Specifically, the detection of the sure such states. The efficiencies of general schemes for vacuum cannot be discriminated from an event where the photon transmitting quantum information via the polarization is lost, or missed by the detector. and phase of optical modes when parties do not share a 11 reference frame for polarization have been fully charac- effect as collective noise on the channel. It is still possi- terized (Ball and Banaszek, 2005, 2006). ble for Alice and Bob to communicate by encoding in the Recently, optical quantum information experiments relational degrees of freedom of the qubits, as we shall have made use of the spatial mode structure of see. light (Langford et al., 2004; Mair et al., 2001; Vaziri The problem of determining the communication capac- et al., 2003); use of this degree of freedom requires a ities in the presence of this restriction is quite simple if shared reference frame for both position and orienta- we decompose the Hilbert space in the manner dictated tion. Using spatial modes, it is possible to restrict at- by Eqs. (2.22) and (2.23). We begin with some simple tention to states of a single photon with a fixed orbital examples, illustrating the basic techniques and some few- angular momentum (the standard basis for which is the qubit schemes for classical and quantum communication, Laguerre-Gauss-Vortex modes (Siegman, 1986)). Encod- before presenting the general results. ings into a subspace of fixed orbital angular momentum will be invariant under rotations about the direction of propagation, and thus will not require a shared refer- a. One transmitted qubit. Given that R(Ω) is an irre- ence frame for orientation about this direction. These ducible representation on 1/2, by Schur’s lemma, the encodings do require a shared reference frame for the di- SU(2)-twirling on one qubitH is equivalent to the com- rection of propagation, and also a precise determination pletely depolarizing operation, of the separation between parties in order to compensate for the relative phase (Gouy shift) acquired between dif- 1 = . (3.3) ferent states of fixed orbital angular momentum during E DH1/2 propagation (Spedalieri, 2004). Thus, if Alice prepares a single qubit in the state ρ and transmits it to Bob, he represents the state of this re- ceived qubit as the completely mixed state 2. Communication without a shared Cartesian frame 1 1[ρ]= I . (3.4) We now turn our attention to the problem of how E 2 Alice and Bob can perform both classical and quan- Consequently, Bob can infer nothing about ρ from the tum communication through the exchange of spin-1/2 outcome of any measurement. So, without a shared RF, systems (qubits) when they lack a shared Cartesian Alice cannot communicate any information to Bob using frame (Bartlett et al., 2003). This problem has a much only a single qubit. richer structure than the phase-reference case investi- gated above, due to the existence of decoherence-free subsystems (rather than subspaces). For simplicity, we b. Two transmitted qubits: a classical channel. The uni- consider a noiseless channel that transmits these spin-1/2 tary representation R(Ω) 2 of SU(2) is reducible. To systems from Alice to Bob; these results can be extended ⊗ decompose it into irreducible representations, we briefly to noisy channels or higher-dimensional (spin > 1/2) sys- review the representation theory of SU(2). tems (Byrd, 2006; van Enk, 2006). The inequivalent representations of SU(2) are labeled The group of transformations of orientation relative 2 to a Cartesian frame is SO(3), which we will extend to by the total angular momentum J quantum number j. SU(2) to allow for spinor representations. We will de- The carrier spaces of these representations are the charge note an element of SU(2) by Ω, which might represent, sectors j . The carrier spaces of the irreducible rep- resentations,H the gauge spaces, are denoted . Such for instance, a set of three Euler angles. In the case of Mj a single spin-1/2 system, the Wigner rotation operators spaces have dimensionality 2j +1, and may be decom- posed into a basis j, m of eigenstates of Jz with eigen- R(Ω) provide an irreducible representation of SU(2). If ~ | i Alice sends N spin-1/2 systems to Bob, and she describes values m where m j, j +1,...,j . The multi- plicity spaces arise∈ when {− − there are different} ways of these, relative to her Cartesian frame, by ρ then Bob de- Nj scribes these same spins relative to his Cartesian frame coupling multiple systems to a given total angular mo- by mentum. A pair of spins with angular momentum num- bers j1 and j2 couple to any total angular momenta j

N N satisfying j1 j2 j j1 + j2. We summarize this as N [ρ]= dΩ R(Ω)⊗ ρR†(Ω)⊗ . (3.2) j j = |j −j |≤ ≤ (j + j ). E 1 ⊗ 2 | 1 − 2|⊕···⊕ 1 2 Z It follows that for a pair of spin 1/2 systems, we have 1 2 1 1 That is, he averages over all passive rotations that might ( 2 )⊗ = 2 2 = 0 1. The possible total angular mo- relate his frame to Alice’s, and every rotation acts on each menta are ⊗j = 0 and⊕ j = 1 and each has multiplicity N of the N spins identically as R(Ω)⊗ because each spin 1. The joint eigenstates of total angular momentum op- 2 experiences the same rotation by virtue of the fact that erators J and Jz, denoted j, m , form a basis of the each is prepared relative to the same Cartesian frame. Hilbert space (the coupled representation).| i We can re- 2 We refer to this representation of SO(3) as collective. late this coupled basis to the joint eigenstates of J1 , J1z, Thus, Bob’s lack of Alice’s Cartesian frame has the same J 2, J , denoted by j ,m j ,m (the uncoupled 2 2z | 1 1i ⊗ | 2 2i 12 representation) by and 1 if the coupling was to j1 =1. These states can be given explicitly in terms of the three spin-1/2 systems as 1, 1 = 00 (3.5) | i | i 1 1, 0 = ( 01 + 10 ) /√2 (3.6) 1 , 1 , 0 = ( 011 101 ) , (3.12) | i | i | i | 2 2 i √2 | i − | i 1, 1 = 11 (3.7) 1 | − i | i 1 , 1 , 0 = ( 010 100 ) , (3.13) 0, 0 = ( 01 10 ) /√2 ψ− (3.8) | 2 − 2 i √ | i − | i | i | i − | i ≡ 2 1 1 1 where 0 ( 1 ) is the quantum information-theoretic 2 , 2 , 1 = (2 110 101 011 ) , (3.14) shorthand| i for| i1/2, 1/2 , and 01 0 1 , etcetera. | i √6 | i − | i − | i | ± i | i ≡ | i ⊗ | i 1 These are the j = 1 (symmetric) triplet states and the 1 , 1 , 1 = ( 2 001 + 010 + 100 ) . (3.15) j = 0 (antisymmetric) singlet state. | 2 − 2 i √6 − | i | i | i Suppressing multiplicity spaces when they are 1- We can then define an isomorphism j=1/2 = j=1/2 dimensional (because j = j C = j ), we have H M ⊗ M 1 H M ⊗ j=1/2 through m λ 2 ,m,λ with m a basis of 2 N | i ⊗ | i ≡ | i | i ( ) = , (3.9) j=1/2 and λ a basis of the multiplicity space j=1/2. 1/2 ⊗ j=1 j=0 M | i N H 4 H3 ⊕ H1 Thus the total Hilbert space decomposes as

3 where the dimensionality of each space is expressed in ( )⊗ = , (3.16) 2 1/2 j=3/2 j=1/2 j=1/2 bold underneath each subspace. Writing R(Ω)⊗ = H 8 H 4 ⊕ M 2 ⊗ N 2 Rj=1(Ω) Rj=0(Ω), and applying Schur’s lemma, we in-
fer that ⊕ where again we have included the dimensions of each sub- system.

2 = ( j=1 j=1)+ j=0 , (3.10) An application of Schur’s lemma along the lines pre- E DM ◦P P sented in Sec. II.C implies that where [ρ] = Π ρΠ and Π is the projector onto the Pj j j j subspace j . This equation asserts that the coherence 3 = j=3/2 j=3/2 + ( j=1/2 j=1/2 ) j=1/2 . E DM ◦P DM ⊗ IN ◦P between theH singlet and triplet spaces is eliminated and (3.17) the triplet space is depolarized. where is the identity map. Note that the operation Thus, if Alice transmits two qubits and she assigns the I is only defined on the space of operators acting DMj ⊗INj state ρ to the pair, Bob describes the pair by on j j = j , but it is always preceded by j, which projectsM ⊗N into thisH space. If Alice prepares threeP qubits in 1 2[ρ]= pj=1( Πj=1)+ pj=0 ψ− ψ− , (3.11) the state ρ, then Bob assigns to them the state E 3 | ih | where pj = Tr(ρΠj ). Note that Bob can distinguish per- 1 1 3[ρ]= p3/2 Πj=3/2 + p1/2 I ρ , fectly between the antisymmetric state ψ− ψ− and a E 4 2 Mj=1/2 ⊗ Nj=1/2 | ih |     state ρS which lies in the symmetric subspace because (3.18) 1 2[ ψ− ψ− ]= ψ− ψ− and 2[ρS]= 3 Πj=1, and these where pj = Tr (ρΠj ) and Etwo| imagesih | are orthogonal.| ih | E Thus, Alice can communicate one classical bit to Bob 1 ρ j=1/2 = p1−/2Tr j=1/2 Πj=1/2ρΠj=1/2 . (3.19) with every two transmitted qubits by implementing the N M following protocol: Alice sends Bob the antisymmetric We note that the subsystem is unaffected
by Nj=1/2 state ψ− to communicate b = 0 and any state in the the decohering superoperator ; i.e., it is a decoherence- | i 3 symmetric subspace (for example, the state 00 ) for b = free subsystem. Thus, Alice canE encode a logical qubit | i 1. Bob then performs a projective measurement onto the into this subsystem (Kempe et al., 2001). That is, she antisymmetric and symmetric subspaces and recovers b can prepare states of the form σ ρ on j=1/2 j=1/2, with certainty. where ρ is the logical qubit state⊗ she wishesM to transmit⊗N to Bob, and Bob can access this decoherence-free subsystem and retrieve the quantum information without a shared c. Three transmitted qubits: a quantum channel. We must RF. Thus, one logical qubit can be transmitted using 3 determine how R(Ω)⊗ is decomposed into irreducible three physical qubits without a shared RF. representations. To see how three spin-1/2 systems cou- ple to total spin, imagine coupling the first pair to a 1 3 spin j1 and then coupling this to the third: ( 2 )⊗ = d. Asymptotic behaviour. The above two schemes (0 1) 1 = 1 1 3 . Note that because the third demonstrate that classical and quantum communication ⊕ ⊗ 2 2 ⊕ 2 ⊕ 2 spin 1/2 can couple to either j1 = 0 or j1 = 1 to yield are possible without a shared RF. The efficiency of j =1/2, the latter representation has multiplicity 2. We the above schemes can be increased through the use of let 1/2, 1/2, λ denote a basis of j=1/2 in the coupled more qubits, because the sizes of the decoherence-free representation,| ± wherei λ is a degeneracyH index which by subsystems can grow exponentially with increasing convention we take to be 0 if the coupling was to j1 =0 number of qubits. 13

For simplicity, we consider only the case where N is qubits encoded per physical qubit in N physical qubits 1 even. The collective (tensor) representation of SU(2) on behaves as 1 N − log2(N), approaching unity for large N − N spin-1/2 systems, R(Ω)⊗ , can again be decomposed N. Full details can be found in Kempe et al. (2001). into a direct sum of SU(2) irreps, each with angular mo- This remarkable result proves that quantum communi- mentum quantum number j ranging from 0 to N/2. That cation without a shared RF is asymptotically as effi- is, we can decompose the Hilbert space as cient as quantum communication with a shared RF, and is the communication analog of “asymptotic universal- N/2 et al. N ity” (Knill , 2000). In addition, we note that the ( 1/2)⊗ = j j , (3.20) algorithm for encoding/decoding quantum information H N M ⊗ N(N) 2 j=0 2j+1 c M j into the decoherence-free subsystems can be done effi- ciently (Bacon et al., 2006a,b). where we have indicated the dimensions of the various spaces. The multiplicity of the irrep j, which is the di- mension of j , is found from representation theory to be N e. Relativistic considerations. The ability to perform quantum information processing in a relativistic setting (N) N 2j +1 has also been of recent interest (see Peres and Terno c = . (3.21) j N/2 j N/2+ j +1 (2004) for a review), and in this context it is natu-  −  ral to consider whether parties who do not share an Relative to this decomposition, the SU(2)-twirling op- inertial frame (i.e., a reference frame for the Poincar´e eration has the form EN group) can still perform quantum communication, etc., and at what efficiency. It has been shown that classi- N = ( ) j , (3.22) E DMj ⊗ INj ◦P cal and quantum communication can be performed at j X the same rate as demonstrated above using indistinguish- as can be inferred from the result for arbitrary groups able massive spin-1/2 particles, or using photons, if ap- in Sec. II.C. The carrier spaces for the irreducible rep- propriately localized wavepackets for these particles are used (Bartlett and Terno, 2005). In addition, continuous- resentations of SU(2), the j , are the decoherence-full M variable quantum information can be shared using re- subsystems for N , while the multiplicity spaces j , which carry theE trivial representation of SU(2), areN the lated methods (Kok et al., 2005). decoherence-free subsystems for N . Alice can choose to transmitE classical messages by preparing orthogonal states as follows: for each irrep j, 3. Consequences for quantum information processing she can choose one arbitrary state from each multiplic- ity. Thus it is possible to transmit, without a shared The communication schemes presented above imply RF, a number of classical messages equal to the number that Alice and Bob can share entangled states in the C(N) of SU(2) irreps in the direct sum decomposition of absence of any particular shared RF. Consider the the tensor representation of SU(2) on N qubits, which is case of lacking a shared Cartesian frame as an ex- given by ample. Denoting the logical qubit that can be en- coded using three physical qubits in Alice’s (Bob’s) N/2 possession by 0 , 1 , a triple of physical (N) N L A(B) L A(B) C(N) = c = . (3.23) qubits in Alice’s{| possessioni | i can} be maximally entan- j N/2 j=0 X   gled with a triple in Bob’s possession using the state 1 ( 0 0 + 1 1 ). Because Alice and Bob In fact, this is the maximum number of classical messages √2 | LiA| LiB | LiA| LiB that can be sent; for a proof, see Bartlett et al. (2003). can perform any measurement in their respective logi- Thus, the number of classical bits that can be transmit- cal qubit Hilbert spaces, they can demonstrate quantum 1 (N) nonlocality (Bell’s theorem) despite having no shared ted per qubit using the above scheme is N − log2 C , 1 Cartesian RF (Bartlett et al., 2003; Cabello, 2003). It which tends asymptotically to 1 (2N)− log2 N; in the large N limit, one classical bit− can be transmitted for also follows that such entangled states can be used for every qubit sent. Remarkably, this rate is equivalent to quantum teleportation of logical qubits, which implies what can be accomplished if Alice and Bob do possess a that the latter does not rely upon the existence of a shared RF. shared Cartesian RF either, contrary to some expecta- To determine the optimal scheme for transmitting tions (van Enk, 2001). In fact, for any quantum infor- quantum (rather than classical) information, again us- mation task that assumes some shared RF, it is possi- ing N qubits and under the restriction of no shared RF, ble to make use of logical encodings to perform the task we identify the largest decoherence-free subsystem for without this shared RF. (Any task, that is, which deals with speakable rather than unspeakable quantum infor- N . This is the subsystem j with the greatest mul- E (N) N mation; the alignment of RFs, for instance, obviously tiplicity cj . Asymptotically, this is found to occur at cannot be achieved in this way.) It should be noted, how- √ 1 (N) jmax = N/2, and the number N − log2 cjmax of logical ever, that although one can achieve quantum information 14 tasks without any particular kind of shared RF, some is used as the quantum channel, the polarization of a form of shared RF is always required. For instance, in transmitted photon is rotated by a random amount due the example just described, Alice and Bob must agree on to optical birefringence. Although this random rotation the ordering of the three physical qubits, and this agree- fluctuates with time, it can be considered constant on a ment constitutes a kind of shared RF (Bartlett et al., short time scale so that all photons in a pulse are sub- 2004a). ject to the same rotation. Thus, the problem becomes Several recent experiments have demonstrated the key equivalent to one in which Alice communicates to Bob techniques required for quantum information processing using a noiseless channel, but in which they do not share without a shared Cartesian frame. These experiments a reference frame for polarization. The communication make use of single-photon polarization qubits. Lacking scenario, then, becomes equivalent to that analyzed in a shared RF for polarization means that Bob’s polariz- Sec. III.A.2. ing elements (such as calcite crystals) are uncorrelated Alice can perform quantum communication with Bob with Alice’s. The relevant group is also SU(2), and thus without a shared RF for polarization through the use the analysis presented above applies to this scenario as of decoherence-free subspaces or subsystems. We now well. Banaszek et al. (2004) have demonstrated that two briefly outline two straightforward and experimentally- orthogonal entangled states of two single-photon polar- accessible QKD protocols using these techniques; the ization qubits remain perfectly distinguishable between first protocol makes use of a four-photon decoherence-free two parties who do not share a reference frame for polar- subspace, and the second makes use of a three-photon ization, thereby demonstrating the classical communica- decoherence-free subsystems. tion protocol in Sec. III.A.2b. In addition, Bourennane The smallest non-trivial decoherence-free subspace for et al. (2004) have demonstrated non-orthogonal entan- the superoperator N of Eq. (3.2) occurs for N = 4. gled states – states of a logical qubit encoded in four It is the two-dimensionalE j = 0 (singlet) subspace. A single-photon polarization qubits – that are identical in simple QKD scheme using this subspace is as follows. any reference frame; see also Zou et al. (2006). These Define the state ψ− µν = ( 0 µ 1 ν 1 µ 0 ν )/√2 to be states demonstrate the basic principles of a decoherence- the two-photon singlet| i state| i of| photonsi − | iµ| andi ν (µ, ν free subsystem that are needed for quantum communica- 1, 2, 3, 4 ). Define three four-photon states as products∈ tion without a shared RF. of{ singlet} states of differing photons, i.e.,

Ψ = ψ− ψ− , | 1i | i12| i34 B. QKD without a shared reference frame Ψ = ψ− ψ− , (3.24) | 2i | i13| i24 Ψ3 = ψ− 14 ψ− 23 . The possibility of performing secure communication | i | i | i through the use of quantum key distribution (QKD) is Clearly, all three states are j = 0 (singlet) states in the one of the most celebrated applications of quantum in- N = 4 decoherence-free subsystem. Thus, each of the formation science (Gisin et al., 2002). Because of its ad- states Ψa prepared by Alice is represented the same vanced state of development, it is also one of the first way by| Bob,i even though they do not share a reference quantum protocols to require explicit consideration of frame for polarization. shared reference frames, or the lack thereof, between Note that these states are also non-orthogonal, satis- communicating parties. All practical QKD protocols are fying Ψa Ψb = 1/2 for a = b. Thus, if Alice restricts based on the exchange of quantum states of light, and her transmitted|h | i| states to a pair6 of these, then they can as discussed in Sec. III.A.1, essentially any identification implement a B92-type QKD protocol (Bennett, 1992). In of a mode structure (either spatial, time-bin, or polar- addition, this protocol can be defined in such a way that ization) requires a reference frame of some sort. For ex- Bob need only perform single-photon measurements in ample, in all single-photon implementations of QKD, a some fixed polarization basis (i.e., without the need for shared clock is necessary in order to agree upon a short entangling measurements); see Boileau et al. (2004) for time window for communication; otherwise, dark counts details. from the photodetectors can greatly reduce security and As noted in Sec. III.A.2, there exists a two-dimensional efficiency (Brassard et al., 2000). decoherence-free subsystem with N = 3. There is a sim- QKD schemes that obviate the need for certain shared ple modification of the above QKD protocol which makes reference frames (and that are robust against other forms use of this subsystem. Define the following three mixed of noise) have recently been developed, and make use of states, obtained from the three pure states of Eq. (3.24) the techniques of decoherence-free subspaces and subsys- by discarding the last photon, i.e., tems (Walton et al., 2003). Consider the following pro- posal of Boileau et al. (2004). Alice (the sender) and ρa = Tr4[ Ψa Ψa ] . (3.25) Bob (the receiver) wish to perform QKD using the po- | ih | larization states of single photons. This choice avoids the In terms of the decomposition of the three-qubit Hilbert stabilization problems inherent in phase-based schemes, space of Sec. III.A.2c, all three of these states lie on the but presents a problem of its own: if an optical fibre subspace, and in terms of the tensor product Hj=1/2 15 structure j=1/2 = j=1/2 j=1/2, these states are defined as in Eq. (2.9) as H M ⊗ N 1 products of the completely mixed state I on j=1/2 2 M [ρ ] ΠAρ ΠA , (3.26) and one of three pure non-orthogonal states on j=1/2. UA A ≡ n A n N n Again, if Alice restricts her transmitted states to a pair X of these, then they can implement a B92-type QKD pro- A where Πn in the projector onto the eigenspace of total tocol without the need for a shared RF for polarization. photon number n on Alice’s local modes. All of Bob’s The unconditional security of the QKD schemes of operations commute with the local map B, defined sim- Boileau et al. (2004), which are based on the use of the ilarly. U above states, has been proven (Boileau et al., 2005). In In such situations, there has been considerable de- addition, a BB84-version of this QKD scheme, which does bate over the entanglement properties of certain types not require a shared reference frame for polarization, has of states, such as the two-mode single-photon state (van been demonstrated experimentally (Chen et al., 2006). Enk, 2005b; Greenberger et al., 1995; Hardy, 1994, 1995; We note that the essential concept of this scheme – to Tan et al., 1991), use the techniques of decoherence-free subspaces or sub- systems to obviate the necessity for a shared reference ( 0 A 1 B + 1 A 0 B)/√2 . (3.27) frame in QKD – can be applied to any system and RF. In | i | i | i | i particular, it has been proposed to use the spatial encod- There is a temptation to say that this state is entangled ings of optical modes discussed at the end of Sec. III.A.1 simply because of its non-product form. However, it is to perform QKD (Spedalieri, 2004). far more useful to consider whether or not this state sat- isfies certain operational notions of entanglement. One such notion is whether a state can be used to violate a Bell inequality. Another is whether it is useful as a re- C. Entanglement without a shared reference frame source for quantum information processing, for instance, to teleport qubits or implement a dense coding proto- Entanglement is often considered the key resource in col. In the context of a local photon-number superse- quantum information processing, and so it is valuable lection rule, this two-mode single-photon state fails to to consider the role of shared reference frames in both satisfy either of these notions of entanglement, because qualitative and quantitative properties of bipartite en- all such tasks would require Alice and Bob to violate the tanglement. As we will demonstrate in this section, the local photon-number superselection rule. A different but very meaning of entanglement between parties who do equally common notion of entanglement is that a state is not share a reference frame must be reassessed, with some entangled if it cannot be prepared by LOCC. The two- surprising results. mode single-photon state certainly does fit this notion because the pure non-product states cannot be prepared by LOCC. Thus we see that operational notions of en- tanglement that coincided for pure states under unre- 1. Entanglement without a shared phase reference stricted LOCC, namely being not locally preparable and being useful as a resource for tasks such as teleportation As an example, we again consider a number of optical or violating a Bell inequality, do not coincide under a lo- modes shared between two parties, Alice and Bob, who cal photon-number superselection rule, and the state in do not share a common phase reference. We will con- question is judged entangled by one notion and not the sider all states and operations to be described relative other.7 to the phase reference of a third party, Charlie, which is Another class of states whose entanglement properties assumed to be uncorrelated with both Alice’s and Bob’s have been discussed in the quantum optics literature are local phase references. (For many of the issues we wish those that are separable but not locally preparable under to consider, we could dispense with Charlie and describe a local photon-number superselection rule (Rudolph and everything relative to either Alice or Bob, but this intro- Sanders, 2001b; Verstraete and Cirac, 2003). Examples duces an artificial asymmetry into the formalism which of such states are easily leads to confusion. We therefore opt to describe all states relative to Charlie, whether he participates in + A + B , A B , (3.28) the protocol or not.) As such, Alice redescribes states | i | i |−i |−i where = ( 0 1 )/√2. (Rudolph and Sanders prepared relative to Charlie’s phase reference by mixing |±i | i ± | i over all possible phase shifts. Bob does the same, and (2001b) and Verstraete and Cirac (2003) considered because Alice and Bob’s phase references are uncorre- lated, the phases over which they mix are independent.

Recalling the results of Sec. II.B, the mixing over phases 7 yields a photon-number superselection rule, and the in- Of course, if there is no local photon-number superselection rule, this state would satisfy all of these notions of entanglement, as dependence implies that Alice and Bob are subject to emphasized by van Enk (2005b). In particular, no such super- local photon-number superselection rules. In this case, selection rule would apply if all parties share a common phase all of Alice’s operations commute with the local map , reference. UA 16 states such as the equal mixture of + + and resource for this process resolves the controversy over | iA| iB A B. For simplicity, we restrict our attention to the use of the state to demonstrate quantum nonlocal- |−ipure|−i states.) Because of the superselection rule, these ity (Greenberger et al., 1995; Hardy, 1994, 1995; Tan states cannot be prepared locally. However, because they et al., 1991). are product states, they clearly cannot be used for tasks As we have shown, this state cannot be used for such as teleportation or violating a Bell inequality. We tasks such as violating a Bell inequality when Alice and will return our attention to states such as these in Sec. IV. Bob do not share a phase reference, i.e., when a lo- In contrast, consider a state of the form cal photon-number superselection rule applies. However, combining ( 0 1 + 1 0 )/√2 with + + , one | iA| iB | iA| iB | iA| iB ( 01 A 10 B + 10 A 01 B)/√2 . (3.29) obtains a state that is useful for such tasks. The | i | i | i | i state + A + B is said to activate the entanglement of This state is certainly not locally preparable. In addi- | i | i ( 0 A 1 B + 1 A 0 B)/√2. This is seen as follows. Let tion, it can be used to violate a Bell inequality, imple- Alice| i | andi Bob| i both| i perform a quantum non-demolition ment dense coding, and so on, despite the superselection measurement of local photon number on both of their lo- rule. This is because Alice and Bob can still implement cal modes, and post-select the case where they both find any measurements they please in the 2-dimensional sub- a local photon number of one. The resulting state is spaces spanned by 01 and 10 . Thus, this state is un- ambiguously entangled| i by any| reasonablei notion. ΠA ΠB[ 1 ( 0 1 + 1 0 ) + + ] We see, then, that the remarkable and often confusing 1 ⊗ 1 √2 | iA| iB | iA| iB | iA| iB entanglement properties of states when parties do not 1 ( 01 10 + 10 01 ) . (3.31) √2 A B A B share a reference frame can be understood by recogniz- ∝ | i | i | i | i ing that different operational notions of entanglement do Violations of a Bell inequality have recently been not coincide in this case. Specifically, for pure quantum- demonstrated experimentally using the state (3.30) by optical states in a situation where Alice and Bob to not Hessmo et al. (2004) and Babichev et al. (2004). One can share a phase reference, there exists a proper gap be- take two different perspectives on such an experiment. It tween states that are locally-preparable under LOCC, is illustrative to consider them both. and states that are useful for performing quantum infor- In Hessmo et al. (2004), in addition to the state (3.30), mation tasks such as teleportation and violating a Bell in- a correlated pair of coherent states α A α B , where α equality. The existence of this proper gap is reminiscent 2 | i | i | i≡ (e α /2αn/√n!) n , are assumed to be shared be- of a similar situation for mixed quantum states: that of n −| | tween Alice and Bob.| Thesei modes are used as the local bound entanglement (Horodecki et al., 1998). This anal- Poscillators in the homodyne detections at each site. Not- ogy can be extended further; in the following section, ing that neither ( 0 1 + 1 0 )/√2 nor α α we demonstrate that some of the strange phenomena A B A B A B can be used individually| i | i for| violatingi | i a Bell inequality,| i | i from mixed-state entanglement – activation, and multi- it is unclear how it is possible to do so using such re- copy entanglement distillation – are present as well in sources. The resolution of the puzzle is that a pair of pure-state quantum optics with a local photon-number correlated coherent states α α , much like the state SSR. This analogy is pursued in detail in Bartlett et al. A B + + discussed above,|activatesi | i the entanglement of (2006a). A B the| i two-mode| i single photon state. An experimental demonstration of nonlocality using 2. Activation and entanglement distillation the two-mode single photon state can also be described as in Babichev et al. (2004). Rather than treating the In this section, we demonstrate that there exist analo- local oscillators as coherent states, they are treated as gous processes of activation (Horodecki et al., 1999) and correlated classical phase references. In this case, they multi-copy entanglement distillation (Watrous, 2004) us- constitute an additional resource that “lifts” the restric- ing pure bipartite quantum-optical states when Alice and tion of the local photon-number superselection rule, and the state ( 0 1 + 1 0 )/√2 becomes unambigu- Bob do not share a phase reference. An understanding of | iA| iB | iA| iB these processes and their relation to the above-mentioned ously entangled. These two alternative descriptions are gap between two commonly-used notions of entanglement equally valid; see Bartlett et al. (2006b). is key to resolving several recent controversies regard- The existence of such activation processes also resolves ing the entanglement of quantum-optical states (Bartlett a controversy concerning the source of entanglement in et al., 2006a; van Enk, 2005a). the experimental realization of Furusawa et al. (1998) of We now demonstrate that, to achieve a Bell inequality continuous-variable quantum teleportation. Again, it is violation with the state illustrative to consider two different perspectives of this experiment.

( 0 A 1 B + 1 A 0 B)/√2 , (3.30) The first perspective is a variant of the one presented | i | i | i | i by Rudolph and Sanders (2001b). In our language, it can it is necessary to use a process that is analogous to ac- be synopsized as follows. Alice and Bob are presumed to tivation. Understanding the necessity of an additional be restricted in the operations they can perform by a local 17 photon-number superselection rule. They share a two- ( 0 A 1 B + 1 A 0 B)/√2 has been distilled by making 2 n | i | i | i | i mode squeezed state γ = 1 γ n∞=0 γ n,n where use of two copies. 0 γ 1. In addition,| i they share− two other| modesi pre- Finally, we consider the analogue of multi-copy en- pared≤ ≤ in a product of correlatedp coherentP states α α .8 tanglement distillation from two copies of the two-mode | i| i 2 n The former is the purported entanglement resource in squeezed state γ = 1 γ n∞=0 γ n,n . Homodyne the teleportation protocol, while the latter is a quantum measurements by| i Alice and− Bob (relative| toi their uncor- version of a shared phase reference. These states are anal- related local oscillators)p can beP performed on one copy ogous to ( 0 A 1 B + 1 A 0 B)/√2 and + A + B respec- of this state to establish a shared phase reference, which | i | i | i | i | i | i tively – neither can be used as a resource for teleportation then lifts the superselection rule and causes the second when considered on its own. So the question arises as copy to become unambiguously entangled. to how teleportation could possibly have been achieved. The answer is that the product of coherent states acti- 9 vates the entanglement in the two-mode squeezed state. 3. Quantifying bi-partite entanglement without a shared The second perspective is one wherein the shared phase reference frame reference is treated classically; this perspective was taken in Furusawa et al. (1998). As described above, this classi- As we have seen above, operational notions of entan- cal shared phase reference acts as a resource that lifts the glement for a bipartite pure state no longer coincide when superselection rule, and causes the two-mode squeezed parties do not share a reference frame. How, then, does state to become unambiguously entangled. one quantify the amount of entanglement of a bipartite An analogue of multi-copy entanglement distillation state in such a situation? Entanglement measures can can also be demonstrated in our quantum optical exam- be defined in the presence of such a restriction again ple. Two copies of the state ( 0 1 + 1 0 )/√2 | iA| iB | iA| iB by being operational. In the following, we discuss one can be used to obtain free entanglement (i.e., not bound) such operational measure which quantifies the distillable in the presence of the SSR, whereas only one copy can- entanglement under a local Abelian superselection rule. not. The protocol, introduced in Wiseman (2003) and (This measure is directly related to the entanglement discussed in greater detail in Vaccaro et al. (2003), is as of particles (Wiseman and Vaccaro, 2003).) We note follows. As in the activation example above, Alice and that these results apply directly to a general (possibly Bob both perform a quantum non-demolition measure- non-Abelian) SSR, with local operations restricted as in ment of local photon number (on both local modes) and Eq. (2.17) (Bartlett and Wiseman, 2003); however, for post-select the case where they both find a local photon simplicity, we focus here on the Abelian case. number of one. The resulting state is We continue with the scenario of the previous section. Consider a bipartite state ρAB shared by Alice and Bob A B 1 2 Π1 Π1 [ ( 0 A 1 B + 1 A 0 B)]⊗ and defined relative to Charlie’s phase reference. We ⊗ √2 | i | i | i | i 1 assume that in addition to this bipartite system, Alice ( 01 A 10 B + 10 A 01 B) , (3.32) ∝ √2 | i | i | i | i and Bob each possess a number of quantum registers, not subject to any SSR, with total Hilbert space dimension where ψ 2 = ψ ψ . A process very similar to this 2- ⊗ equal to or greater than that of their respective systems. copy entanglement| i | i| distillationi has been demonstrated in (For example, these registers could be standard qubits quantum optics experiments (c.f. Ou and Mandel (1988); over which Alice and Bob have complete control.) These Shih and Alley (1988)), where correlated but unentangled registers are initiated in a pure product state ̺ . photon pairs from parametric downconversion were made AB The entanglement in the presence of an SSR of the incident on the two input modes of a beamsplitter, so state ρ is quantified through a measure E , which each photon transforms to a state of the form ( 0 1 + AB SSR | iA| iB is defined by the maximum amount of entanglement that 1 0 )/√2. Subsequently, measurements on the two A B Alice and Bob can produce between their quantum reg- output| i | i modes are postselected for one photon detection isters using local U(1)-invariant operations and classical at each output mode. The fact that their postselected communication (U(1)-LOCC). The latter can be quan- results are consistent with a description of an entangled tified by an appropriate standard measure E; it seems state demonstrates that the entanglement of the state most appropriate to use the entanglement of distillation. We now prove that the entanglement in the pres- ence of an SSR, ESSR(ρAB), is given by the entan- 8 glement E( [ρ ]) that they can produce from the The state assigned by Rudolph and Sanders (2001b) is simply loc AB state [ρU ] by unconstrained LOCC, where a mixed version (mixed over the phase of the pump beam) of Uloc AB Uloc ≡ |γi|αi|αi. A B. 9 U ⊗U van Enk and Fuchs (2002a) suggest a similar protocol to the The proof is illustrative, so we present it here. Let one we describe here, for the mixed states discussed in the pre- O = be the set of all LOCC operations by Alice and vious footnote. Homodyne measurement is performed on the {O} pump beam with respect to an external phase reference, and the Bob that commute with loc. Note that, for any quantum operation , the compositeU operation ¯ measurement result will yield a two-mode squeezed state that is E E≡Uloc ◦E◦Uloc unambiguously entangled with respect to this external RF. is in the set O. Let O be some operation on the O ∈ 18 initial state ρ ̺ . The final state of the registers coherent state for fermions. See Dowling et al. (2006).) AB ⊗ AB is given by ̺AB′ = Trsys [ρAB ̺AB] , where the trace Moreover, entangled states between angular momentum is over the shared system.O The⊗ maximum entanglement degrees of freedom of different particles will yield no real produced between the registers is given
by maximizing advantage over the two-mode single-electron Fock state E(̺AB′ ) over all operations in O. Thus, in situations wherein there is a local SU(2) superselec- tion rule. Such a superselection rule will be in force, for ESSR(ρAB) instance, if the parties fail to share a Cartesian frame for = max E Tr [ρ ̺ ] spatial orientations. As with quantum optical systems, sys O AB ⊗ AB O such considerations emphasize the need to be operational = max ETr (

)[ρ ̺ ] when classifying or quantifying entanglement. sys Uloc ◦O◦Uloc AB ⊗ AB O The theory of entanglement for indistinguishable par- = max ETr ( )[ρ ̺ ]

sys Uloc ◦E◦Uloc AB ⊗ AB ticles is another situation where considerations of en- E tanglement without a shared reference frame are rele- = max ETr [ρ ] ̺ , (3.33)

sys loc AB AB vant. States of indistinguishable particles can appear E E U ⊗  

entangled due to the necessary symmetrization or anti- where the second line follows from the properties of the symmetrization of the wavefunction. For example, in the trace and by applying the definition (2.7) to and , UA UB position representation of two indistinguishable particles, and the last line follows from the properties of the trace. a wavefunction of the two particles is expressed as The latter maximization is over all LOCC (not just op- erations that commute with loc), and gives the entan- 1 U ψ12(x1, x2)= ψ1(x1)ψ2(x2) ψ1(x2)ψ2(x1) , glement E( loc[ρAB]) that Alice and Bob can produce √2 ± between theirU registers from the state [ρ ] by un- Uloc AB (3.34)
constrained LOCC. where the cases correspond to bosons and fermions. The entanglement± properties of such a state are the sub- ject of some debate (Dowling et al., 2006; Paskauskas and 4. Extensions and application to other systems You, 2001; Schliemann et al., 2001; Wiseman and Vac- caro, 2003). From the perspective of this review, one The general perspective discussed above for investigat- can view the indistinguishability of particles as a lack ing entanglement without a shared reference frame can be of a reference ordering, i.e., lack of a reference frame to applied to other situations, although for the most part, uniquely label the particles (Bartlett and Wiseman, 2003; this issue has not been explored. Condensed matter sys- Eisert et al., 2000; Jones et al., 2005, 2006; von Korff and tems is one area where these results can be directly ap- Kempe, 2004). For example, if the particles described plied, because these systems possess a number of prac- in the above state were distinguishable through another tical restrictions on operations. Local particle-number degree of freedom, such as their spin, then the entangle- superselection rules often apply in practice; for example, ment of the above state would be unambiguous. Thus, in as noted by several authors, the single-electron two-mode many condensed matter systems, it may be worthwhile Fock state ( 0 1 + 1 0 )/√2 has ambiguous en- | iA| iB | iA| iB to consider the possibility of “lifting” the restriction of tanglement properties under this restriction (Beenakker, indistinguishability, viewed as a lack of a reference or- 2005; Dowling et al., 2006; Samuelsson et al., 2005; Wise- dering, through an appropriate reference frame. (Such a man and Vaccaro, 2003). For this reason, most proposals reference frame would necessarily make use of some phys- for creating bi-partite entangled states make use of spin ical degrees of freedom to uniquely label the particles.) or orbital angular momentum degrees of freedom of mul- tiple particles (Beenakker et al., 2003; Samuelsson et al., 2003, 2004). We note, however, that the two-mode single- D. Private shared reference frames as cryptographic key electron Fock state is an entanglement resource akin to the two-mode single-photon state, which we have shown Two parties, Alice and Bob, are said to possess a pri- to be useful through activation or multi-copy entangle- vate shared RF for some degree of freedom if their refer- ment distillation; also, a suitable shared U(1) reference ence frames are perfectly correlated with each other, and frame could “lift” the restriction of the superselection are completely uncorrelated with any other party. Such rule, and the two-mode single-electron Fock state would private shared RFs can be used as a novel kind of key be unambiguously entangled with such a resource. (De- for cryptography. To illustrate the general idea, consider termining a suitable quantum state of such a shared U(1) the case where Alice and Bob share a private Cartesian RF consisting of fermions is an outstanding problem in frame.10 They can achieve some private classical commu- general. The 2-copy entanglement distillation protocol described above applies equally well to fermionic states of the form ( 0 1 + 1 0 )/√2. The activation | iA| iB | iA| iB protocols described above, however, do not appear to 10 Although it is difficult to imagine how a Cartesian frame de- have precise fermionic analogues; specifically, there are fined by the fixed stars might be made private, it is clear that several challenges in defining an analogue of the optical if the Cartesian frame is defined by a set of gyroscopes, privacy 19 nication as follows: Alice transmits to Bob an orientable that define the SRF or the bounded degree of correla- physical system (e.g., a pencil or a gyroscope) after en- tion in the SRF). For instance, the parties can measure coding her message into the relative orientation between the Euler angles relating the private SRF to the public this system and her local reference frame (for instance, SRF and then express these in binary to obtain secret by turning her bit string into a set of Euler angles). Bob bits. Nonetheless —and this is the critical point— in can decrypt the message by measuring the relative orien- the absence of either a public SRF or public communi- tation between this system and his local reference frame. cation of unspeakable information, there is no procedure Because an eavesdropper (Eve) does not have a reference for interconverting secret key and private SRF. Thus the frame correlated with theirs, she cannot infer any infor- two resources are not equivalent. Similarly, one can show mation about the message from the transmission. that the resource of a private SRF is distinct from that We shall consider the quantum version of this exam- of entanglement. ple, where Alice sends spin-1/2 particles to Bob via a noiseless channel, as in the communication problem of Sec. III.A.2. Note that whereas in that problem Bob a. One qubit. Consider the transmission of a single qubit lacked the RF with respect to which the spins were pre- from Alice to Bob. As they share an RF, Bob represents pared, here Bob shares the RF and it is Eve who lacks it. states of this single qubit in the same way as Alice. On the other hand, Eve, who does not share Alice’s RF, de- Thus, the superoperator N of Eq. (3.2) now describes E scribes the state ρ as (ρ) = 1 I, as in Eq. (3.4). She the restriction that Eve faces by virtue of lacking the pri- E1 2 vate shared RF. In Sec. III.A.2 we sought to determine consequently cannot correlate the outcomes of her mea- how Alice could encode information in such a way that surements with Alice’s preparations. It follows that using it remained accessible to someone who lacked her RF, this single qubit and their private shared RF, Alice and whereas here we are interested in the opposite problem: Bob can privately communicate one logical qubit, and how to encode information in such a way that it is inac- thus also one logical classical bit. cessible to someone who lacks her RF (but accessible to someone who has it). We follow Bartlett et al. (2004a), to which the reader is directed for a more complete analysis. b. Two qubits: Decoherence-full subspaces. If multiple qubits are transmitted, it is possible for Eve to acquire A few points are worth noting before presenting the some information about the preparation even without results. First, private communication using a private access to the private shared RF by performing rela- SRF is similar in some ways to private-key cryptography, tive measurements on the qubits. For two transmit- specifically, the Vernam cipher (one-time pad). For ex- ted qubits in the state ρ, Eve’s description is 2(ρ) = ample, the secret key in the Vernam cipher can be used 1 E pj=1( 3 Πj=1)+ pj=0Πj=0 as in Eq. (3.11). Despite not only once to ensure perfect security. Similarly, for our sharing the RF, Eve can still discriminate the singlet and communication schemes, only a single plain-text (classi- triplet subspaces and thus acquire information about the cal or quantum) can be encoded using a single private preparation. Nonetheless, Alice can achieve some pri- SRF. If the same private SRF is used to encode two vate quantum communication by encoding the state of a plain-texts, then the relation that holds between the two qutrit (a 3-dimensional generalization of the qubit) into cipher-texts carries information about the plain-texts, j=1, the triplet subspace. Bob, sharing the private RF, and because it is possible to learn about this relation canH recover this qutrit with perfect fidelity. However, Eve without making use of the SRF, Eve can obtain this in- 1 identifies all such qutrit states with 3 Πj=1, and therefore formation. This is akin to the fact that in our example cannot infer anything about Alice’s preparation. of the classical pencil or gyroscope, Eve can measure the The property of j=1 that is key for this scheme is angular separation of the two pencils. H that the two-qubit superoperator 2 is completely depo- This analogy prompts us to raise and dismiss the pos- larizing on it, as seen explicitly fromE Eq. (3.10), which we sibility that a private SRF is equivalent, as a resource, repeat: 2 = ( j=1 j=1)+ j=0. We define subspaces to some amount of secret key or entanglement. It is true with thisE propertyDM to◦P be decoherence-fullP subspaces, con- that a private SRF may, through public communication, sistent with the terminology presented in Sec. II.C. yield secret key. Conversely, as will be seen in Sec. V.J, Now consider how many classical bits of information a secret key may, through public communication, yield Alice can transmit privately to Bob. An obvious scheme a private SRF. Moreover, if, contrary to what has been is for her to encode a classical trit as three orthogonal assumed here and in Sec. V.J, the parties possess a public states within the triplet subspace. However, this is not SRF, then a private SRF is equivalent to an unbounded the most efficient scheme. Suppose instead that Alice amount of secret key (in practice, the size of the key encodes two classical bits as the four orthogonal states is limited by the bounded size of the physical systems 1 √3 i = ψ− + n n , i =1,..., 4 , (3.35) | i 2 2 | ii | ii

where ψ− is the singlet state and the n n are four amounts to no other party having gyroscopes that are correlated | i | ii | ii with those of Alice and Bob. states in the triplet subspace with both spins pointed in 20 the same direction, with the four directions forming a orthogonal maximally entangled states on the virtual tetrahedron on the Bloch sphere; specifically, tensor product j=1/2 j=1/2, because the depolar- ization on M is sufficient⊗ N to map all of these to n Mj=1/2 1 = 0 , (3.36) 1 I 1 I , making them indistinguishable to | i | i 2 j=1/2 2 j=1/2 i Eve.M ⊗ N n2 = 0 + √2 1 , (3.37) | i √3 | i | i It turns out that the optimally efficient scheme for   private classical communication uses both the j = 3/2 n i 2πi/3 3 = − 0 + e √2 1 , (3.38) and j = 1/2 subspaces. Let j=3/2, µ , µ = 1,..., 4 | i √3 | i | i | i   be four orthogonal states on the j = 3/2 subspace, and i 2πi/3 n4 = 0 + e− √2 1 , (3.39) let j=1/2, µ , µ =1,..., 4 be four maximally entangled | i √3 | i | i states| (as describedi above) on the j =1/2 subspace. De-   fine the eight orthogonal states as in Massar and Popescu (1995). It is straightforward to verify that 1 b, µ = j=3/2, µ + ( 1)b j=1/2, µ , (3.43) | i √2 | i − | i ( i i )= 1 I , i , (3.40) E2 | ih | 4 ∀
where b =1, 2 and µ =1,..., 4. Alice can encode 3 bits i.e., all four states are represented by Eve as the com- into these eight states, which are completely distinguish- pletely mixed state. As these four states are orthogonal, able by Bob. It is easily shown that the decohering su- they are completely distinguishable by Bob and so pro- peroperator 3 maps all of these states to the completely vide an optimal private classical communication scheme. mixed state onE the total Hilbert space; thus, these states are completely indistinguishable from Eve’s perspective. c. Three qubits: Decoherence-full subsystems. Consider the case where Alice transmits three qubits to Bob. The d. General results. 3 In general, an optimally efficient pri- Hilbert space ⊗ and the superoperator decompose H1/2 E3 vate quantum communication scheme for N spin-1/2 sys- into irreps as tems is given by encoding into the largest decoherence-

3 full subsystem for N of Eq. (3.2). The largest is j=N/2 ( 1/2)⊗ = j=3/2 j=1/2 j=1/2 , (3.41) and has dimensionE N + 1. Thus, given a privateM Carte- H 8 H 4 ⊕ M 2 ⊗ N 2
sian frame and the transmission of N qubits, Alice and and Bob can privately communicate log2(N + 1) qubits. The general results for private classical communication are much more complex, and beyond the scope of this 3 = j=3/2 E DMj=3/2 ◦P review. (Observe the complexity of even the three-qubit + ( ) j=1/2 , (3.42) DMj=1/2 ⊗ INj=1/2 ◦P example above.) Here, we simply state the result, which is that the number of private classical bits that can be as in Eqs. (3.16) and (3.17). Clearly, the four-dimensional communicated using a private shared Cartesian frame subspace j=3/2 is a decoherence-full subspace. We also and N qubits is 3 log N (Bartlett et al., 2004a). see that anyH state on that is of the product form 2 Hj=1/2 These results show that, asymptotically, the private ρ σ with respect the factorization j=1/2 = j=1/2 classical capacity (3log N) is three times the private ⊗ H 1 M ⊗ 2 j=1/2 is mapped by 3 to the state I j=1/2 σ (see quantum capacity (log N). By relaxing the requirement N E 2 M ⊗ 2 also Eq. (3.18)). Thus, every state of the virtual subsys- of perfect privacy, it is possible to use the properties of tem j=1/2 is mapped to the completely mixed state on random subspaces to nearly triple the private quantum M that subsystem. Such a subsystem is an example of a capacity, almost closing the gap between the private clas- decoherence-full subsystem. sical and quantum capacities (Bartlett et al., 2005). Fi- Alice can therefore achieve private communication of nally, we note in passing that bipartite entangled states of two qubits using the decoherence-full subspace j=3/2 2N spins, completely mixed on the total-J=0 subspace, H or a single qubit using the decoherence-full subsystem have been identified as a resource for private quantum j=1/2. Note, however, that for greater numbers of and classical communication; it is illustrative to view M transmitted qubits, the decoherence-full subsystems typ- such states as quantum private shared RFs (Livine and ically have greater dimensionality than the decoherence- Terno, 2006). full subspaces, and schemes that encode within them are necessary to achieve optimal efficiency, as discussed be- low. IV. QUANTUM TREATMENT OF REFERENCE FRAMES For private classical communication, the question of optimal efficiency is much more complex. One scheme As we have seen in the previous two sections, the lack would be for Alice to encode two (classical) bits into of a reference frame has the effect of inducing a superse- four orthogonal states within the j = 3/2 decoherence- lection rule. We have explored examples of how the lack full subspace. Alice can also encode two bits into four of a phase reference in quantum optics experiments leads 21 to an Abelian SSR and how the lack of a Cartesian frame and Braunstein, 2003). We consider the pair consisting leads to a non-Abelian SSR. of our original system, which we denote by S, and a new However, some SSRs are typically viewed as being system, which we denote by R. Defining the states axiomatic; a canonical example is a SSR for electric charge, which forbids superpositions of eigenstates of dif- χ0(π) R = n 1 R n R /√2 , (4.3) | i | − i ± | i ferent charge (Strocchi and Wightman, 1974; Wick et al., on R (with n 1), we may then define
states on the pair 1952, 1970). In a classic paper, Aharonov and Susskind ≥ (1967) challenged the necessity of this SSR, and outlined with relative phases 0 and π respectively, a gedanken experiment for exhibiting a coherent super- Ψ = χ ψ χ ψ /√2 , (4.4) position of charge eigenstates as an example of how this | 0iRS | 0iR| 0iS − | πiR| πiS SSR can be obviated in practice. This gedanken exper- Ψπ RS = χ0 R ψπ S χπ R ψ0 S
/√2 . (4.5) iment highlights the requirement of an appropriate ref- | i | i | i − | i | i erence frame in order to exhibit superpositions between Noting that these states can also be expressed
as eigenstates of superselected quantities, and as a result it can be argued that an SSR is simply a practical limita- Ψ0 RS = n R 0 S + n 1 R 1 S /√2 , (4.6) | i | i | i | − i | i tion due to the lack of such a reference frame. This point Ψπ RS = n R 0 S n 1 R 1 S
/√2 , (4.7) has been repeated by several authors (Giulini, 2000a,b; | i | i | i − | − i | i Lubkin, 1970; Mirman, 1969, 1970). it is clear that both of these states are
eigenstates of In this section, we demonstrate that this result is gen- total number with eigenvalue n and are therefore valid eral: any SSR associated with a unitary representation preparations under the SSR. of a compact group can be viewed as the lack of an ap- Moreover, within the eigenvalue n eigenspace, one can propriate reference frame, and can be overcome by using measure the basis Ψ0 , Ψπ in order to statistically an appropriate quantum system to serve as a reference distinguish states with{| ai well-defined| i} relative phase from frame. 1 1 those, like 2 n +1 n +1 0 0 + 2 n n 1 1 , which do not| have ai h well-defined| ⊗ | i hrelative| | phase.i h | ⊗ Clearly, | i h | this measurement is also valid within the constraints of A. Relational descriptions of phase the SSR. In fact, for every preparation, operation and measure- 1. Quantization of a phase reference ment of the system that is not U(1)-invariant, one can find an equivalent preparation, operation and measure- Suppose we have a system that transforms under U(1) ment for the relation between the pair of systems that is and where the associated eigenstates are denoted by n . U(1)-invariant. To do so, we simply use the map For concreteness, we shall imagine these to be eigenstates| i of photon number, or of number of bosonic atoms, in 0 n 0 | i → | iR | iS some mode. If there is no SSR for U(1), then we can 1 n 1 R 1 S , (4.8) prepare states such as | i → | − i | i so that in particular, we have ψ0 = 0 + 1 /√2 , (4.1) | i | i | i a 0 + b 1 a n R 0 S + b n 1 R 1 S , (4.9) ψπ = 0 1
/√2 , (4.2) | i | i → | i | i | − i | i | i | i − | i for a 2 + b 2 = 1.
which differ only in their phases. We distinguish such a It| is| straightforward| | to generalize the quantization map state, which has coherence between 0 and 1 , from the of Eq. (4.8) to the case of a system which may have more incoherent mixture I/2= 1 0 0 + 1| 1i 1 by| i measuring 2 | ih | 2 | ih | than one photon. If it has at most mmax photons, we an ensemble of such systems in the basis ψ0 , ψπ and simply use the map observing whether the outcome is random{| ori not.| i}

Now suppose instead that there is a SSR for U(1) in m n m R m S (4.10) force for this system. For optical systems, this corre- | i → | − i | i sponds as discussed above to the situation where one where we require that n mmax. In this case, ≥ lacks the phase reference with which these states were mmax mmax prepared. If the states n are eigenstates of bosonic atom cm m cm n m R m S . (4.11) number (such as are used| i in describing Bose-Einstein con- | i → | − i | i m=0 m=0 densates), then such a SSR is often assumed to be an ax- X X iomatic restriction (cf. Cirac et al. (1996); Leggett (2001); This extension of the Hilbert space corresponds phys- Wick et al. (1952); Wiseman and Vaccaro (2003)). In ei- ically to incorporating the phase reference into the quan- ther case, it becomes impossible to prepare a coherent su- tum formalism. In other words, it describes the inter- perposition of eigenstates of total number. Nonetheless, nalization or quantization of the reference frame. To see it is still possible to prepare a pair of systems in such a this, consider the following analogy with classical me- way that they have a well defined relative phase (Nemoto chanics. Suppose a ball is bounced off of a wall. If we do 22 not treat the wall as a dynamical entity, but rather as an are simultaneous eigenstates of NˆR, the number operator external potential that appears in the equations of mo- for R, and NˆS, the number operator for S with eigenval- tion of the ball, then the solutions to the equations of mo- ues n and m respectively. The operators NˆR and NˆS form tion are not translationally-invariant. Specifically, if we a complete set of commuting operators for the Hilbert take a given bouncing trajectory for the ball and trans- space . late it in such a way that the bounce no longer coincides By choosingH a different complete set of commuting op- with the location of the wall, we do not obtain another erators, we can define an alternate tensor product struc- solution – the external potential breaks the translation- ture for the Hilbert space. Specifically, we choose NˆS, invariance. However, if we internalize the wall, that is, the number operator for S, and Nˆtot = NˆS + NˆR, the to- treat its position as a dynamical degree of freedom, then tal number operator. The state m R n S is also a joint we find that the equations of motion, and the solutions, eigenstate of this pair, with eigenvalues| i | i m and m + n will be invariant under translations of the entire system respectively. Given that NˆS and Nˆtot form a complete (consisting of the ball and the wall). set of commuting observables, we may label an element Similarly, when one writes down a state such as of the basis n R m S instead by the eigenvalues of NˆS {| i | i } ψ = ( 0 1 )/√2, the phase of this state is only and Nˆ , that is, n m = N = m + n,N = m . | 0(π)i | i ± | i tot R S tot S defined relative to an external phase reference. We can Now, if it were| thei | casei that| any pair of values, onei view this external phase reference as a type of external drawn from the spectra of NS and the other drawn from potential, which provides the means for preparing states the spectra of Ntot, could be simultaneous eigenvalues of and performing operations (i.e., giving solutions to the NS and Ntot, then we could define a new tensor product quantum-mechanical equations of motion) that are not structure by Ntot = l,NS = m = l m . However, invariant under phase shifts. However, if we incorporate any pair (l,m|) with m>l cannoti be| simultaneousi ⊗ | i eigen- the phase reference as an internal system (and we do values. This problem can be resolved by restricting our not compare our internal systems to any other external attention to states n R m S where the minimum value phase reference), then the only empirically meaningful of n is larger than| thei | maximumi value of m. Recall- states and operations are invariant under phase shifts of ing the physical significance of these eigenvalues, we see the entire system (including the internalized phase refer- that this corresponds to assuming that the RF has more ence). excitations than the system. Whether one treats the wall in our classical example Assuming a system with at most mmax excitations, as an external potential or an internal dynamical system and a reference with a number of excitations that is is a choice of the physicist. Similarly, one can treat a ref- at least m , we may focus upon the subspace ′ = max H erence frame internally or externally; with either choice, span n R m S ,m =0,...,mmax,n mmax . It is then one can obtain an empirically adequate description of the straightforward{| i | i to introduce a tensor≥ product} structure experiment (Bartlett et al., 2006b). on ′ as follows. We define an mmax-dimensional Hilbert spaceH with an orthonormal basis m labeled by Hrel | irel the eigenvalue m of NˆS. We call this the relational 2. Dequantization of a phase reference Hilbert space. We also define a Hilbert space gl with an H orthonormal basis l gl labeled by the eigenvalue of Nˆtot. It is useful to consider the opposite problem to the We call this the global| i Hilbert space. We then have a one considered above, namely, given a description of an vector space isomorphism experiment wherein the phase reference is being treated internally, how does one obtain a description wherein it ′ = gl rel , (4.12) H ∼ H ⊗ H is treated externally? In our classical example of a ball bouncing off a wall, this involves finding the equations of which is made by identifying motion for the relative position of the ball to the wall. l m N =l,N =m , (4.13) We would like to determine the quantum analogue of | igl| irel ≡ | tot S i this process. In the context of a quantum reference frame for all m mmax and l mmax. for spatial location, it is relatively straightforward. To We can≤ therefore define≥ a linear map from the subspace externalize the reference frame, one defines a novel ten- ′ of R S to gl rel in terms of their respective sor product structure of the Hilbert space in terms of Hbasis statesH ⊗ H as H ⊗ H the commuting pair of observables qR qS and pR + pS, where q ,p and q ,p are the position− and momentum R R S S n R m S m + n gl n rel . (4.14) operators for the reference frame and system respectively. | i | i 7→ | i | i The procedure is a bit more subtle in our U(1) exam- Under this map, we have ple, but also involves identifying a novel tensor product a n +1 0 + b n 1 n (a 0 + b 1 ) . structure of the Hilbert space. The original tensor prod- | iR | iS | iR | iS 7→ | igl ⊗ | irel | irel uct structure, corresponding to the reference frame and (4.15) system division, will be denoted = . The Any U(1)-invariant state on R S will lead to a state R S H ⊗ H product states with respect to thisH structure,H ⊗ Hn m , on that commutes with Nˆ , i.e., the state will | iR | iS Hrel ⊗Hgl tot 23 be diagonal in the number basis of gl. By discarding two cases. For instance, if the state of S in the external- the global degrees of freedom and consideringH only the R paradigm is ψ = ( 0 + 1 )/√2 of Eq. (4.1), after | 0i | i | i reduced density matrix on rel, we are essentially mov- internalizing R, the joint state is Ψ0 of Eq. (4.6), and H 1 1 | i ing to a paradigm of description wherein the RF is not the state on S is 2 0 0 + 2 1 1 . treated within the quantum formalism. We call this pro- H | ih | | ih | cedure externalizing or dequantizing the reference frame. Thus, just as the classical and quantum treatments of For instance, if we follow the map of Eq. (4.15) by a trace the gain medium in the generation of laser light led to over gl, we obtain the map distinct state ascriptions for the radiation field, our clas- H sical and quantum treatments of the phase reference R a n +1 0 S + b n R 1 a 0 rel + b 1 rel , (4.16) lead to distinct state ascriptions for the system S. It is | iR | i | i | iS 7→ | i | i a mistake however to conclude that the two descriptions which is the inverse of Eq. (4.9), the map describing the are inconsistent. As we saw in the previous subsection, internalization or quantization of the phase reference. both descriptions are valid. To resolve the confusion ex- plicitly, we elaborate on the physical interpretation of the states in these Hilbert space. 3. The optical coherence controversy In the external-R paradigm, we saw that the phase of This simple analysis of the quantum treatment of refer- the quantum state of S (that is, the phase of the ratio ence frames is useful for resolving a controversy concern- of amplitudes of 0 and 1 ) can only be given meaning ing whether quantum coherences between photon number relative to the external| i phase| i reference R. So it is clear eigenstates are fact or fiction (Bartlett et al., 2006b; van that the state on S describes not just the intrinsic prop- Enk and Fuchs, 2002a,b; Fujii, 2003; Gea-Banacloche, erties of S, but someH of its extrinsic properties as well, 1998; Molmer, 1997, 1998; Nemoto and Braunstein, 2002, specifically, its relation to R. 2003, 2004; Rudolph and Sanders, 2001a,b; Sanders et al., 2003; Smolin, 2004; Spekkens and Sipe, 2003; Wiseman, In the internal-R paradigm, any phase of the quantum 2003, 2004). It is standard practice in quantum optics state of S can also only be given meaning relative to an to model the state of the electromagnetic field generated external phase reference, but R is no longer an external by a laser to be a coherent state, which is a coherent RF, and any phase reference that is still treated exter- superposition of photon number eigenstates. One justifi- nally, say R′, has been assumed not to be correlated with cation that may be given for such an approach is that if S. Thus, we expect S to not have a well-defined phase in one imagines the source of the radiation to be a classical this case. The point is that in the internal-R paradigm, oscillating dipole (which seems a reasonable assumption) also describes extrinsic properties of S, but in this HS then a simple calculation shows that the field is left in case it is the relation of S to R′, rather than R. a coherent state. On the other hand, if one quantizes the dipole moments in the gain medium and assumes Thus, the fact that the quantum states on S are dis- that these are initially in a thermal state (which must tinct in the two paradigms is not an inconsistencyH be- have zero expectation value of the dipole moment opera- cause despite the common notation, they describe differ- tor), and that the coupling between the gain medium and ent degrees of freedom: one describes the relation of S to the radiation field conserves photon number (which again R and the other the relation of S to R′. seem like reasonable assumptions), then the reduced den- sity operator of the field is found to be in an incoherent Moreover, if one wishes to recover the quantum state mixture of photon number eigenstates (Molmer, 1997). describing the relation of S to R in the internal-R The fact that distinct states are obtained by the two paradigm, it is clear that one should not look at the quan- analyses has led many researchers to conclude that the tum state on S because this amount to tracing over R two descriptions are inconsistent and that one must be which correspondsH to ignoring R, and one clearly cannotH wrong. ignore a system when one seeks to find the relation be- To gain insight into this controversy, it is useful to con- tween it and another. But where then is information sider the gain medium as a phase reference for the radia- about the relation between S and R found in S R? tion field. Rather than considering this case in detail, we The answer is that it is found in a virtual subsystemH ⊗ H , return to the example of the previous section, which pro- specifically in the Hilbert space rel. The resolution of vides a simplified version of the controversial phenomena. the optical coherence controversyH is achieved in an anal- Recall that we also considered two distinct paradigms of ogous manner. description for a system S and a phase reference R. In the first description – the external-R paradigm – only The key insight for resolving these sorts of confusions S was treated quantum mechanically, so that the total is that quantum states of systems in an external RF Hilbert space was S . In the second description – the paradigm do not simply describe its intrinsic properties, internal-R paradigmH – both S and R were treated quan- but also the relation of the system to the external RF; tum mechanically, so that the total Hilbert space was further discussion on this issue can be found in (Bartlett . Moreover, the state on is different in the et al., 2006b; Wiseman, 2004). HS ⊗ HR HS 24

4. Generalization to composite systems and finally a measurement associated with the POVM E , the probability of the measurement outcome k is { k}k The generalization of the quantization procedure in Tr[ (ρ)Ek]. Sec. IV.A.1 to the case of multiple systems (i.e., modes) NowE consider a quantum description for the same sys- is not so straightforward. The problem is that if we wish tem, but where a SSR for G is in force. This SSR implies to describe a pair of systems S1 and S2 relative to an a restriction on the states, transformations, and measure- RF R, then the reduced density operators on RS1 and ments. However, as we now demonstrate, it is possible on RS2 cannot both be pure entangled states (Coffman to append this system with another quantum system R et al., 2000). This fact is known as the monogamy of which serves as a quantum reference frame in such a way pure entanglement. As a result, the quantum description that, although a SSR for G applies to the entire compos- of RF and system that was presented above is only ade- ite system RS, the RF allows us to effectively describe quate if the system in question is the only one that will the system as if the SSR did not exist (Kitaev et al., ever be compared to the RF. However, the most general 2004). notion of an RF is something with respect to which the Our aim is to map the elements of the old representa- orientation of many systems can be defined. We consider tion (of preparations, operations, measurements) to ele- such a generalization presently. ments in a new, G-invariant representation on R S. First, note that if we demand that there be no limit Thus, we seek a map H ⊗ H on the number of systems that can be correlated with ρ ρinv , (4.19) the RF, and that the degree of correlation with the RF → inv be equal for all the systems, then the reduced density Ek k Ek k , (4.20) operator on RS for an arbitrary system S must be un- { } →{ } inv . (4.21) entangled, that is, a separable state. At first glance, this E→E might seem problematic, because it might seem that the inv inv inv inv such that ρ = (ρ ) and Ek = (Ek ) are both G- entanglement in Eqs. (4.9) and (4.11) is critical for the G G inv invariant operators on R S, and is a G-invariant RF quantization procedure to work. It is true that if we superoperator on ( H) ⊗H( ) (thisE implies that when B HR ⊗B HS restrict ourselves to a subspace of R of the same dimen- acting on a G-invariant operator A˜, inv satisfies ( sion as the system of interest, as weH did in Eqs. (4.9) and E G ◦ inv )[A˜]= inv[A˜]). (4.11), then we cannot obtain a faithful representation. E In◦ addition, G E we would like this map to preserve the However, by allowing ourselves to make use of a larger statistical predictions of the old representation; if all subspace, we can obtain a good representation, and in the statistics of the Born rule can be reproduced in this the limit of arbitrarily large dimension, we obtain a per- new representation, then it is equivalent to the old one. fect representation, as before. Defining the unnormalized Specifically, we want this map to be such that states inv inv inv TrRS [ (ρ )Ek ] = TrS[ (ρ)Ek] , (4.22) ∞ iφn E E χφ R = e n R , (4.17) | i | i for all states ρ, operations and measurements Ek k. n=0 E { } X Such a map does exist (assuming we can allow dR, the di- mension of , to be arbitrarily large), as we now demon- which have well-defined phase, the quantization map HR takes the form strate. First, the quantum system R that will constitute the RF for G must clearly transform under G in some non- ρ dφ χ χ U(φ)ρU †(φ) . (4.18) → | φiRh φ|⊗ trivial manner. Thus, must carry a representation Z HR of G, denoted UR, which in general will be reducible. In We must demonstrate that this is a faithful representa- order for R to serve as a complete quantum RF for G, tion of the system that satisfies the U(1)-SSR. Rather the state g on R corresponding to the configuration than doing so for the phase reference case individually, g G must| i not possessH a non-trivial invariant subgroup, ∈ we proceed directly to present the generalization of this i.e., if UR(g′) g g then g′ must be the identity. It quantization map to an arbitrary group, and prove that follows that the| i states ∝ | i of R transform as, the latter has the properties we desire. U (g′) g = g′g , g,g′ G . (4.23) R | i | i ∀ ∈ B. Quantization of a general reference frame For this quantum system to function as a perfect refer- ence frame for G, the different configurations g must | i Consider the quantum description of a system with all be distinguishable. Thus, we require that states for Hilbert space , in the case where one possesses an different configurations are orthogonal HS external reference frame for a degree of freedom associ- 1 g g′ = δ(g− g′) , (4.24) ated with the group G. The Born rule predicts that, h | i for a preparation associated with density operator ρ fol- where δ(g) is the delta-function on G defined by lowed by a transformation associated with operation dgδ(g)f(g) = f(e) for any continuous function f of E R 25

G, where e is the identity element in G. The above re- so is Einv , and that if is a superoperator with Kraus { k } E quirements are the defining properties of the left regular operators K satisfying K† K = E, then the su- { µ} µ µ µ representation of G. In the case of a Lie group, the di- peroperator inv having Kraus operators Kinv satis- P µ mensionality of R must be infinite for such states to invE inv inv { } fies (K )†K = E . Most importantly, one can exist. We refer toH such an infinite-dimensional quantum µ µ µ prove that the new representation satisfies Eq. (4.22) and RF as unbounded.11 thereforeP reproduces the quantum statistics: We now present the map from operators on to G- HS invariant operators on R S : inv inv H ⊗ H TrRS[ρ Ek ] 1 = dR− TrRS[$(ρ)$(Ek)] $ : A dg g g US(g)AUS† (g) , (4.25) 7→ G | ih |⊗ 1 Z = dR− TrRS[$(ρEk)] where US is the representation of G on the system. 1 = dR− TrRS dg g g US(g)ρEkUS† (g) Using this map $, we define the invariant versions of G | ih |⊗ density operators, elements of POVMs and Kraus oper- hZ i 1 ators respectively as = dR− TrR dg g g TrS[ρEk] G | ih | hZ i 1 = Tr [ρE ] . (4.31) ρinv = $(ρ) , (4.26) S k dR Einv = $(E) , (4.27) The case where there is a nontrivial operation can be dealt with similarly. E inv K = $(K) , (4.28) This is a remarkable result. It proves that supers- election rules cannot provide any fundamental restric- where d is the dimensionality of the Hilbert space R R′ tions on quantum theory. This has particular implica- span g ,g G spanned by the orbit of the RF states,H ≡ tions for quantum cryptography as we discuss below. It which{| mayi be∈ a} subspace of . (One can easily check R also proves that all superselection rules associated with that Tr [ρinv]=1 if Tr [ρ] =H 1.) RS S unitary representations of compact groups result from a The following are properties of the $ map: lack of an appropriate reference frame, because, as we 1. $(A) is G-invariant; have shown, including an unbounded quantum reference frame reproduces a quantum theory that is equivalent to 2. $(A + B) = $(A)+$(B) and $(AB) = $(A)$(B), one in which the superselection rule does not apply. so the algebra of operators is reproduced. The G-invariance of $(A) follows from C. Are certain superselection rules fundamental?

(UR(g′) US(g′))$(A)(UR† (g′) US† (g′)) ⊗ ⊗ We now return to the question, introduced at the be- = dg g′g g′g U (g′g)AU † (g′g) ginning of this section, of whether certain superselection | ih |⊗ S S ZG rules are more fundamental than others. That is, are = $(A) , (4.29) certain SSRs axiomatic, as opposed to those which arise in practice when there is not an appropriate RF? This where the final equality follows from the invariance of the issue bears on several controversies that are the coun- Haar measure dg. To prove property 2, we note that $ is terparts of the optical coherence controversy in other linear by definition, and that contexts. It has arisen in the context of coherence be- tween charge eigenstates in superconductivity (Anderson, $(A)$(B)= dg dg′ g g g′ g′ 1986; Haag, 1962; Kershaw and Woo, 1974) and of coher- | ih | ih | Z ence between atom number eigenstates in Bose-Einstein US(g)AU † (g)US(g′)BU † (g′) condensation (Castin and Dalibard, 1997; Hoston and ⊗ S S You, 1996; Javanainen and Yoo, 1996; Leggett, 2001; Yoo = dg g g U (g)ABU † (g) | ih |⊗ S S et al., 1997). Here, however, the intuition for the coher- Z ences being a fiction is based on the notion that the su- = $(AB) , (4.30) perselection rule for charge and for baryon number are where we have used Eq. (4.24). axiomatic, so that any quantum state that violates this From these properties, one can show that if ρ is a den- SSR does not represent reality. inv sity operator, then so is ρ , if Ek is a POVM, then To make the discussion definite, let us compare on the { } one hand quantities such as charge and baryon number, for which axiomatic SSRs are conventionally assumed to apply, and on the other, quantities such as linear momen- 11 ′ For finite groups, one need only assume that hg|g i = δg,g′ where tum, angular momentum and photon number, for which δg,g′ is the Kronecker-delta. SSRs are generally not assumed to apply. We wish to 26 consider whether our conclusion, that it is possible to ef- D. Superselection rules and quantum cryptography fectively lift a superselection rule, should apply to both of these equally. Information-theoretic security is a form of security that does not rely on assumptions about the computational Certainly, the example of the phase reference provided capabilities of one’s adversary. The appeal of quantum in Sec. IV.A applies equally well in the case of atom num- cryptography is that it offers protocols achieving this sort ber (and thus baryon number) as it does to the case of of security where classical protocols fail. Quantum key photon number. In both cases, one can certainly create distribution is the primary example of a task for which well-defined relative phases between a pair of systems. this is the case. On the other hand, there exist crypto- Moreover, the reasons for interpreting the larger of the graphic tasks, such as bit commitment, for which it has two systems as a reference frame for the other are just as been shown that even quantum protocols cannot achieve valid in the case of atom number as they are in the case information-theoretic security. The possibility of quan- of photon number. Finally, in both cases one can recover tum key distribution arises ultimately from a restriction a description of the relational degree of freedom, wherein imposed by the laws of quantum mechanics on would-be one effectively has lifted the SSR. eavesdroppers – namely, that quantum information can- Given the generalization to non-Abelian groups, pro- not be cloned. By definition, SSRs also impose restric- vided in Sec. IV.B, it would appear that all such su- tions on the accessible quantum states and operations. perselection rules may be lifted in practice. Of course, For instance, a SSR for charge forbids the creation of the technical challenge in doing so is to build a refer- superpositions of eigenstates of differing total charge. It ence frame for the degree of freedom in question. Admit- is conceivable therefore, as first suggested by Popescu tedly, it may be more difficult to construct good reference (2002), that SSRs could place restrictions on would-be frames for some degrees of freedom, but there is nothing cheaters and thereby achieve greater security for some in principle preventing their construction. For instance, tasks (for instance, unconditional security for bit com- to lift the superselection rule associated with charge, one mitment) (DiVincenzo et al., 2004; Kitaev et al., 2004; must simply have a large reference system with respect Mayers, 2002; Verstraete and Cirac, 2003). to which one can coherently exchange charge, as argued To motivate the intuition that SSRs might improve the by Aharonov and Susskind (1967). As another example, security of quantum protocols, we consider the case of the experimental realization of Bose-Einstein condensa- a partially binding and partially concealing bit commit- tion in alkali atoms provided a reference frame for the ment protocol (Spekkens and Rudolph, 2001) in the pres- phase that is conjugate to atom number (Dowling et al., ence of a superselection rule for SO(3). Alice prepares 2006). We see no obstacle in principle to lifting more two qubits in either the singlet state ψ− , which has to- general sorts of superselection rules as well. tal spin 0, or the triplet state 11 , which| i has total spin 1, according to whether she wants| i to commit a bit b =0 What sets the two categories apart in practice seems or 1 respectively. She sends Bob one of the two qubits to be the fact that some reference frames, such as those as a token of her commitment. Bob cannot distinguish for spatial location or angular position, are ubiquitous, the reduced states I/2 and 1 1 with certainty and so whereas others, such as a frame for the quantity conju- the protocol is partially concealing.| i h | At a later stage, she gate to charge, tend not to arise through natural causes sends him the second qubit, at which point Bob checks and are difficult to prepare and maintain. But this may her honesty by performing a projective measurement to be only a practical and not a fundamental difference. discriminate ψ− from 11 . There is no cheating strat- | i | i Another motivation might be given for treating the two egy that allows Alice to unveil an arbitrary bit value, so categories differently, specifically, that a superselection the protocol is partially binding. Clearly each step in rule for linear momentum would seem to imply that ob- the honest protocol respects the SSR. However it is quite jects could not be localized in space, and this, one might plausible, at first sight, that an optimal cheating strat- think, would be contrary to what is observed. However, egy for Alice will not respect the SSR – either because all that is ever observed empirically is the localization of she must prepare a state which is a superposition of two different angular momenta, such as ( ψ− + 11 )/√2, or systems relative to other systems, and this is consistent | i | i with a superselection rule for total linear momentum. If because prior to sending the second qubit to Bob she one seeks to describe the entire universe quantum me- must apply to it some local operation that violates the chanically, as is typically done in quantum cosmology and SSR. If all of Alice’s optimal cheating strategies required some approaches to quantum gravity, then it is natural to SSR violation, then the degree of bindingness against Al- assume SSRs for all global transformations, so that there ice and thus the security of the protocol would be greater is no distinction between charge and linear momentum. by virtue of the SSR. One can reach this conclusion by noting that all physi- Despite the plausibility of this notion, it turns out that cal systems that could serve as RFs have been quantized. SSRs do not, in general, offer the possibility of cryp- Alternatively, one can appeal to one of the central lessons tographic protocols with greater security (Kitaev et al., of general relativity: that all observable quantities ought 2004). This result can be proven using the general frame- to be relational. work of Sec. IV.B. We begin by demonstrating this for 27 the case of arbitrary two-party cryptographic protocols. that $ preserves the algebra of operators (property (2) Such protocols can be formulated as follows. Alice and of the $ map), we have $(VB )$(V ′ ) $(VB )$(V ′ )= n An · · · 1 A1 Bob each hold a local system in their laboratories, called $(VBn VA′ n VB1 VA′ 1 ) = $(V ′). The initial state and A and B respectively, and exchange a message system Bob’s final· · measurement · are also trivial on R and G- 1 M back and forth. At the outset, they share a product invariant. It follows that dR− IR ρA ρM ρB = 1 ⊗ ⊗ ⊗ state ρA ρM ρB and in each round of the protocol, d− $(ρ ρ ρ ) and I E = $(E ). ⊗ ⊗ R A M B R B,k B,k one of the parties applies a joint operation on their local Thus,⊗ the probability⊗ of outcome⊗ k in Alice’s SSR- systems and the message system and then sends the mes- respecting cheating strategy is sage system to the other party. At the end, both parties 1 d− Tr $(E )$(V ′)$(ρ ρ ρ )$(V ′†) perform a measurement on their local system. R B,k A ⊗ M ⊗ B Security in this context is a restriction on the degree = Tr EB,kV ′ρA ρM ρBV ′† = pB′ (k
) , (4.34) to which a cheating Alice can influence the probability ⊗ ⊗ where we have used Eq. (4.31). Thus, any probability distribution over the outcomes of the final measurement
distribution achieved by a cheating strategy that violates of an honest Bob, and a similar restriction with the roles the SSR can also be achieved by one that respects it. of Alice and Bob reversed. (no restrictions are guaran- It is straightforward to generalize this result to the case teed for the case where both parties cheat.) We consider of an n-party protocol with k cheating parties. We be- the case of a cheating Alice here. gin with the case of a pair of cheating parties (Alice and Because we may include any ancillas used by Alice and David). If their SSR-violating cheating strategies consist Bob in the local systems A and B, we can assume that all of unitaries V ′ and V ′ , then, by the same reasoning as operations are unitary. In the honest protocol, the first Aj Dj operation implemented by Alice is V , the first imple- applied above, they can achieve an equivalent degree of A1 success using SSR-respecting cheating strategies consist- mented by Bob is VB1 , the second by Alice is VA2 , and ing of unitaries $(V ′ ) and $(V ′ ) which are nontrivial so forth. We denote the POVM associated with Bob’s Aj Dj on RA and RD respectively. Because only one of these final measurement by EB,k where k labels the possible outcomes. The probability{ of} outcome k is parties is ever implementing an operation at a given time, they can achieve this strategy by passing the RF R back pB(k) = Tr EB,kV (ρA ρM ρB)V † , (4.32) and forth between them. Moreover, even if Alice and ⊗ ⊗ David are prevented from implementing such transmis- where
sions during the protocol, there is a resource they may share prior to the protocol, namely, a shared RF, which V = V V V V V V . (4.33) Bn An · · · B2 A2 B1 A1 allows them to do just as well. Suppose their shared RF is constituted of a pair of systems, R and R , in the Suppose that the honest protocol respects the SSR, 1 2 state and that the SSR is associated with a group G. In this case, all the states, unitaries and POVM elements de- ρ = dg g g g g . (4.35) R1R2 | iR1 h | ⊗ | iR2 h | scribed above are G-invariant operators. We now show Z that a cheating strategy that violates the SSR can al- We show that by using R2 alone, David can achieve the ways be simulated by a cheating strategy that respects same operations on MD as could be achieved if Alice had the SSR, and consequently a cheater that faces a SSR passed him R1. We define $1 and $2 as the generalizations does not suffer any disadvantage in cheating ability com- of $ for R = R1 and R = R2 respectively. If Alice had pared to one who does not. passed David a copy of R1, he could replace VD′ by the Suppose that Alice’s optimal SSR-violating cheating j operation $1(V ′ ). But given that strategy is one wherein she replaces each G-invariant op- Dj eration VAj with an operation V ′ that need not be G- $ (V ′ )ρ ρ $ (V ′† ) Aj 1 Dj R1R2 MD 1 Dj invariant. She thereby can cause the probability of out- ⊗ come k to be p′ (k) = Tr E V ′(ρ ρ ρ )V ′† = dg g g g g B B,k A ⊗ M ⊗ B | iR1 h | ⊗ | iR2 h | where V ′ = VBn VA′ VB2 VA′ VB1 VA′ . We now demon- Z n · · · 2 1
strate that there is an SSR-respecting cheating strategy (U(g)VD U †(g))ρMD(U(g)V † U †(g)) ⊗ j Dj that also leads to pB′ (k). The trick is to use the construc- =$2(V ′ )ρR R ρMD$2(V ′† ) , (4.36) tion of Sec. IV.B. Alice simply extends her local system Dj 1 2 ⊗ Dj A to RA, where R is a system that will play the role of a it follows that David achieves the same effect by replac- local reference frame. She replaces the G-noninvariant ing VD′ by the operation $2(VD′ ), which is something operation V , which acts nontrivally on AM, with j j A′ j that he can achieve locally. The generalization of this

$(VA′ j ), where $ is the map defined in Eq. (4.25). $(VA′ j ) argument to an arbitrary number of cheating parties is is a G-invariant unitary operator that acts nontrivially straightforward. on RAM. Bob’s operations must be trivial on R, so It should be noted that for Lie groups, the states we that we can write these as I V . It is useful to have been considering are, strictly speaking, not normal- R ⊗ Bj note however that because the VBj are G-invariant, it izable. However, one can introduce a sequence of nor- follows that I V = $(V ). Moreover, given malizable approximations to these states, parametrized R ⊗ Bj Bj 28 by an integer N, corresponding to RFs of bounded size, are bounded, the end result of an alignment scheme is such that the results described here are reproduced in partial correlation between the local reference frames. the limit N , that is, the limit of an unbounded RF. The operational consequence of not having complete cor- See Kitaev et→ al. ∞(2004) for details. relation is that the parties must contend with the deco- Superselection rules that are not associated with a herence that arises from the weighted G-twirling opera- compact symmetry group have also been considered, for tion discussed in Sec. II.C. As the imprecision in this instance, the superselection rule for univalence which de- alignment approaches zero, the weighted G-twirling op- nies the possibility of a coherent superposition of boson eration approaches the identity map. We shall be con- and fermion (Doplicher and Roberts, 1990) and supers- cerned here with schemes that minimize this imprecision. election rules that can arise in two-dimensional systems Consider an alignment scheme for some form of refer- that admit non-Abelian anyons. Using different methods, ence frame, which makes use of a number N of trans- it has been shown that such SSRs also fail to yield any mitted quantum systems. The expected error in align- advantages for two-party cryptography (Kitaev et al., ment, measured by the variance, can be theoretically de- 2004), but the question remains open for multi-party pro- termined as a function of N. The problem has close tocols. connections with the field of quantum parameter esti- mation (Holevo, 1982) and quantum metrology (see Gio- vannetti et al. (2004a)), and a general feature of this be- V. ALIGNING REFERENCE FRAMES haviour is closely related to well-studied results in phase estimation. Specifically, if the N quantum systems are Separated parties often require the use of a shared used independently (i.e., entangled signal states are not reference frame. For instance, they may require their used, and measurements on individual systems are in- clocks to be synchronized or their Cartesian frames to be dependent) one can only achieve an error, quantified by aligned. Furthermore, although it was shown in Sec. III the variance, that scales as 1/N. This result, which is that lacking a reference frame does not prevent one from a consequence of the central limit theorem, is commonly achieving information-theoretic tasks such as communi- known as the standard quantum limit. In contrast, strate- cation, cryptography and computation, this restriction gies which make use of entanglement between the N sys- can decrease the (non-asymptotic) efficiency with which tems, as well as joint measurements, can achieve an error 2 they can be achieved and often requires more sophisti- (variance) that scales as 1/N . This result, commonly cated encodings. Thus, separated parties might opt to known as the Heisenberg limit, represents the fundamen- initially devote their communication resources to setting tal limit to the scaling of accuracy as allowed by the laws up a shared reference frame and thereafter use a standard of quantum physics (Giovannetti et al., 2006). encoding, rather than perpetually circumventing the lack We begin with a discussion of a simple example to pro- of such an RF with a relational encoding. vide some intuition about what sorts of states are optimal We refer to the process by which observers correlate for the alignment problem. Heuristically, they are states their local reference frames, that is, by which they re- that, when mixed over the action of the group, have sig- fine their knowledge of the relation between them, as nificant support on the largest possible dimensionality of reference frame alignment. In order to do so, the parties Hilbert space, thereby making them as distinguishable as must exchange systems with the relevant degrees of free- possible. This intuition can be made rigorous in the con- dom, which serve as finite samples of the sender’s local text of a simple figure of merit: the maximum likelihood RF and can be compared to the receiver’s local RF to of a correct guess. After introducing a more useful fig- obtain some information about the relative orientation ure of merit, the fidelity, we describe in detail strategies of the two frames. For example, through the exchange for the alignment of phase references, spatial directions, of spin-1/2 particles, Alice and Bob can align their lo- and Cartesian frames, and demonstrate how the Heisen- cal Cartesian frames. Exchanging quantum states of an berg limit can be achieved. We also overview results on optical mode allows them to align their phase references. the alignment problem for a few other sorts of reference We discussed in Sec. IV how a quantum system of un- frames. For alternate overviews of techniques for align- bounded size can play the same role as a classical refer- ing directions and Cartesian frames, see Peres and Scudo ence frame. By the transmission of such a system, one (2002b) and Bagan and Munoz-Tapia (2006). can achieve perfect alignment of separated classical refer- ence frames. However, one is often restricted to sending systems of bounded size, either to economize on com- A. Example: sending a direction with two spins munication resources or because of the impracticality of encodings that require a joint preparation of too many Suppose Alice and Bob have uncorrelated Cartesian systems. It is therefore of great interest to determine the frames and they wish to align their z axes by Alice trans- fundamental quantum limits on the alignment precision mitting a pair of spin-1/2 particles to Bob. What state that can be achieved for given communication resources. of these two spins should Alice prepare, and what mea- This is the question we shall address in this section. surement should Bob perform, in order to optimize their It should be noted that if the communication resources expected success in this task? 29

A seemingly reasonable strategy would be for Alice to to send parallel spins aligned with her z axis. Assuming 1 π 1+cos β 3 Alice’s z axis points in the n direction relative to Bob’s F = dβ sin β 3cos4(β/2) = . (5.6) 2 2 4 frame, this strategy corresponds to sending Bob the state Z0 n n , where (Sˆ n) n = ~ n , relative to his local frame.
| i| i · | i 2 | i We note that the same average fidelity can be achieved Bob’s task is now one of state estimation – to optimally with a finite (4-element) PVM in the basis given by estimate the pure state n given two copies (Massar and | i Eq. (3.35) (Massar and Popescu, 1995). Popescu, 1995). First, we note that the set of states from Remarkably, this method where Alice sends two par- which Bob must measure are all on the three-dimensional allel spins is not optimal; a higher average fidelity symmetric j = 1 subspace of two spins; he can Hj=1 can be achieved if Alice instead sends two anti-parallel thus restrict his measurement to a POVM on this Hilbert spins (Gisin and Popescu, 1999), as we now demonstrate. space. We consider the case where Bob performs a co- Let n n be the two-qubit state transmitted by Alice; variant measurement, i.e., a continuously-parametrized again,| i|− Bobi must perform a type of state-estimation to POVM of the form determine n. Note that the set of possible states is no 2 2 longer contained within the j = 1 symmetric subspace, E = R(Ω)⊗ E R†(Ω)⊗ , Ω SU(2) , (5.1) Ω 0 ∈ and thus Bob must now perform a measurement on the entire two-spin Hilbert space. Again, choosing a covari- where E0 = e e is a positive rank-1 operator. (Any POVM of higher| ih | rank can be simulated by a rank-1 ant POVM of the form (5.1), the new normalization con- POVM followed by classical post-processing of the re- dition now becomes sult.) To form a POVM, the vector e must satisfy the normalization condition | i dΩ EΩ = I , (5.7) Z dΩ EΩ = Ij=1 , (5.2) where I is now the identity on the full two-spin Hilbert Z space. Choosing E0 = e e to be rank-1, this normal- ization again completely| constrainsih | the POVM (up to an where Ij=1 is the identity operator on j=1. It is straightforward to show (for example,H by using arbitrary choice of single-spin basis by Bob) to be Eq. (3.10)) that this condition completely constraints the + e = √3 ψ + ψ− , (5.8) form of the POVM, i.e., it requires that e = √3 00 up | i | i | i | i | i to an arbitrary choice of single-spin basis 0 , 1 . Let 1 where ψ± = ( 01 10 ). z {| i | i} | i √2 | i ± | i Bob choose 0 to be aligned with his + direction. Again, the probability that Bob obtains the measure- If Alice sends| i two spins in the state n n , and Bob | i| i ment outcome Ω given that Alice prepared n n is a performs the measurement (5.1), then the probability of function only of the angle β between n and| Ωzi|−, giveni in Bob obtaining the measurement outcome Ω is given by this case by the Born rule, (1 + √3cos β)2 p(Ω n) = Tr[EΩ n n n n ] , (5.3) p(Ω n)= . (5.9) | | i| ih |h | | 2 = 3cos4(β/2) , (5.4) This leads to a fidelity of F = (1+ √3)/(2√3) 0.789, 1 ≃ where β = cos− (n Ωz) is the angle between n and Ωz. If which is greater that that achieved for the parallel spin Bob obtains the measurement· outcome Ω, then his best case. (We note that this fidelity can also be achieved guess as to the direction n is ng = Ωz. A natural way with a finite (4-outcome) PVM of the form with which to quantify the quality of Bob’s guess is to 2 √ n n n n use the fidelity (1 + n ng)/2 = cos (β/2), which gives 3 i i + i i 1 · i = | i|− i |− i| i + ψ− , (5.10) a value of 1 if he guesses correctly (ng = n), a value of | i 2 √2 2| i 0 if he guesses the opposite direction (ng = n), and − where n are along the four direction of the tetrahedron, which decreases monotonically between these two limits. | ii A random guess would give an average fidelity of 1/2. given in Eqs. (3.36)-(3.39). An experiment demonstrat- The average fidelity of Bob’s guess, then, is given by ing this protocol has been performed by Jeffrey et al. averaging over the distribution of transmitted states by (2006), wherein it was referred to as “quantum orien- Alice (chosen to be uniformly sampled from the sphere) teering”.) and all possible measurement outcomes by Bob, weighted A heuristic explanation of why the anti-parallel spins by the fidelity, are superior to the parallel spins for this task is obtained by investigating the orbits of the transmitted states un- 1+ n Ωz der the relevant group, in this case, the group of rotations F = dn dΩ p(Ω n) · . (5.5) | 2 SU(2). Consider the orbit under the group of a state of Z Z parallel spins n n , As the probability p(Ω n) and the fidelity depend only | i| i 1 | on the angle β = cos− (n Ωz), this expression simplifies M = R(Ω) R(Ω) n n , Ω SU(2) . (5.11) · par ⊗ | i| i ∈ 
30

This orbit has support entirely on the three-dimensional be a Schmidt decomposition of this state on q q, symmetric j = 1 subspace of the two-spin Hilbert space. where M ⊗ N In contrast, the orbit under the group of a state of anti- parallel spins n n , dq min dim q, dim q . (5.16) | i|− i ≡ { M N } n n Manti = R(Ω) R(Ω) , Ω SU(2) , (5.12) We note that, if ψq does not have maximal Schmidt ⊗ | i|− i ∈ | i (q) has support on the full four-dimensional
two-spin Hilbert rank (meaning some of the λm are zero), then the Schmidt vectors are not unique. Let ˜ be the space. The latter orbit spans a larger space, therefore Nq ⊆ Nq its elements are, loosely speaking, more orthogonal and dq-dimensional space spanned by the Schmidt vectors (q) (q) consequently easier to discriminate. We will see in the rm . Let φm be a basis for q, obtained us- following that this heuristic idea can be formalized to {|ing thei} Schmidt{| vectorsi} from (5.15)M and, if necessary, provide optimal methods for the alignment of any type completing this set arbitrarily to a basis. of reference frame. A general expression for the fiducial state is thus

dq B. General approach to aligning reference frames ψ = β λ(q) φ(q) r(q) , (5.17) | i q m | m i ⊗ | m i q m=1 X X Consider the general problem of aligning an RF asso- ciated with the group G using the one-way transmission which lies on the subspace ˜ given by of a quantum system (composed, say, of a number of el- H ⊂ H ementary systems) with Hilbert space . For instance, ˜ ˜ . (5.18) H H≡ Mq ⊗ Nq one might be trying to communicate information about q X a Cartesian frame, associated with the group SU(2), us- ing N spin-1/2 particles and corresponding Hilbert space In addition, the support of the orbit of the fiducial N = ( 1/2)⊗ . The problem is to devise an optimal state, i.e., the space protocolH H for this task, given the allowed communication ψ = span U(g) ψ , g G resources, for some given figure of merit. H | i ∈ The most general statements we can make about op- = supp ( ψ ψ ) , (5.19) timal RF distribution schemes concern the form of an G | ih | optimal POVM for a given covariant set of signal states, will also lie within ˜,   provided the figure of merit satisfies some very general H and natural properties. Optimal states for a general ψ ˜ . (5.20) task of this sort can be taken to be pure, given that any H ⊆ H mixed state scheme is a convex sum of pure state schemes Thus, for any choice of a fiducial state ψ , the mea- | i and therefore can do no better than the best pure state surement may be described by a POVM that is restricted scheme. Thus, the signal states form an orbit of pure to ˜. In addition, if the figure of merit we are attempt- H states ing to optimize satisfies some general conditions (which we discuss below), the optimal POVM can be chosen to ψ(g) = U(g) ψ , (5.13) -covariant 12 | i | i be G . Moreover, its elements can be taken to be rank-1.13 Thus, the optimal POVM must have the where U(g) is the representation of G on . We call ψ H | i form E(g) , given by the fiducial state. { } We now consider a general form for the fiducial state E(g)= U(g) e e U(g)† . (5.21) in terms of the decomposition of into irreps of G, given | ih | H by Eqs. (2.22) and (2.23). In terms of the charge sectors We call e e the fiducial POVM element. Given that , we express the fiducial state as | ih | Hq these elements form a resolution of identity on ˜, we have dg E(g)= I , or equivalently, H ψ = β ψ , (5.14) ˜ | i q| qi H q R X [ e e ]= I ˜ . (5.22) G | ih | H 2 with βq satisfying q βq = 1, and where ψq Πq ψ is the normalized component| | of the fiducial state| i∝ on each| i P charge sector q (where we recall that Πq is the pro- 12 jector onto theHqth charge sector). Each state ψ can We note that choosing a covariant POVM is sufficient to obtain | qi an optimal protocol, but not necessary. For practical schemes, be viewed as a (generally entangled) state on the tensor it may be valuable to identify finite-element POVMs that also product decomposition q = q q of Eq. (2.23). Let obtain the optimum, as we saw in Sec. V.A. H M ⊗N 13 For a proof of this, see Chiribella et al. (2005). In general, any dq non-rank-1 POVM can be simulated by a rank-1 POVM followed ψ = λ(q) φ(q) r(q) , (5.15) by classical post-processing of the result; however, this simulation | qi m | m i ⊗ | m i m=1 need not be covariant. X 31

Thus an RF alignment scheme of this sort is specified by a a continuous group, the likelihood of a correct guess is fiducial state ψ and a fiducial POVM element e e . In infinitesimal, and so we must look at the maximum like- order to determine| i an optimal scheme, we first determine| ih | lihood density – the probability density µ of obtaining the optimal e e for a given ψ . the POVM outcome E(g) given that the signal state is The constraint| ih | Eq. (5.22) completely| i fixes the form of ψ(g) , averaged over the prior distribution over signal e to be states,| i which we take to be uniform. (A more simple | i analysis is possible for finite groups.) This density takes dq the simple form e = dim( ) φ(q) r(q) . (5.23) | i Mq | m i ⊗ | m i q m=1 X q X µ = dg Tr E(g) ψ(g) ψ(g) | ih | This vector e of Eq. (5.23) can be described as follows: Z it is a coherent| i superposition across the charge sectors = e ψ 2 .  (5.29) |h | i| where ψ has support, with the amplitude squared in each such| i charge sector given by the dimensionality of As the fiducial POVM element e is fixed, optimization | i , and the projections in each such charge sector given is achieved by taking ψ to be parallel to e , Mq | i | i by maximally entangled states across q ˜q. M ⊗ N e We now demonstrate why e must take this form. ψ = | i , (5.30) Eq. (5.22) can be expressed as | i | i e k k

q q Πq e e Πq = I q I ˜ . (5.24) where e = e e . It follows from Eq. (5.23) that D ⊗ I | ih | M ⊗ Nq q q k k h | i X   X p In terms of the charge sectors, we write e = dim( )d = dim ˜ . (5.31) k k Mq q H s q p e = c e , (5.25) X | i q| qi q The optimal fiducial state, which has the form of X (q) Eq. (5.17), must have all Schmidt coefficients λm equal where the cq are nonzero and eq Πq e is a normalized state in the jth sector. Projecting| i≡ Eq. (5.24)| i onto a single and non-zero, and the coefficients βq are completely fixed charge sector and tracing over , we find by the optimal e ; i.e., the optimal fiducial state is Mq | i dq Tr q eq eq = I ˜ /dq , (5.26) M | ih | Nq dim( q) (q) (q) ψ = M φm rm . (5.32)   | i dim ˜ | i ⊗ | i which tells us that the reduced density operator on ˜ j s m=1 Nq X H X of eq is the completely mixed state, and consequently | i Note that this state satisfies that eq is a maximally entangled states across q ˜q. This| mayi be written as M ⊗N [ ψ ψ ]= I ˜ / dim ˜ . (5.33) d G | ih | H H 1 q e = φ(q) r(q) . (5.27) We can conclude that the maximum likelihood density of | qi | m i ⊗ | m i dq m=1 a correct guess takes the general form X Projecting Eq. (5.24)p onto a single charge sector and trac- 2 µmax = e = dim ˜ = rank( [ ψ ψ ]) ing over both ˜q and q, we conclude that k k H G | ih | N M = dim q min dim q, dim q . (5.34) 2 M × { M N } cq = Tr[I q ]Tr[I ˜ ] = dim( q)dq . (5.28) q | | M Nq M X (q) We may define φm in such a way that the coefficients Given Eq. (5.33) and Eq. (5.34), we can interpret this | i cq can be taken to be real and positive. Combining result as follows: we maximize the likelihood of a cor- Eqs. (5.25), (5.27) and (5.28), we recover Eq. (5.23). rect guess by choosing the fiducial signal state ψ to be | i The covariant POVM, then, is fixed by the problem, such that, under G-averaging, the weights of the state and it remains only to determine the optimal fiducial [ ψ ψ ] are spread uniformly over the largest possible G | ih | state ψ . To do so, one needs to specify a figure of merit. space. Thus, at least for the case of maximum likelihood | i estimation, the intuition behind why antiparallel spins do better than parallel spins is found to have a rigorous C. Maximum likelihood estimation counterpart, and indeed this intuition is found to gen- eralize to the alignment of any RF whose configurations We now consider a particular choice for a figure of correspond to the elements of a group. By choosing a merit: the maximum likelihood of a correct guess (Chiri- fiducial state in this way, the signal states are made as bella et al., 2004c). Because our standard example is distinguishable as possible. 32

1. Maximum likelihood estimation of a phase reference presently, whereas the optimal strategies for aligning RFs associated with non-Abelian groups can also be under- With the general results above, the optimal perfor- stood in terms of the support of the group orbit of the mance of any particular alignment protocol quantified signal state, it is at present unclear whether an argument by maximizing the likelihood can be directly and sim- in terms of an uncertainty principle can be provided in ply calculated. Suppose, for example, one seeks to align such cases. phase references by transmitting at most nmax photons in a single mode. The relevant group in this case is U(1). The charge sectors correspond to total photon number, 2. Maximum likelihood estimation of a Cartesian frame so we use n rather than q to denote them. Because the irreps of U(1) are one-dimensional, we have dim n = 1. We now consider the task of optimally aligning a full In this case, the optimal fiducial POVM elementM and the spatial (Cartesian) frame through the exchange of spin- optimal fiducial signal state are 1/2 particles, based on maximizing the likelihood of a correct estimation. For this example, we will be required n n max 1 max to use the multiplicity of irreducible representations of e = n , ψ = n . (5.35) N | i | i | i √n +1 | i SO(3) that occur in ( 1/2)⊗ to obtain the optimal n=0 max n=0 X X scheme. H Clearly, We restrict ourselves to the case of N an even num-

n ber for simplicity. N spin-1/2 particles carry a tensor max 1 N ψ ψ = n n , (5.36) representation R⊗ of SO(3); this representation is re- G | ih | n +1| ih | ducible, and the irreducible representations (labeled by n=0 max   X j) appear with nontrivial multiplicities. As analyzed in so that the maximum likelihood density of a correct guess Sec. III.A.2, the total Hilbert space of N spin-1/2 parti- is cles can be decomposed as

µmax = rank ψ ψ = nmax +1 . (5.37) N/2 G | ih | N ( 1/2)⊗ = j j , (5.39) Note that for multiple  modes,
the subspaces n may be H M ⊗ N H j=0 multi-dimensional, and in this case the basis states can M ˆ be chosen to be any set of eigenstates of Ntot, i.e., any where carry irreducible representations R of SO(3) multi-mode states that are eigenstates of total photon Mj j and have dimensionality 2j + 1, and j carry the trivial number. representation of SO(3) and have dimensionalityN given For comparison, it is useful to consider the maximum by Eq. (3.21). For this example, the dimension of each likelihood that could be achieved using a coherent state decoherence-free subsystem is greater than or equal to α with mean photon number n /2 (we cut off the Nj | i max the dimension of the corresponding decoherence-full sub- amplitude for n>nmax which is negligible for sufficiently system j for all j except j = N/2 (where dim j = 1). large values of nmax), Thus, theM maximum dimension of ψ is, from Eqs.N (5.20) H n 2 max α /2 n 2 and (5.34), 2 e−| | α µCS = α e = , (5.38) |h | i| √n! N/2 1 n=0 − X dim ψ = (N +1)+ (2j + 1)2 H which behaves as α = nmax/2 for large values of nmax. j=0 | | X Thus, the phase eigenstate offers a quadratic improve- 1 3 5 p = 6 N + 6 N +1 . (5.40) ment in nmax over the coherent state. Heuristically, this is due to the fact that for the Poissonian number distribu- For each j

D. General figures of merit and it is for this reason that we focussed our attention on covariant POVMs early in this section. As discussed above, maximizing the likelihood of the Any function f(g′,g) that is left-invariant can be writ- 1 correct guess led us directly to a general principle for ten as a function f(g′,g) = f˜(g′− g); if this function 1 1 choosing the fiducial signal state. However, as a fig- is also right-invariant, then it satisfies f˜(hg′− gh− ) = ure of merit, the maximum likelihood density is not a 1 f˜(g′− g), and thus is a class function, i.e., a function on very practical choice – it rewards only a perfectly correct the conjugacy classes of G. (Recall two group elements, guess. In many situations, one would desire a figure of g1 and g2, are in the same conjugacy class if there ex- merit that would quantify the performance of a scheme 1 ists another group element h such that g1 = hg2h− .) by the amount of Shannon information gained by the re- Any class function f˜ can be expanded as a sum of the cipient about the sender’s reference frame. However, such characters14 χ (g) of G as figures of merit usually lead to intractable optimization q problems. f˜(g 1g)= a χ (g 1g) , (5.46) A more common and tractable approach is to introduce ′− q q ′− q a payoff function f(g′,g) which specifies the payoff for X guessing group element g′ when the actual group element where the aq are arbitrary coefficients. We restrict our is g (Chiribella et al., 2005). Assuming a uniform prior attention to real, positive-valued payoff functions, which for the signal states, the figure of merit for the alignment will allow us to perform a simple maximization. scheme can then be the average payoff We note that the maximum likelihood estimation task described above corresponds to choosing a payoff func- 1 f¯ = dg dg′ p(g′ g)f(g′,g) , (5.43) tion f(g′,g) = δ(g′− g), a delta function. This payoff | Z function is both left- and right-invariant, and its expan- sion in terms of characters as in Eq. (5.46) corresponds where p(g′ g) is the probability of guessing g′ when the signal state| is g for the scheme in question. In particular, to choosing all aq positive and equal to the dimension of the commonly-used fidelity, which quantifies the variance the irrep. of the average guess and which leads to direct compar- As a consequence of the covariance of both the set of signal states and the POVM, the probability p(g′ g) is isons with the standard quantum limit, is one choice of 1 | payoff function; we will determine this fidelity and ex- also a function of g′− g, i.e., plore protocols that optimize its average in the examples 1 2 p(g′ g)= e U(g′− g) ψ that follow. | | h | | i | 1 The task of reference frame alignment imposes some p˜(g′− g) . (5.47) ≡ natural constraints on the form of payoff functions. First, The average payoff of Eq. (5.43) then simplifies as we note that the group elements g and g′ denote an orien- tation relative to some background RF, i.e., the identity 1 1 group element corresponds to “aligned with this back- f¯ = dg dg′ p˜(g′− g)f˜(g′− g) ground RF”. However, it is desirable to construct pro- Z tocols that are independent of any background RF; for = dg p˜(g)f˜(g) , (5.48) example, if the background RF was transformed by a Z group element h G, and g and g′ were now defined with respect to this transformed∈ background RF, the payoff which follows from the invariance of the measure dg. Us- function should be the same. Such protocols are associ- ing the explicit form of Eq. (5.47), we have ated with a payoff function f(g′,g) that is right-invariant, i.e., which satisfies f¯ = dg ψ U †(g) e e U(g) ψ f˜(g) . (5.49) h | | ih | | i 1 1 Z f(g′h− ,gh− )= f(g′,g) , h G . (5.44) ∀ ∈ Defining In addition, the payoff function should be a function only of the relative transformation relating the transmitted M dgU †(g) e e U(g)f˜(g) , (5.50) ≡ | ih | state (determined by g) and the measurement outcome Z (determined by g′). This requirement of a protocol de- we may rewrite f¯ as mands that the payoff function be left-invariant, f¯ = ψ M ψ . (5.51) f(hg′,hg)= f(g′,g) , h G . (5.45) h | | i ∀ ∈ Payoff functions that are left-invariant are also referred to as covariant. It is always possible to find a covariant 14 The characters χq (g) of a group G form a basis of class functions; POVM that is optimal for any estimation problem with a they are given by the trace of the irreducible representations Tq covariant (left-invariant) payoff function (Holevo, 1982), of G, i.e., χq(g) = Tr[Tq(g)]. 34

This expression is the generalization of Eq. (5.29) to As mentioned above, for any alignment scheme based an arbitrary covariant payoff function. As the fiducial on independent uses of N modes with at most a single POVM element is completely constrained to be of the photon in each, that is, N single-rail qubits, the average form of Eq. (5.23), the operator M is therefore deter- fidelity will approach f¯ = 1 as1/N, from the central mined by the figure of merit. In order to maximize the limit theorem. This scaling is referred to as the standard average payoff f¯, then, one must find a fiducial state ψ quantum limit; the optimal scheme outperforms this scal- of the form (5.17) that lies in the eigenspace of M with| i ing, as we will now demonstrate. We now optimize over the largest eigenvalue. Specifically, we solve the eigen- choices of signal state ψ(N) in order to maximize the value equation expected payoff, quantified| byi the average fidelity f¯ of Eq. (5.51). M ψ = λmax ψ , (5.52) Let n , n = 0, 1,...,n be an arbitrary set of | i | i {| i max} eigenstates of the total number operator Nˆ tot; the de- and the use of this state yields a maximal average payoff tails of these states, including their mode structure, is of irrelevant to the task. The fiducial POVM element is of the form e = nmax n . max max n=0 f¯ = λ . (5.53) The operator| i M of| Eq.i (5.50) is given in this instance by the matrix P For the problem of optimally aligning reference frames using a left- and right-invariant payoff function, we will M ′ = n M n′ nn h | | i use the following result of Chiribella et al. (2005) with- 2π 15 dθ i(n n′)θ 2 out proof : that the optimal fiducial signal state can be = e − cos (θ/2) chosen to have the form 0 2π Z1 1 1 = 2 δn,n′ + 4 δn,n′+1 + 4 δn+1,n′ . (5.55) dq ψ = β φ(q) r(q) , (5.54) Note that | i q | m i ⊗ | m i q m=1 X X 1 ˜ 1 M = 4 M + 2 I , (5.56) for coefficients β satisfying β 2 = 1. These coef- q q | q| ficients are determined by the specific choice of payoff where M˜ nn′ = δn,n′+1 +δn+1,n′ . As any eigenvector of M˜ function. We note, however, thatP this result greatly sim- is an eigenvector of M, it suffices to find the eigenvalues plifies the optimization problem: the number of coeffi- and eigenvectors of M˜ . The maximum average fidelity is cients is now given by the number of irreps appearing in then the decomposition of U, rather than by the dimension of ¯max 1 1 max ˜ the Hilbert space. f = 2 + 4 λ (M) , (5.57)

and is achieved when ψ is the eigenvector of M˜ associ- | i max 1. Fidelity of aligning a phase reference ated with the maximum eigenvalue λ (M˜ ). The characteristic equation we must solve is We now reconsider the problem of aligning a phase det(M˜ λI)=0 , (5.58) reference, as in Sec. V.C.1, but with an alternate (and − commonly used) payoff function: the function f(θ′,θ)= 2 where I is the identity. Defining Gk M˜ λI, where cos [(θ′ θ)/2], which takes the value 1 for the correct − k is the dimension of the vector space≡ on which− M˜ acts, guess (θ′ = θ) and 0 for θ′ = θ + π. Note that this payoff function is left- and right-invariant, and can be written one finds that as f˜(θ) = cos2(θ/2), where θ now denotes the relative det Gk = λ det Gk 1 det Gk 2 , (5.59) angle between the signal and guess. This figure of merit − − − − is commonly referred to as the fidelity. for which the solution is The Hilbert space for this task will be restricted as

follows: we allow arbitrarily few or many modes, but the det Gk = Uk( λ/2) , (5.60) maximum total photon number is restricted to nmax. (An − alternate approach would be to bound the mean photon where the Uk are the Chebyshev polynomials of the sec- number; however, this adds considerable complexity to ond kind, given by the problem.) sin[(k + 1)θ] U (cos θ)= . (5.61) k sin θ

15 Given that U (x)= U ( x), it follows that the charac- The proof relies on determining an upper bound on the average k ± k − payoff, and then demonstrating that states of the form (5.54) teristic equation is UN+1(λ/2) = 0, and thus the largest saturate this bound. eigenvalue is λmax = 2 cos(π/(N + 2)). The maximum 35 average fidelity for the distribution of a phase reference Conjugation by another rotation in SO(3) simply changes is thus the axis, not the value of ω. Thus, conjugacy classes are labeled by an angle ω (a rotation by ω about some axis), ¯max 1 f = 2 1 + cos(π/(N + 2)) . (5.62) and characters being class functions are functions only of ω. Explicitly, they are given by To find the eigenvector ψ associated with
the largest | i eigenvalue, we must solve sin[(2j + 1)ω/2] χj (Ω) = χj (ω)= . (5.69) M˜ ψ = λmax ψ . (5.63) sin(ω/2) | i | i Let β be the coefficients of ψ = nmax β n . The The payoff function can be expressed in terms of the char- n | i n=0 n| i definition of M˜ leads to acter χj=1 of Rj=1 (the representation of SO(3) that acts P on spatial vectors), as max βn+1 + βn 1 = λ βn , (5.64) − 1 1 1 max f(Ω′, Ω) = + χ1(Ω′− Ω) . (5.70) for 1 j N 1. At n = 0, we have β1 = λ β0, 4 4 ≤ ≤ − max and at n = N, we have βN 1 = λ βN . The solution is max − βn = Un(λ /2), and the coefficients βn fall to zero at As it is a covariant function only on the conjugacy class, N + 1 as required. The optimal state thus has the form we can express it as

N (n + 1)π ˜ 1 1 ψ(N) = sin n , (5.65) f(ω)= + χ1(ω) . (5.71) | i N N +2 | i 4 4 n=0 X   The fiducial POVM element can be written as where the normalization is approximately (N/2+ 1/2 N N ≃ 1)− in the large-N limit (Berry and Wiseman, 2000). N/2 1 − In the limit of large N, the average fidelity behaves as e(N) = √N +1 N , N + (2j + 1) e , (5.72) | i | 2 2 i | j i 2 j=0 ¯max π X f 1 2 , for N 1 . (5.66) ≃ − 4N ≫ where Thus, this optimal protocol for the alignment of a phase j reference has an error (variance) which decreases as 1 2 e j, m j, α(m) , (5.73) 1/N , i.e., at the Heisenberg limit. | j i≡ √2j +1 | i ⊗ | i m= j X−

2. Fidelity of aligning a Cartesian frame are maximally entangled. From Eq. (5.54), the optimal fiducial signal state has We now consider the task of aligning a Cartesian frame the form through the exchange of spin-1/2 particles, using the fi- N/2 1 delity as the figure of merit (Bagan et al., 2004b; Chiri- − (N) N N et al. ψ = βN/2 , + βj ej , (5.74) bella , 2004b, 2005). | i | 2 2 i | i j=0 We first develop the payoff function, which we require X to be both left- and right-invariant. One such possibility is to use the mean deviation between Alice’s coordinate where the coefficients βj are to be determined. For sim- axes and Bob’s, i.e., plicity and brevity, we will only solve this eigenvalue problem in the limit of large N. In this limit, the βN/2 1 2 term (the only exceptional term) can be ignored. f(Ω′ Ω)=1 Ωn Ω′n , (5.67) | − 8 iA − iB i=x,y,z As with the phase distribution problem, the goal is to X find the state ψ that maximizes | i where Ωn denotes the vector obtained by rotating the vector n by Ω SO(3). This function can be expressed f¯ = ψ M ψ ∈ h | | i in terms of the characters χj (Ω) for SO(3); because these = β β ′ M ′ , (5.75) characters will be useful in the following, we briefly re- j∗ j jj ′ view them here. The characters of SO(3) are given by Xj,j the trace of the irreps Rj as where Mjj′ is the matrix

χj(Ω) = Tr[Rj (Ω)] . (5.68)

M ′ = dΩ e R (Ω) e e ′ R ′ (Ω)† e ′ f˜(Ω) . Recall that any rotation Ω in SO(3) can be expressed as a jj h j | j | j ih j | j | j i rotation by ω, in the range 0 ω< 2π, about some axis. Z (5.76) ≤ 36

We note that a reference frame for the coset space G/G0. We may in- corporate such cases into the framework specified above ej Rj (Ω) ej by choosing our figure of merit to reflect the unimpor- h | | i j tance of the subgroup in the estimation task; i.e., choose 1 ˜ 1 = j, m R (Ω) j, m′ α(m) α(m′) a payoff function f(g′− g) that satisfies 2j +1 h | j | ih | i m,m′= j X− ˜ 1 ˜ 1 j f(g′− gg0)= f(g′− g) , g0 G0 . (5.81) 1 ∀ ∈ = j, m R (Ω) j, m 2j +1 h | j | i In other words, we imagine choosing signal states and m= j X− POVMs that are covariant for a group G that is a cover- 1 = χ (Ω) . (5.77) ing group for the coset space in question. Let z be a set 2j +1 j of coset representatives, i.e., z G/G , and let dz be a ∈ 0 left-invariant measure on G/G0. Then, using Eq. (5.48), Thus, 1 1 f¯ = dg p˜(g)f˜(g) M ′ dωχ (ω)χ∗′ (ω)( + χ (ω)) . (5.78) jj ∝ j j 4 4 1 Z Z = dz dg0 p˜(zg0) f˜(z) To evaluate this integral, one can make use of the or- ZG/G0 ZG0 thogonality properties of group characters; see Chiribella   ˜ et al. (2004b) for details. We find that = dz p˜inv(z)f(z) . (5.82) ZG/G0 1 1 Mjj′ δj,j′ + δj,j′ 1 + δj 1,j′ . (5.79) Here, we have defined ∝ 4 4 − −
The eigenvalue problem, then, is essentially identical to p˜ (z) dg p˜(zg ) inv ≡ 0 0 that solved for the distribution of a phase reference in ZG0 the previous section. In this limit, then, this maximum = Tr dg U(zg ) e e U †(zg ) ψ ψ average fidelity scales as 0 0 | i h | 0 | i h | ZG0 h
i π2 = Tr U(z)EinvU †(z) ψ ψ , (5.83) f¯max 1 , for N 1 . (5.80) | ih | ≃ − N 2 ≫ where   Thus, this scheme also scales at the Heisenberg limit. We note that this particular task has given rise to some Einv = dg0 U(g0) e e U †(g0) , (5.84) G | i h | controversy and errors in the literature. In particular, Z 0 a mistaken claim of optimality for this task in Bagan is G0-invariant. Thus, for any covariant measurement et al. (2001b), which resulted from a failure to include the with fiducial POVM element e that achieves the opti- | i multiplicity of irreducible representations, led to some mum figure of merit, there exists a G0-invariant covari- confusion over the use of covariant measurements in this ant measurement with fiducial POVM element Einv that task (Peres and Scudo, 2002a). achieves the same optimum. For this reason, we may as well restrict the fiducial signal state and fiducial POVM element to be G0-invariant. E. Reference frames associated with coset spaces We note that, if the group G0 is non-Abelian, it may not be possible to find a pure state that is invariant un- A directional RF, for the z-axis say, can be obtained der the subgroup. In such a situation, if one wishes to from a full Cartesian RF by throwing away the informa- work with G0-invariant states and measurements, then tion about the azimuthal angle. To specify a direction, one will have to use mixed fiducial states and POVM therefore, it is sufficient to specify an equivalence class elements (Chiribella and D’Ariano, 2004a). We now con- of Cartesian frames, those related by an SO(2) transfor- sider an example with an Abelian group G0, for which mation about this axis. Hence, a directional RF is asso- these complications do not arise. ciated with an element of the coset space SO(3)/SO(2). This coset space is equivalent to S2, the space of points on a three-dimensional sphere, which corresponds to the 1. Aligning a direction possible directions in space. Thus, certain reference frames have distinct configu- Consider the task of optimally aligning a direction in rations which do not correspond to the elements of a space through the exchange of spin-1/2 particles. This group, but rather those of a coset space of a group. If was first considered for just two particles by Gisin and we consider a reference frame for a group G but are un- Popescu (1999); Massar (2000), as discussed in Sec. V.A. concerned about the difference between those related by The problem was subsequently considered for an arbi- a subgroup G0 of transformations, then we can speak of trary number of particles by Bagan et al. (2000, 2001a); 37

Peres and Scudo (2001a). (For a related investigation, that this is the generalization of the example provided wherein it is addressed how to perform this task using in Sec. V.A from two to an arbitrary number of spin-1/2 product states, see Bagan et al. (2001b).) systems. Let ψ(N) be the fiducial signal state. Again, we re- Again, the fiducial POVM element is essentially16 con- | i strict our attention to the case of N even. Because we are strained to be that of Eq. (5.87), and now the signal state concerned only with aligning a direction and not a full takes the general form Cartesian frame, we can choose ψ(N) to be invariant un- der rotations about the z-axis without| i loss of generality. N/2 (N) Any such pure invariant state is an eigenstate of Jz; thus, ψ = bj j, 0 , (5.90) (N) | i | i j=0 choose ψm to be an eigenstate of Jz with eigenvalue X ~m. Clearly,| im must be in the range N/2,...,N/2. − First, some notation. It is standard to express a rota- where the coefficients bj are to be determined. The Born tion in SO(3) in terms of its Euler angles (α,β,γ). Specif- rule yields ically, a unitary rotation operator can be expressed as (N) N (N) 2 p˜(α, β, 0) = ψ R⊗ (α, β, 0) e . (5.91) |h | | i| R(α,β,γ)= Rz(α)Ry(β)Rz(γ) , (5.85) This quantity is independent of α, and thus the relevant where Ry and Rz are SO(2) rotations about the y- and conditional probability is z-axes, respectively. For any element in SO(3), a set of Euler angles can be found in the range 0 α,γ < 2π p˜(β)= ψ(N) R N (0, β, 0) e(N) 2 . (5.92) ≤ ⊗ and 0 β < π. The invariant subgroup is G0 =SO(2) |h | | i| in this≤ problem, rotations about the z-axis; thus, the N N Note that R⊗ (0, β, 0) = Ry⊗ (β) (a rotation about the parameters (α, β) provide coordinates for the coset space j y-axis) and the reduced Wigner matrix d00(β) is given SO(3)/SO(2). by

dj (β) j, 0 R (β) j, 0 a. Maximum likelihood. We now maximize the likelihood 00 ≡ h | y | i of a correct guess. Restricting the fiducial POVM ele- = Pj (cos β) , (5.93) ment to be SO(2)-invariant, it takes the form where Pj (x) is a Legendre polynomial. N/2 The operator M of Eq. (5.50) is given in this instance e(N) = 2j +1 j, m . (5.86) by the matrix | m i | i j=m X p Mjj′ = j, 0 M j′, 0 As we wish to include all possible irreps j, following the h |π | i 1 2 general construction of Sec. V.B, we should choose m = = 2 sin β dβ Pj (cos β)Pj′ (cos β)cos (β/2) 0, i.e., a fiducial POVM element 0 Z 1 1 N/2 = 4 dx Pj (x)Pj′ (x)(P0(x)+ P1(x)) (N) 1 e = 2j +1 j, 0 . (5.87) Z− | i | i 1 2 2j j=0 = δj,j′ + ′ δj,j′+1 X p 4 2j+1 (2j+1)(2j +1)  2j′ The signal state should then be parallel to this vector, of + (2j+1)(2j′ +1) δj,j′ 1 , (5.94) the form −  N/2 where we have expanded the payoff function in terms of 1 ψ(N) = 2j +1 j, 0 , (5.88) Legendre polynomials. | i (N/2+1)2 | i j=0 This eigenvalue problem is essentially the same as those X p solved in the previous section. The maximum average and the maximum likelihood density of a correct guess is fidelity is given by

µ = (N/2+1)2 . (5.89) 1+ x max f¯max = N/2+1 , (5.95) 2 b. Fidelity. A natural payoff function for this problem is the inner product between Bob’s guess direction ng and 16 The choice of m = 0, necessary to optimize the maximum likeli- Alice’s transmitted n, given by f˜(θ) = (1+ ng n) = 2 · hood problem, is not a priori optimal for maximizing the aver- cos (θ/2), where θ is the angle between their directions. age fidelity. However, m can be left free and then optimized at This payoff function is also known as the fidelity. We the end, with the result that m = 0 is indeed optimal for this provide the details for this optimization as well. Note task (Peres and Scudo, 2001a). 38 where x is the largest zero of the Legendre polyno- An optimal state ψ on for maximizing the like- N/2+1 | i H⊗K mial PN/2+1(x). In the limit of large N, this maximum lihood of a correct guess, up to a normalization constant, average fidelity scales as is 2 ζ N/2 1 j cj f¯max 1 , for N 1 , (5.96) − 1 ≃ − N 2 ≫ ψ j, m, α j, m, α , | i∝ (2j + 1)c | iH| iK where ζ 2.4. Thus, this optimal scheme also scales j=0 j m= j α=1 ≃ X X− X at the Heisenberg limit (Bagan et al., 2001a; Peres and p (5.97) Scudo, 2001a). where cj is the multiplicity of the jth representation. The fiducial POVM element will be parallel to this vector. We note that this state is a superposition over irreps j F. Relation to phase/parameter estimation of a maximally-entangled state between an irrep j on and an equally-sized space on . The optimality of thisH We note that the task of aligning a phase reference state for alignment follows fromK the general arguments is essentially equivalent to the task of estimating an un- presented in Sec. V.C: the optimal fiducial state is the known phase. Specifically, instead of viewing the problem one that maximizes the dimension of the group orbit. of noiseless transmission of a quantum system between Without the help of an ancilla, this is achieved within a parties who do not share a phase reference, the prob- given irrep j by entangling the gauge space j (on which lem could instead be viewed as one of transmission of the group acts nontrivially) with the multiplicityM space the same quantum system between parties who do share j (on which it acts trivially). In the present context, it a phase reference, but where the transmitting channel isN achieved by entangling the system with the ancilla. induces an unknown phase shift on the system. As such methods for parameter estimation have im- In this light, we note that the protocol presented in portance for quantum computing in terms of the charac- Sec. V.D.1 is equivalent to the optimal solution for phase terization of quantum gates, it is interesting to consider estimation using the same figure of merit (Berry and how the methods of reference frame alignment may be Wiseman, 2000). Techniques for quantum-limited phase applied to such characterization problems as well. estimation have been well studied, and there exist a wide Finally, work on magnetometry – the use of magnets as variety of alternate methods that could each be applied, direction indicators to determine the strength and direc- in some form, to the task of aligning a phase reference. tion of a magnetic field – is also a problem of parameter For an overview of phase estimation techniques from dif- estimation, closely related to the problem of reference ferent viewpoints, we refer the reader to the review article frame distribution. Practical proposals for quantum- on quantum metrology by Giovannetti et al. (2004a), or limited magnetometry make use of spin-squeezed clouds the text of Nielsen and Chuang (2000) which discusses of cold atoms to measure the three components of an phase estimation techniques from a quantum algorithm unknown magnetic field through a form of phase esti- perspective. Also, see Giovannetti et al. (2006) for a uni- mation (Petersen et al., 2005). It would be interesting to fied framework of these techniques. investigate whether spin-squeezed states of indistinguish- Similarly, the task of aligning a reference frame for able particles (e.g., atoms) can be used for the distribu- G through the transmission of a quantum system is es- tion of a direction or frame as efficiently as the optimal sentially equivalent to estimating an unknown element protocols derived above, which make use of (distinguish- g G given a quantum channel that acts on the same able) qubits in highly-entangled states and corresponding ∈ quantum system with the unitary U(g). This latter task entangling measurements. is generally referred to as parameter estimation. We note, then, that the scheme for aligning a Carte- sian frame presented in Sec. V.D.2 is closely related to a G. Communication complexity of alignment method for estimating an unknown SU(2) (or more gener- ally SU(d)) transformation (Acin et al., 2001). We briefly We have thus far only considered protocols for RF review this latter scheme, because of its close relation to alignment wherein there is a single round of commu- the topic at hand. Let R(Ω) be the unitary representa- nication from Alice to Bob. We now consider multi- tion of an unknown rotation Ω SU(2), which acts on round protocols (de Burgh and Bartlett, 2005; Giovan- states of a Hilbert space ; the∈ task is to estimate Ω netti et al., 2006; Rudolph and Grover, 2003). Whereas through one application ofHR(Ω) to some quantum state. the single-round protocols generally require entangle- For this problem, we allow the use of an ancillary system, ment between the transmitted systems to achieve the with Hilbert space of arbitrary dimension; this ancilla Heisenberg limit, multi-round protocols have the advan- is assumed to transformK trivially under SU(2). (That is, tage that they can achieve this limit despite using no SU(2) acts as R(Ω) I on .) Without loss of gener- entanglement. ality, one can choose⊗ dim H⊗K= dim , and choose a basis With multi-round protocols, it is natural to frame the for with the same labelsK as . WeH choose the standard problem as one of communication complexity, wherein SU(2)K angular momentum basisH j, m, α , where α labels one investigates the resources of rounds of communi- the multiplicity. | i cation along with the standard resources of number of 39 transmitted quantum or classical bits. To conform with uncertainty of standard notions of communication complexity, it is use- ful to consider the alignment problem with two depar- ∆O = O2 O 2 = sin (2θ ) . (5.99) tures from the approach adopted in analyzing the pre- A h Ai − h Ai BA vious alignment protocols. First, we consider the worst q Expressing O in terms of T , we have case scenarios (rather than the average case considered h Ai above); second, we avoid the use of payoff functions, such O = cos(2θ ) = cos(2π0.t t ) . (5.100) as the fidelity, with a view to obtaining a more precise h Ai BA 1 2 · · · estimate of how well any given instance of the protocol has performed. As such, we consider strategies for align- By repeating this procedure n1 times, i.e., sending n1 ing spatial reference frames that allow Bob to directly independent qubits and averaging the results, Alice ob- tains O , the estimate of O . If n is chosen such determine the angle which relates his and Alice’s RFs to h Ai h Ai 1 that O O 1/2 with some error probability, some specified accuracy with a bounded probability of |h Ai − h Ai| ≤ error in the worst case scenario. More precisely, if θ is then T T 1/4, thus determining the first bit t1 with | − |≤ an angle relating Alice and Bobs’ RFs, and θ′ is the es- this same probability. The required number of iterations timation of θ inferred by Bob, then we will be interested n1 to achieve the desired error is given by the Chernoff in the amount (and type) of communication required for bound, with δ = 1/4. That is, the probability that the protocols that achieve P = Pr[ θ θ δ] ǫ, for first bit of Alice’s estimate O differs from the first bit error ′ h Ai some fixed ǫ,δ > 0. By setting δ =| 1−/2k+1| ≥we say≤ that of the actual value O decreases exponentially in the h Ai with probability (1 ǫ) Bob has a k-bit approximation number of repetitions n1, and is bounded explicitly by to θ. − n /32 We now describe such a protocol for the case of sharing Pr O O 1/2 ǫ/k 2e− 1 . (5.101) |h Ai − h Ai| ≥ ≤ ≤ a phase reference through the exchange of qubits, i.e., the h i same task as investigated in Sec. V.D.1. This protocol Thus, allowing a probability of error ǫ/k in this bit, we can also be applied to the task of aligning a Cartesian require n 32 ln(2k/ǫ) iterations. 1 ≥ frame (Rudolph and Grover, 2003). The effects of deco- Now we define a similar procedure for estimating an herence on these protocols has been shown to be equiv- arbitrary bit, t . Alice prepares the energy eigenstate alent to that of decoherence on the “standard” protocol j+1 0 , and performs her HA operation. Alice and Bob then of Sec. V.D.1 (Boixo et al., 2006). |passi the qubit back and forth to each other 2j times, Let θBA be the unknown angle (misalignment) that each time Bob performs his XB operation and Alice per- relates Bob’s phase reference to Alice’s. In this protocol, forms her XA operation. That is, they jointly implement Alice and Bob use an algorithm that estimates each bit of 2j the operation (XAXB) . Finally Alice performs her HA the phase angle θBA independently. We define the phase operation. Expressing these operations in Alice’s frame, angle θBA = πT , where T has the binary expansion T = the protocol to estimate tj+1 produces the state 0.t t t . Alice and Bob will attempt to determine T 1 2 3 · · · to k bits of precision, and accept a total error probability 2j ψj A = HA(XAXB) HA 0 Perror ǫ. If the total error probability is to be bounded | i | i ≤ +iωt Z 2j by ǫ, then each ti, i = 1,...,k, must be estimated with = H (e AB ) H 0 A A| i an error probability of ǫ/k. (An error in any one bit j = H e+i2 ωtBAZ H 0 causes the protocol to fail, so the total error probability A A| i k j j in estimating all k bits is Perror =1 (1 ǫ/k) ǫ.) = i sin(2 ωt ) 0 + cos(2 ωt ) 1 . (5.102) − − ≤ BA | i BA | i A To estimate the first bit t , Alice prepares a single 1   qubit in the state ( 0 + 1 )/√2 (relative to her phase Alice then measures the observable OA = Z. The ex- A A − reference) and sends| i the| qubiti to Bob. Bob then per- pected value of this observable is: forms his operation XB and sends the qubit back to Al- j+1 ice, where X is the Pauli bit-flip operator according OA = cos(2 ωtBA) B h i to Bob. She then performs her operation XA. The re- j = cos(2 [2π0.t1t2 ]) sulting combined transformation X X is described in · · · A B = cos(2πt t t .t t ) Alice’s frame as 1 2 · · · j j+1 j+2 · · · = cos(2π0.t t ) . (5.103) j+1 j+2 · · · iθ Z/2 +iθ Z/2 X X = X (e− BA X e BA ) A B A A This expression has the same form as one iteration of the +iθBAZ = e , (5.98) scheme to estimate the first bit t1; Alice and Bob sim- 2j ply require more exchanges to implement (XAXB) . To where Z is the Pauli phase-flip operator. Finally, Alice get a probability estimate for each bit tj+1, this more performs a Hadamard transformation HA (in her frame) complicated procedure is repeated nj+1 times. Because and measures the observable OA = Z. The expected we require equal probabilities for correctly estimating value of this observable is O = cos(2− θ ), with an each bit, we can set all n equal to the same value, h Ai BA j+1 40 n 32 ln(2k/ǫ). Thus the total amount of qubit commu- and Bob’s clocks of k bits. That is, the synchroniza- ∼ nication Nc required to obtain bits t1 through tk by this tion accuracy scales as the standard quantum limit. In procedure is comparison, a protocol by Chuang (2000) makes use of the Quantum Fourier Transform and an exponentially k j 1 k k large range of qubit ticking frequencies. This protocol N = n 2 − = n(2 1) = O(2 ln(2k/ǫ)) . (5.104) c × − requires only O(k) quantum messages to achieve k bits j=1 X of precision, an exponential advantage over the standard To facilitate comparison with the previous sections, quantum limit. Although this protocol gives insight into wherein the focus was on maximizing the average fidelity, the ways that quantum resources may allow an advan- we imagine that Alice and Bob use the above protocol tage in clock synchronization, it is unsatisfactory for two reasons: (1) its use of exponentially demanding physi- to obtain, with probability (1 ǫ), an angle θ′ which is a “k-bit” estimator of the true− angle θ, i.e., they obtain cal resources is arguably the origin of the enhanced effi- k+1 ciency (Chuang, 2000; Giovannetti et al., 2001); and (2) θ θ′ 2π/2 with probability (1 ǫ). The fidelity of | − |≤ 2 − 2 Alice and Bob need to a priori share a synchronized clock this estimate is f = cos ((θ θ′)/2). Since cos x 1 x ¯ 2π −2 ≥ − in order to implement the required operations as defined we have that f 1 ( 2k ) . To compare with the av- erage case fidelity≥ computed− previously, we assume that in Chuang (2000). These problems are not present in when the protocol fails (which happens with probability subsequent protocols based on ticking qubits, which used ǫ) the fidelity obtained is 0, i.e., worse than a random the techniques for phase estimation presented in Sec. V.G guess. We have then that the expected fidelity from this to design a clock synchronization protocol that operates protocol satisfies near the Heisenberg limit (de Burgh and Bartlett, 2005). The Heisenberg limit for clock synchronization can be 2π 2 achieved by making use of the phase estimation protocol f¯ (1 ǫ) 1 . (5.105) ≥ − − 2k of Sec. V.D.1. h   i If we take ǫ = 1/22k, then the total number of qubit Distinct from the approaches based on phase estima- communications scales as k2k for large k, while the ex- tion, there has been considerable interest in another class pected fidelity f¯ scales as 2 2k 1 (log N )/N 2. Thus, of clock synchronization protocols which make use of − c c et al. et al. remarkably, this protocol beats≃ the− standard quantum entanglement (Jozsa , 2000); see also Burt (2001); Jozsa et al. (2001); Preskill (2000); Yurtsever limit of 1/Nc, yet does not require entangled states or collective measurements. This is achieved at the cost of and Dowling (2002). In a variant of this approach, a an increased complexity in coherent qubit communica- third party Charlie distributes a large number of boxes tions. to Alice and Bob, where each box contains one spin of a spin singlet. Each box also contains a classical mag- netic field aligned in the z-direction, such that the free H. Clock synchronization Hamiltonian for each spin is H0 = χσz for some con- stant energy χ. (We note that this establishes a shared If Alice and Bob share a common frequency standard, RF between Alice and Bob for this particular direction.) then they can use techniques for phase reference align- Clock synchronization can be achieved by Alice perform- ment that were outlined above to perform clock synchro- ing measurements on her spins in her x-direction at time nization, i.e., aligning a temporal reference (Jozsa et al., t = 0. By informing Bob (via any classical channel) 2000). To see the relation between phase alignment and as to the sub-ensemble of the singlets for which she ob- clock synchronization, it is easiest to work in a rotating tained the +x outcome, Bob can identify the subset of frame (interaction picture) in which states are described his particles which are all precessing (via the free Hamil- iH0t/~ tonian) around the common z-direction in phase with as ψ I = e ψ , and observables and transforma- | i iH| 0t/i ~ iH0t/~ Alice’s clock. However because Bob does not necessarily tions as AI = e Ae− , where H0 is the free Hamiltonian. In this rotating frame, states are stationary share a common x-direction with Alice, he cannot actu- under free evolution, and the problem of clock synchro- ally read out this phase information. This complication nization is reduced to one of aligning a phase reference. can be neatly circumvented with a slight modification In one class of clock synchronization protocols based – on half of the particles, Alice and Bob use a differ- on phase estimation, the systems exchanged are ticking ent magnetic field strength in the z-direction to establish qubits: nondegenerate two-level quantum systems that two different precession frequencies. Bob can now choose undergo time evolution (Chuang, 2000). Much like phase any x-direction to measure each subensemble, because estimation, the use of entangling operations and/or mea- both ensembles of precessing spins are offset by the same surements can lead to different scalings in the synchro- unknown phase shift with respect to Alice’s spins, and nization accuracy. For example, a protocol that uses only he can achieve synchronization by merely observing the separable (unentangled) ticking qubits and single-qubit beats between the oscillations. measurements requires O(22k) ticking qubit communica- In the language of the final section of this review, we tions (a coherent transfer of a single qubit from Alice to can understand this protocol as one in which standard Bob) to achieve an accuracy in the time offset of Alice’s refbits (see Sec. VI.D for more details) are being distilled 41

from the initial singlets and put to use as a bounded types of reference systems, it is possible to distribute shared RF. Note that if such synchronization was Char- more general and exotic types of reference frames through lie’s intention all along, then such a protocol would not the exchange of appropriate quantum systems. be a particularly efficient use of resources – he could just For example, consider the distribution of a reference as simply have distributed to Alice and Bob a shared ordering, i.e., a labeling of N objects (von Korff and RF state (such as those given in Eq. (3.28), for exam- Kempe, 2004). Through the use of techniques similar ple) which require no entanglement whatsoever (Preskill, to those described in Sec. V.D.2 for the distribution of a 2000).17 This fact, together with the result that this Cartesian frame, one can construct an optimal protocol entanglement cannot be purified (an issue we return to that distributes an ordering of N particles using N sys- in Sec. VI.E), suggest that shared entanglement between tems with dimensionality N/e. (In contrast, the classical two parties does not provide an advantage for clock syn- problem requires N systems each with N distinguishable chronization (or other forms of RF alignment). states.) A third class of protocols for clock synchronization Another problem is the distribution of a reference make use of precise timing of light signals exchanged be- frame for chirality (Collins et al., 2005; Diosi, 2000; tween parties, and for which the quantum limits have Gisin, 2004). Such quantum systems have been given recently been investigated. Instead of classical coherent the moniker “quantum gloves”. Clearly, a full Carte- state light pulses for the signals, one can use highly en- sian frame includes a reference for chirality; however, dis- tangled states of many photons and beat the standard tributing a full Cartesian frame purely for the purposes quantum limit (Giovannetti et al., 2001). Essentially, of distributing a reference chirality is not very econom- the advantage is due to entanglement-induced bunching ical, and more efficient methods are possible. Also, a in arrival time of individual photons, enabling more ac- chiral reference can be distributed perfectly, i.e., with no curate timing measurements. The key disadvantage of error, with only finite quantum resources. Methods for this technique is that the loss of a single photon de- the distribution of a chiral reference using only two kinds stroys the entanglement and renders the measurement of particle (i.e., a proton and an electron) and only four useless (Giovannetti et al., 2001, 2004a), although tech- spinless particles, along with other interesting combina- niques have been developed to “trade off” the quantum tions, can be found in Collins et al. (2005). advantage in return for robustness against loss (Giovan- One can also consider the problem of secret sharing of netti et al., 2002). Furthermore, the effect of dispersion unspeakable information (Bagan et al., 2006b). In such is known to be an important issue with such protocols, a protocol, a quantum state is shared between several with the use of entanglement possibly offering an advan- parties with the aim that a reference frame can only be tage here as well (Fitch and Franson, 2002; Giovannetti determined if all of the parties come together (to per- et al., 2004b). We note that such protocols differ from form collective operations and measurements). Parties those based on phase estimation in that they make use working alone, or together using only LOCC, cannot de- of relativistic principles (specifically, the constancy of the termine the reference frame with the same precision. speed of light).

J. Private communication of unspeakable information I. Other instances of alignment With the development of optimal schemes for the dis- Above, we considered the alignment of Cartesian tribution of a reference frame, it is natural to consider frames using N spin-1/2 systems. A different approach how two parties, Alice and Bob, can perform such a to this alignment problem is to use a single Hydrogen distribution privately by using some number of shared atom (Peres and Scudo, 2001b). The analysis of this secret bits – a classical key – to randomize the signal task is similar to that presented above, with the notable state (Chiribella et al., 2006). In other words, we con- difference that multiplicities of representations of SO(3) sider the problem of how well two parties can convert a not are available with the hydrogen atom. Also, the use private classical key into a private shared RF (of bounded of elliptic Rydberg states of the hydrogen atom have been size) using a public channel and given that they do not considered for this problem, with resulting fidelity com- previously possess a shared RF (private or public). For parable with that of the optimal scheme for a hydrogen concreteness, we investigate the private communication et al. atom (Lindner , 2003). of a Cartesian frame. Although a phase reference (clock) and spatial direc- Consider the optimal scheme for the distribution of tion (or Cartesian frame) may be the most ubiquitous a Cartesian frame using fidelity as the figure of merit (which achieves the Heisenberg limit). The fiducial signal state for this scheme is given in Eq. (5.74). (The scheme will be essentially identical for any figure of merit.) As 17 An analogue of this scheme for Cartesian frame alignment has been proposed (Rudolph, 1999a), and similar reservations apply; with any private quantum communication, Alice and Bob however, at least this latter approach was motivated by a funny can choose unitary operators from a set of unitaries, story (Rudolph, 1999b). based on their classical key, to randomize any quantum 42 state as viewed by an eavesdropper Eve who does not case) if use of the channel at the early time allowed Bob share this key, using the techniques of Ambainis et al. to estimate Alice’s direction with greater accuracy at the (2000). In general, to completely randomize a state on later time. N a Hilbert space of dimension d = 2 , Alice and Bob re- The following is such an analogue. Alice prepares a quire a key consisting of 2N secret bits. However, we pair of spin-1/2 systems in a singlet state, and in the note that Alice and Bob do not share a reference frame first use of the channel, sends one of these to Bob. Later, to begin with and thus they can only perform correlated when she has a sample of the classical direction nˆ that operations on the multiplicity spaces . Despite this Nj she would like to send to Bob, she implements a unitary restriction, a complete randomization can be achieved rotation of π degrees about nˆ on her spin-1/2 system and for the fiducial state of Eq. (5.74). To see this, recall sends it to Bob. Through Alice’s operation, the singlet is that this state can be expressed as a coherent superpo- transformed into ( +nˆ nˆ + nˆ +nˆ )/√2, which is a sition, over all representations (charge sectors) of SO(3), two-spin state that| cani|− bei used|− toi| indicatei the direction maximally-entangled of states across the representation nˆ; in fact, given that the image of such a state under space and an equally-sized subspace ˜ of the multi- Mj Nj SU(2)-averaging covers the entire symmetric subspace, plicity space. Because of this particular structure of the this state is as good a direction indicator as the parallel fiducial state, a complete randomization over just the spin state of Sec. V.A. In her second use of the quantum subsystems ˜ will take every signal state to the same Nj channel, she sends her spin to Bob, and Bob estimates mixed state (and thus achieves a complete randomization the direction. The optimal average fidelity that can be on the Hilbert space that is the span of the supports of ˜ achieved for such a state and measurement was shown in the signal states). The dimension of each subsystem j Sec. V.A to be 3/4, which is greater than the fidelity of is equal to that of , namely, d =2j + 1, thus requir-N Mj j 2/3 that could be achieved using a single spin-1/2 sys- ing 2 log2(2j + 1) secret bits to completely randomize. tem.18 Thus, this scheme provides an analogue of dense The total number of secret bits required to completely coding for unspeakable quantum information. The opti- randomize all the signal states is mization of this sort of dense coding scheme has not been N/2 investigated to date. log (2j + 1)2 3log N . (5.106) A slightly different analogue of dense coding of un- 2 ≃ 2 j=0 speakable information was considered in Bagan et al. hX i (2004a), building on the results of Acin et al. (2001). Thus, through the transmission of N qubits on a pub- This protocol involves Alice initially sending Bob half of lic channel and using 3 log2 N classical bits of private an entangled state over multiple spin systems. It is as- key, one can achieve the distribution of a private Carte- sumed that subsequently the entire lab of the sender is sian frame at the Heisenberg limit, which is to say with subject to the same SU(2) transformation that her half an error that scales as 1/N 2. We note that this num- of the entangled pairs are subject to. Under this as- ber 3log2 N is identical to the number of classical bits sumption the three parameters describing the relation of that can be transmitted privately given a private shared her spin-1/2 system to that of the receiver also describe Cartesian frame as key, as discussed in Sec. III.D. the relation of her local Cartesian frame to that of the receiver. In this scenario, the sender is essentially pas- sive: both the spin and the local Cartesian frame must K. Dense coding of unspeakable information be acted upon by some external agency. Unfortunately, it is not clear whether this is of practical significance in Consider the following problem. Alice wants to send the most common case where the SU(2) transformation Bob classical information, but at the time that Alice acting on the local Cartesian frame is a rotation in space. learns which message she would like to send Bob, the For instance, if a rotation of the entire laboratory is real- cost of using the quantum channel is very high, whereas ized by an external torque, it is not clear that the state of earlier, before Alice learns the message, the cost of using a spin-1/2 system stored in this laboratory (i.e., in some the channel is low. Dense coding allows Alice to make trapping potential) will necessarily undergo the same ro- use of the channel at the early time, prior to learning tation.19 Nonetheless, the optimal solution for this sort the message she wishes to send, in order to increase the of scheme has been provided for an arbitrary number amount of information she succeeds at transmitting to Bob at the later time. Suppose that Alice wishes to send to Bob a direction in space rather than a classical message. Suppose more- 18 The optimal average fidelity that can be achieved for a symmetric over that at the time where Alice learns the direction product state consisting of N spin-1/2 systems (a parallel state) she would like to send Bob, the cost of using the quan- is (N + 1)/(N + 2), and this result generalizes to any symmetric tum channel is very high, whereas earlier, before Alice pure state (Massar and Popescu, 1995). 19 That is, unless the spin degrees of freedom are coupled to other learns the direction, the cost of using the channel is low. fields in the lab. This coupling itself would negate the protocol One would have a natural analogue of dense coding to however, as it implies some ongoing active transformations on unspeakable information (directional information in this the stored spins. 43 of spin-1/2 systems (Bagan et al., 2004a). The optimal VI. QUANTUM INFORMATION WITH BOUNDED state bears a strong similarity to the optimal state for REFERENCE FRAMES aligning Cartesian RFs, presented in Sec. V.D.2. In the reference frame alignment schemes of Sec. V, we determined which quantum states of a given bounded size L. Error correction of unspeakable information? were optimal in serving as a sample of the sender’s clas- sical reference frame. The systems were ultimately mea- We end this section with a cautionary note on the po- sured relative to the receiver’s classical reference frame, so that the unspeakable information that they contained tential use of quantum methods for aligning reference frames, first made by Preskill (2000) for the specific task was essentially amplified to the macroscopic scale with some associated uncertainty. However, there will be sit- of clock synchronization: that the standard techniques of quantum error correction cannot be directly applied to uations for which this amplification process is not ideal, and instead one should make direct use of the quan- unspeakable information. tum RF itself.20 Whatever purpose the recipient had Consider a situation wherein Alice and Bob wish to in mind in trying to align his classical RF with that of align their respective frames by exchanging quantum sys- the sender’s, one can ask to what extent he could achieve tems via some noisy quantum channel. Let be the this same purpose by storing his quantum sample of the decohering superoperator describing the channel.F The sender’s reference frame in his lab and thereafter using it form of this noise is critical to their ability to complete in place of his classical RF. this task; here, we consider only two extreme cases. If Furthermore, many quantum experiments involve the noise is of the form = qI in terms of the Mq Nq mesoscopic or even microscopic systems that can be un- decomposition of Eq. (2.24),F ⊕ where ⊗D is the completely N derstood as playing the role of a reference frame. For depolarizing superoperator on . ThisD noise affects only instance, a Bose-Einstein condensate may act as a ref- the multiplicity subsystems; inN other words, it acts only erence frame for the phase conjugate to atom number, on the relational degrees of freedom of the transmitted even though it may contain a relatively small number of systems. In such a case, RF alignment is still possible atoms. We are therefore led to consider the question of (although possibly at a decreased efficiency, as the opti- how well a bounded-size quantum system may stand in mal protocols took advantage of these multiplicity sub- for a classical reference frame. systems). Alice and Bob can choose to transmit states In Sec. IV we have already considered the problem that are encoded entirely within the gauge subsystems of treating reference frames within the quantum formal- , as these subsystems are decoherence-free in terms q ism, but the system instantiating the reference frame Mof the noise. was assumed to be of unbounded size. Here we shall On the other hand, if the noise is of the form = be interested in bounded-size quantum reference frames. I in terms of the decomposition of Eq. (2.24),F q q q We shall focus in particular on the implications of such ⊕thenDM the⊗ gaugeN subsystems will experience com- Mq RFs on one’s ability to perform quantum-information plete decoherence. However, Alice and Bob cannot processing tasks, specifically: the fundamental primitive choose to execute their alignment protocol entirely within of quantum state estimation, operations and measure- the decoherence-free multiplicity subsystems , because Nq ments in quantum computation, and the quantum cryp- these subsystems cannot carry unspeakable information tographic protocols of data hiding and bit commitment. (at least, not of this type). Whereas speakable infor- Furthermore, we demonstrate that for bounded shared mation can be encoded into any desired subsystem, un- RFs, like entanglement, it is possible to develop a gen- speakable information must be encoded into subsystems eral theory of the manner in which this resource is dis- carrying the appropriate degree of freedom. tributed, transformed from one form to another, distilled, This latter case can be worded as a simple physical ex- degraded with use, quantified, etcetera. ample. Consider the alignment of a phase reference, using a noisy channel that simply adds a constant but unknown phase shift. If Alice and Bob use one of the techniques A. Measurements and state estimation with bounded of this section to attempt to align their phase references reference frames using this channel, Bob will acquire an estimate of the phase difference between his and Alice’s RFs. However, State estimation is a fundamental primitive of quan- because Bob knows this estimate may differ from the ac- tum information processing. In this section, we discuss tual difference by some unknown shift, caused by the the role of reference frames in performing measurements channel, he in fact has learnt nothing about the rela- required for state estimation, and the effect of bounding tion between his phase reference and Alice’s. There is no the size of this RF. protocol that they can perform that will distinguish the unknown phase shift relating their RFs and the unknown phase shift applied by the channel, and thus alignment cannot be performed using this channel (Preskill, 2000; 20 For example, Janzing and Beth (2003) consider the constraints Yurtsever and Dowling, 2002). on amplifying and copying quantum RFs for phase. 44

1. A directional example yields 2j +1 1 p(+ +) = , p( +) = , Consider the task of estimating whether the state of a | 2j +2 −| 2j +2 spin-1/2 system is aligned or anti-aligned with some per- p(+ )=0 , p( )=1 . (6.2) fect (unbounded) directional RF, given the promise that |− −|− it is one of the two. If one is able to compare the system Assuming equal prior probabilities for +z and z , the with this RF, then this task can be easily achieved, as it average probability of successful discrimination| i |− is i corresponds simply to discriminating a pair of orthogonal states, +z and z , where we take z to be the axis de- 1 1 1 | i |− i psuccess = p(+ +)+ p( )=1 . (6.3) fined by the directional RF. Specifically, a measurement 2 | 2 −|− − 4(j + 1) of S z, the spin along z, determines the answer with certainty.· In contrast, if one is not able to make use of The smallest possible RF corresponds to taking j =1/2, this RF, then a superselection rule is in force, the mea- in which case psuccess = 5/6; see Pryde et al. (2005) for surement of S z becomes impossible, and the states +z an experiment based on this example. For large j, we and z become· completely indistinguishable. | i have a probability of success that approaches 1 linearly in |− i 1/j, and we recover perfect distinguishability as j , There is an intermediate scenario between these two corresponding to the case of an unbounded RF. → ∞ extreme cases, however, wherein one only has access to This example can be extended to the problem of esti- a sample of the RF – one that is of bounded size. In mating the relative angle between a spin j1 and a spin this case, +z and z become partially distinguishable, j2; the optimal measurement is the projective measure- as we now| demonstratei |− i with an example. We consider ment ΠJ , J = j1 j2 ,...,j1 + j2 onto the subspaces the case wherein the directional RF is a spin-j system, of total{ angular| momentum− | J, for} the same reasons as for some arbitrary but finite j, prepared in an SU(2) above (Bartlett et al., 2004b). The optimization prob- coherent state jz (the eigenstate of J z associated with | i · lem becomes nontrivial when we allow for states of the the maximum eigenvalue). bounded RF and/or the system that span multiple irreps, Because the task is to estimate the relations between i.e., states that are not eigenstates of Jˆ2. Measuring a the bounded RF and the system, it is possible to restrict spin-j system relative to a bounded directional RF con- the measurement to one that is invariant under collec- sisting of a pair of spins is considered in Bagan et al. tive rotations (i.e., rotations of both the bounded RF (2006a); Lindner et al. (2006). and the spin-1/2 system by the same amount). In other These sorts of results serve to illustrate how, for Lie words, one can consider a global superselection rule as- groups at least, a measurement relative to a bounded RF sociated with the group SU(2) to apply, because the sys- cannot perfectly simulate one relative to an unbounded tem serving as an RF for direction is treated internally. RF. Note that we have only considered the inferential As a result, the form of the measurement is highly con- but not the transformative aspect of the measurement, strained. Note that the joint Hilbert space j 1/2 that is, we have not considered how the quantum state of the bounded RF and system decomposes intoH ⊗ a H sum of the system is updated as a result of the measurement. j+1/2 j 1/2 of a J = j +1/2 and a J = j 1/2 The work of Wigner (1952) and of Araki and Yanase Hirreducible⊕ H representation− of SU(2), the group of− col- (1960) demonstrates, however, that for rank-1 projective lective rotations. By Schur’s lemmas (see the Proof in measurements one cannot perfectly simulate a von Neu- Sec. II) a positive operator on this space that is SU(2)- mann update rule when an unbounded RF is replaced by invariant must have the form p+Πj+1/2 + p Πj 1/2, a bounded one. − − where Πj 1/2 is the projector onto j 1/2. Thus, a rotationally-invariant± measurement isH represented± by a POVM with elements of this form. However, any such 2. Measuring relational degrees of freedom POVM may be obtained by classical post-processing of the outcome of the two-element projective measurement We note that measurements relative to a bounded quantum RF are example of measurements of relational Πj+1/2, Πj 1/2 , so that the latter is the most informa- tive{ rotationally-invariant− } POVM. The POVM elements degrees of freedom. While a complete discussion of re- lational formulations of quantum theory is beyond the Πj+1/2 and Πj 1/2 are associated with the measurement outcomes “aligned”− and “anti-aligned,” respectively. scope of this review, we briefly make some connections between problems involving reference frames and rela- Denote the probability that the state z is found tional ones. |± i to be aligned with the bounded RF by p(+ ) and the The estimation of relative parameters for various de- |± probability that it is found to be anti-aligned by p( ). grees of freedom encompasses such natural tasks as esti- −|± The Born rule mating the distance between two massive particles, the phase between two modes of an electromagnetic field (the essential aim of a homodyne measurement), or the angle p( )= jz z Πj 1/2 jz z , (6.1) between a pair of spins as described above, all of which ±|± h |h± | ± | i|± i 45

are clearly related to issues of bounded reference frames. If the reference frame is bounded – quantified in this Such measurements have been discussed recently in con- example as a finite mean photon number of the laser nection with their ability to induce a relation between field – the operations performed with respect to this quantum systems that had no relation prior to the mea- bounded RF may have imperfect precision, and gener- surement, e.g., inducing a relative phase between two ally the system and the field become entangled. A sim- Fock states (Javanainen and Yoo, 1996; Molmer, 1997; ple and standard model of a single two-level atom reso- Sanders et al., 2003) or a relative position between two nantly interacting with a single mode (cavity) field via momentum eigenstates (Cable et al., 2005; Rau et al., a Jaynes-Cummings interaction serves to illustrate the 2003). basic idea (van Enk and Kimble, 2001). The interaction Also, measurements of relative parameters are crit- Hamiltonian takes the form ical for achieving programmable quantum measure- + ments (Duˇsek and Buˇzek, 2002; Fiur´aˇsek et al., 2002). H = i~g Sˆ aˆ aˆ†Sˆ− , (6.4) − Such measurements use the state of a quantum system
(the reference system) to determine the form of a mea- wherea ˆ is the field mode annihilation operator, and surement performed on another system. For example, a Sˆ± are the atomic raising and lowering operators. If reference system in an SU(2) coherent state as above can the field mode is initially prepared as a coherent state 1 2 2 α n √ be used to “program” the choice of measurement basis α = e− | | n(α / n!) n with very large amplitude of a spin-1/2 system. The problem of optimizing pro- |αi2 , it is common to| i replace the field operators | | → ∞ P grammable quantum measurements, including determin- a,ˆ aˆ† with classical c-numbers α and α∗. In the lan- ing the form of the joint measurement and the optimal guage of this review, this is the process of externalizing state of the reference system, is directly related to the the reference frame. For an atom initially in the excited problem of optimizing measurements of relative parame- state, evolution under this classical field then yields the ters between a system and a bounded reference frame. well-known Rabi oscillations between the ground 0 and | i Note that problems of relative parameter estimation excited state 1 . Specifically, the state at time t is | i are complementary to those of determining the optimal measurement schemes for estimating collective parame- ψC (t) = sin(g α t) 0 + cos(g α t) 1 . (6.5) | i | | | i | | | i ters for a rotational degree of freedom, the subject of Sec. V. Consider now what occurs if we choose not to exter- nalize the driving field, and in particular describe it via a finite amplitude coherent state. The evolution of the atom and field under the Hamiltonian of Eq. (6.4) can be B. Quantum computation with bounded reference frames solved exactly, yielding 1. Precision of quantum gates ∞ ψQ(t) = An(t) 0 n + Bn 1(t) 1 n 1 , (6.6) | i | i| i − | i| − i In the majority of architectures proposed for quantum n=1 computation, external classical fields are utilized to im- X plement single qubit logical operations. As an example, where we will focus on the use of coherent states of the elec- 1 α 2 n 1 tromagnetic field, which are particularly ubiquitous for e− 2 | | α − An(t)= sin(g√nt) , (6.7) quantum computing architectures – in the form of either (n 1)! − lasers or radio frequency fields generated by an oscillating 1 α 2 n 1 e−p2 | | α − current. From the perspective of this review, we inter- Bn 1(t)= cos(g√nt) . (6.8) pret such fields as defining a reference frame – in this − (n 1)! − case, a clock – with respect to which coherent super- p positions of the computational basis (energy eigenstate) To compare this evolution under a bounded refer- states are necessarily defined. Note that the fact that ence frame to the ideal (unbounded) case we should de- the RF is interacting directly with the qubits does not quantize the reference frame22 using the techniques of weaken such a viewpoint – at some stage in the quantum Sec. IV.A.2. Following the procedure outlined therein, we information processing a clock must physically interact now move into the tensor product structure induced by with the quantum computer (perhaps via intermediary systems); if it did not, then there would be no opera- tional difference if we enforced a superselection rule for energy.21 in this section, as we are discussing the coupling of atoms to the optical fields under (additively) energy conserving Hamiltonians, we thus extend the type of SSR under consideration to one for total energy. 22 This dequantization was not carried out in van Enk and Kimble 21 Previously, we have concerned ourselves primarily with SSRs for (2001), although for this example it does not make a quantitative photon number when dealing with optical examples. However, difference to the overall conclusions about gate fidelity. 46 energy difference (relational) versus total energy (global). perform operations and measurements on quantum sys- In terms of this tensor product structure we have tems, and thus limitations on quantum information pro- cessing tasks such as quantum computing. However, this ∞ imprecision is not the only limitation enforced by quan- ψ (t) = φ (t) n 1 , (6.9) | Q i | n irel| − igl tum mechanics. In addition, any measurement that ac- n=0 X quires information about the relations between the sys- tem and RF must necessarily disturb them uncontrol- where φn(t) rel = (An(t) 0 rel + Bn 1(t) 1 rel) is unnor- − lably. The resulting disturbance to the RF can be un- malized.| i | i | i derstood as a measurement back action. The effect of The reduced density matrix of the relational system this back-action has been studied for reference frames (i.e., we trace out the global degree of freedom) is there- for spatial position (Aharonov and Kaufherr, 1984), for fore a mixed state directional reference frames (Aharonov et al., 1998) and for clocks (Casher and Reznik, 2000). Here, we investi- ∞ ρ (t)= φ (t) φ (t) , (6.10) gate how measurement back-action on a bounded RF can Q | n ih n | n=0 lead to its degradation, i.e., a reduction of its suitability X to perform future measurements. and it is this mixed state we wish to compare with the The conventional approach wherein reference frames pure state (6.5) expected in the unbounded, externalized suffer no back action may yield a poor approximation to description. the full quantum treatment, as suggested above. This We perform the comparison by computing the fidelity issue may be particularly important for quantum com- F (t)= ψ (t) ρ (t) ψ (t) between the mixed state ob- h C | Q | C i putation, where a large number of high-precision mea- tained through the full quantum treatment and the pure surements must be performed. In some implementa- state of Eq. (6.5) obtained via the above approxima- tions, such measurements are performed relative to a tion. If we consider the specific choice of evolution time reference frame that is usually described by a finite t = π/(2 α g), which corresponds to performing a σ | | x quantum system; for example, the proposed single-spin gate, then this fidelity is very well approximated (even measurement technique using magnetic resonance force for small values of α ) by (Haroche, 1984) | | microscopy (Rugar et al., 2004), or the single-electron 2 transistors used for measurement of superconducting 1 π√1+ α 2 π F t = π 1 cos | | e− 8(|α|2+1) qubits (Makhlin et al., 2001). We now demonstrate that 2 α g ∼= α | | 2 − | | the number of measurements for which a quantum refer-
 π2 1
 =1 16 α 2 + O α 4 . (6.11) ence frame can be used scales quadratically rather than − | | | | linearly in the size of the reference frame, which is a
We see that the gate operation results in a state that is promising result for the prospect of using microscopic in error (as quantified by the fidelity) by an amount that or mesoscopic reference frames in performing repeated is inversely proportional to the mean number of photons high-precision measurements. in the driving field. In the following example, we investigate the degrada- The extent to which such a model captures the essen- tion of a quantum reference direction as it is used for tial features of currently-proposed quantum computing repeated measurements. We use a spin-j system for our architectures has been the subject of considerable de- quantum reference direction (the RF), with Hilbert space bate, cf. van Enk and Kimble (2001); Gea-Banacloche . We choose the initial quantum state of the spin-j Hj (2002a,b); Gea-Banacloche and Ozawa (2005); Itano system to be ρ(0) = j, j j, j ; this choice of initial state (2003); Nha and Carmichael (2005); Silberfarb and simplifies the analysis,| andih in| addition it has been deter- Deutsch (2003). What is clear is that such effects are mined to be the initial state that maximizes the initial generally about two orders of magnitude smaller than success probability (Bartlett et al., 2006c). This quantum the typical spontaneous emission rates in these systems. RF is aligned in the +z direction relative to a background However, in situations wherein the reference frame is frame. small (for example, if quantum computers together with The systems to be measured will be spin-1/2 systems, the control fields were to be built on chips in an inte- each with a Hilbert space 1/2. We choose the initial grated manner) or in systems which have negligible spon- state of each such systemH to be the completely mixed taneous emission, then it is not unreasonable that such state I/2, and our quantum RF will be used to measure considerations will have to be incorporated into analyses many such independent spin-1/2 systems sequentially. of fault tolerance. We shall assume trivial dynamics between measurements, and thus our time index will simply be an integer speci- fying the number of measurements that have taken place. 2. Degradation of a quantum reference frame The state of the RF following the nth measurement is de- noted ρ(n), with ρ(0) denoting the initial state of the RF As we have demonstrated, a bounded reference frame prior to any measurement. We consider the state of the can result in non-trivial limitations on one’s ability to RF from the perspective of someone who has not kept a 47 record of the outcomes of previous measurements. Thus, Let ǫ < 1 be a fixed allowed error probability for the at every measurement, we average over the possible out- spin-1/2 direction estimation problem. After n measure- comes with their respective weights to obtain the final ments, the probability of successful estimation is lower density operator. bounded by 1 + nR, so the number of measurements re- The measurement which optimally determines whether quired to ensure that this bound be greater than 1 ǫ is a spin-1/2 particle is aligned or anti-aligned to a spin-j ǫ/R. Consequently, the number of measurements− that system was determined in Sec. VI.A.1 to be the two- can− be implemented relative to the spin-j RF with prob- outcome projective measurement Π+ Πj+1/2, Π ability of error less than ǫ is { ≡ − ≡ Πj 1/2 on j 1/2. We use this measurement here. − } H ⊗ H 2 It can be shown that of the many ways of implement- nmax ǫj . (6.18) ≃ ing this measurement, the update rule that degrades the reference frame the least is the standard L¨uders update This result implies that the number of measurements for rule (Bartlett et al., 2006c). Thus, the resulting evolution which an RF is useful, that is, the longevity of an RF, in- of the quantum RF as a result of the nth measurement creases quadratically rather than linearly with the size of is the RF. Thus, in order to maximize the number of mea- surements that can be achieved with a given error thresh- ρ(n+1) = (ρ(n)) , (6.12) old, one should combine all of one’s RF resources into a E single large RF and perform all measurements relative where the superoperator is given by to it, rather than use a number of smaller RFs individu- E ally. We note that this degradation, as quantified by the (ρ) = TrS Πc(ρ I/2)Πc , (6.13) decreasing average probability of success P (n), can be E ⊗ s c +, modeled precisely as the distribution of a classical refer-  ∈{X−}  ence direction undergoing a random walk (Bartlett et al., with TrS the partial trace over 1/2. 2006d). The map can be written usingH the operator-sum rep- E Using similar methods, it has also been demonstrated resentation as that a bounded quantum phase reference, realized as a

1 c c single-mode quantum state of the electromagnetic field (ρ)= E ρE † , (6.14) E 2 ab ab with bounded photon number, also leads to a longevity c +, a,b 0,1 ∈{X−} X∈{ } that scales quadratically in this size (mean photon num- ber) (Bartlett et al., 2006c). It is an open problem to where Ec a Π b is a Kraus operator on and ab c j determine if this quadratic scaling is a general result. 0 , 1 is≡ a basish | | fori . These operatorsH can be {| i | i} H1/2 We have discussed the degradation associated with a straightforwardly determined in terms of Clebsch-Gordon loss of purity of the reference frame state. Another mech- coefficients. anism of degradation is for the reference frame to become We quantify the quality of a quantum RF as the aver- misaligned with the background reference frame of which age probability of a successful estimation of the orienta- it is a token. Poulin and Yard (2006) have shown how tion of a fictional “test” spin-1/2 system which is, with a reference frame can suffer this sort of misalignment equal probability, either aligned or anti-aligned with the when the systems being measured have a non-zero po- background +z-axis. Denote the pure state of the test larization (that is, are not described by the completely spin-1/2 system that is aligned (anti-aligned) with the mixed state), and that in the presence of such drift, the initial RF by 0 ( 1 ). For a spin-1/2 system prepared longevity scales linearly, rather than quadratically, in the in the state 0| ori |1 i with equal probability, the average size of the reference system. probability of| i success| i using a quantum RF state ρ is

1 + P s = TrR(ρ(E + E− )) . (6.15) C. Quantum cryptography with bounded reference frames 2 00 11 The solution for ρ(n), given the initial state ρ(0) = In Sec. IV.D, we demonstrated that SSRs cannot pro- j, j j, j , yields an average probability of success P s(n) vide any fundamental limitations on quantum crypto- that| ih decreases| with n as graphic protocols, essentially because quantum reference systems which obey the SSR can enable it to be effec- 1 j 2 n tively lifted. However, this result does not mean that P s(n)= + 1 2 . (6.16) 2 2j +1 − (2j + 1) SSRs are uninteresting for cryptography. In quantum   cryptography, it is typical to focus on unconditional secu- The initial slope R of this function bounds the rate of degradation. It is rity – security not premised upon assumptions about the resources of one’s adversaries, but only upon the validity 3 R P s(1) P s(0) = 2j/(2j + 1) . (6.17) of the laws of quantum mechanics. In classical cryptog- ≡ − − raphy, in contrast, security is typical conditional – it is Thus, in the large j limit, we have the rate of degradation generally premised upon assumptions about the compu- with n satisfying R 1/(4j2). tational capabilities of one’s opponents. Other types of ≥− 48

conditional security can be premised upon assumptions 2. Ancilla-free bit commitment about other non-computational capabilities or resources available to the adversarial parties. In this section, we A particularly simple class of bit commitment proto- consider the specific case where the physical resource cols (Spekkens and Rudolph, 2001) involve Alice prepar- about which assumptions are being made is some kind of ing one of two orthogonal states χ , according to | 0,1i RF, the lack of which in turn induces an effective SSR. whether she wishes to commit a bit b =0, 1. Here χ0,1 This is effectively an assumption of bounded resources, are (generally entangled) states over a “proof” system| i because given unlimited resources the SSR can be lifted and a “token” system. She sends the token system to as in Sec. IV.B. We use the specific examples of data Bob as her commitment. To unveil the bit, she sends hiding and bit commitment to illustrate protocols that Bob the proof system, and he verifies her commitment 23 achieve this sort of security. by projecting onto χ0,1 . A simple version of such a protocol was discussed| ini Sec. IV.D above. Consider the situation wherein Alice and Bob are con- 1. Data hiding with a superselection rule strained such that they cannot make use of a reference frame, either shared or local. This constraint enforces the In a quantum data hiding protocol, one party (Charlie) protocol to be ancilla free – for instance we do not al- wants to share a single bit of data by distributing systems low either party to prepare ancillary systems which could amongst two other parties (Alice and Bob) in such a way then act as an effective RF in the manner described in that the bit can only be recovered if the parties have Sec. IV.B. It turns out that under such a constraint, ar- some mechanism for performing joint measurements on bitrarily secure bit commitment is possible (DiVincenzo the distributed systems. Such measurements could be et al., 2004). performed by the parties coming together, or by using It is illustrative to first consider a case where such a a quantum channel, or by performing teleportation (us- restriction does not help. Consider ancilla-free bit com- ing prior entanglement) with a classical channel. These mitment in the case that Alice and Bob lack a phase reference. As Alice must prepare the initial states χ , possibilities are generally considered equivalent. Thus, | 0,1i Charlie must assume that the two parties have no access they must each lie completely in a single superselection to the specific physical resource of a quantum channel. It sector, i.e., eigenstates of total photon number, and take has been proven that perfect quantum data hiding is not the form possible even with this assumption (Terhal et al., 2001). χ = cb N n n . (6.19) If, in addition, Charlie can assume that the two parties | bi n| − iP | iT n do not share a phase reference (that is, they are subject X to a local Abelian SSR) then perfect data hiding can Because the reduced density matrices of the token system be achieved (Verstraete and Cirac, 2003). For example, b 2 ρb = n cn n T n are diagonal in the number basis, without a shared phase reference for their optical modes the fact that| | Bob| i canh | only perform measurements diago- as in Sec. III.C.1, Alice and Bob cannot distinguish the nal inP this basis actually is no restriction on him at all – pair of orthogonal pure states ψ± = ( 01 10 )/√2 he can cheat (by trying to distinguish these states) just | +i | i ± | i using LOCC because ( ψ ) = ( ψ− ), UA ⊗UB | i UA ⊗UB | i as well as he could in an unconstrained protocol. and so this pair of states could be used to encode the Consider now if Alice is cheating, i.e., she attempts classical bit. to commit her bit only after the commitment stage. An As per the discussion in Sec. IV.B, it is also clear that optimal cheating strategy for Alice is to prepare a state shared reference systems could be used by Alice and Bob χ˜ χ0 + UP IT χ1 , where the unitary matrix UP to break such a data hiding protocol. Such reference sys- |oni the ∝ | proofi system⊗ is| onei which maximizes the overlap tems need not be entangled, which shows that breaking χ0 UP IT χ1 (Spekkens and Rudolph, 2001). If, af- of the data hiding in this case is quite different to the case terh | the commitment⊗ | i stage, she decides to commit b = 0, of using entanglement to implement a quantum channel. then she simply sends the proof system as is. If instead However, if Charlie has reason to believe that the refer- she decides to commit b = 1 then she applies UP† to the ence systems shared by Alice and Bob are bounded in proof system before sending it to Bob. The question is size, then it is still possible for him to achieve data hid- whether Alice can perform this optimal cheating strat- ing. He does this by using such a large number of systems egy despite the SSR. Consider first the unitary UP which to encode the bit that any bounded shared reference does maximizes χ0 UP IT χ1 . If UP N n AN n , not suffice to extract all the required data. h | ⊗ | i | − i ≡ | − i where Ai is a state not necessarily respecting the SSR, | i b 2 then χ0 UP IT χ1 = n cn AN n N n . Clearly the maximizationh | ⊗ | ofi this expression| | h − will| − bei achieved PiφN−n by choosing AN n = e N n , i.e. for a uni- 23 We note in passing that a different type of assumption, namely | − i | − i tary UP which is diagonal in the number state basis. that Alice and Bob share partially misaligned reference frames, can be used as a kind of guaranteed noisy channel, and, as in Furthermore the state χ˜ then takes the generic form c N n n which| i respects the SSR. Under the classical cryptography, such channels can be used for secure two n n| − iP | iT party protocols (Harrow et al., 2006). assumptions of an ancilla-free SSR protocol, any state P 49 prepared by Alice or any unitary operator she applies is Generalizations of the above pair of states χ can | bi constrained to be diagonal in the number basis – as we be defined for which, as j becomes large, F (ρ0,ρ1) have seen, in this case she can still achieve the optimal 0, implying perfect security against Alice (DiVincenzo→ cheating strategy despite such a constraint. et al., 2004). We believe that a fruitful avenue for future The Abelian SSR induced by lack of a phase reference research would be to determine if such constraints on therefore does not help devise a more secure ancilla-free ancilla-free bit commitment can be achieved using a non- bit commitment. It can be shown, however, that a dif- Abelian superselection rule of the form discussed in this ferent type of Abelian SSR does lead to arbitrarily se- review. cure ancilla-free bit commitment. We follow DiVincenzo et al. (2004). Consider a number of spin systems, with a local Abelian SSR given as follows: all local operations D. Quantifying bounded shared reference frames must commute with the total local angular momentum 2 operator J . Thus all states and operations must be di- Much of quantum information theory is concerned with agonal in total spin quantum number j. However, they tradeoffs in the utilization of various types of fundamen- can have coherence between the different m eigenvalues tal resources. The canonical example is quantum tele- of Jz. This superselection rule is distinct from any that portation, which demonstrates that one ebit (Bell pair) we have considered in this review, and does not appear to of shared entanglement plus two communicated classi- be related to the lack of an appropriate reference frame. cal bits are equivalent to the communication of a single The key property of this Abelian SSR that will be use- qubit. In this section, we demonstrate that a shared ref- ful for ancilla-free bit commitment, and is distinct from erence frame is also a quantifiable resource, akin to en- the other Abelian SSRs we consider, is that the quan- tanglement, which allows parties to perform tasks that tum number j labeling the local superselection sectors is they were unable to perform without it, or to perform non-additive . tasks more efficiently. Using the standard j, m notation for the uncoupled Consider the “activation” example of Sec. III.C.2, basis, consider the bit commitment| i protocol which is de- which involved two parties (Alice and Bob) who do not fined by the following two states χ of total spin j = 1 : | bi share the phase reference of a third party (Charlie) or χ = 1 1, 1 φb each other. In this context, the two-mode single-photon b √2 P 0 T | i | i | i state ( 1 A 0 B + 0 A 1 B)/√2 could not be used to per- + 1 1, 0 φb + 1 1, 1 φb , (6.20) form quantum| i | i teleportation| i | i or to violate a Bell inequal- √3 P 1 T √6 P 2 T | i | i | − i | i ity. However, if Alice and Bob were also provided with where the bipartite state + + , where + = ( 0 + 1 )/√2 | iA| iB | i | i | i √ described relative to Charlie’s phase reference, they could φb = 2 0, 0 + ( 1)b 1 1, 0 + 2 2, 0 , (6.21) | 0iT 3 | iT − 2 | iT 6 | iT activate the entanglement in the former state through φb = ( 1)b √3 1, 1 1 2, 1 , (6.22) LOCC. Although Alice and Bob do not share Charlie’s | 1iT − 2 | iT − 2 | iT b phase reference, clearly the state + A + B provides a φ2 T = 2, 2 T , (6.23) | i | i | i | i bounded version of it. This bounded shared phase ref- We note that, although the proof system is also an eigen- erence can be used to activate the entanglement of the 2 state of JT with eigenvalue j = 1, the token system is not; two-mode single-photon state, as can an unbounded clas- this is a result of the non-additive nature of this SSR. If sical shared phase reference. However, unlike the latter, we look at the reduced density matrices ρ on the to- the bounded shared phase reference + A + B can only 0,1 | i | i ken system in the uncoupled basis, we see that they are activate the entanglement with probability 1/2, and in block-diagonal (incoherent mixtures) in the eigenspaces addition, is consumed in the process; it is a shared ref- of Jz, with eigenvalues m = 0, 1, 2. Within each block, erence frame that can be depleted, in this case, through b b the diagonal elements of φ φ are the same – that a single use. The state + A + B can be considered an m m | i | i is, they are indistinguishable| ih by their| total spin. Under elementary unit of Charlie’s phase reference, much like the SSR, Bob is restricted to performing measurements an ebit (a Bell pair) is considered an elementary unit which are diagonal in total spin, and so these two states of entanglement. As a result of this analogy, the state are completely indistinguishable. + A + B has been denoted a refbit. | i | i In a general bit commitment scenario, indistinguisha- Continuing this example, we note that the two-mode bility of the token systems by Bob would imply that Al- single-photon state ( 1 A 0 B + 0 A 1 B)/√2 can also be ice has complete control – that she should be able to viewed as a resource| fori “activating”| i | i | anotheri copy of this perfectly change her commitment after the commitment same state. (This process can alternatively be viewed as stage. However, this is not the case for this example – 2-copy entanglement distillation, as in Sec. III.C.2.) This the two states ρ0,1 have a non-unit fidelity F (ρ0,ρ1) < 1. state is invariant under global phase changes, and thus Because the fidelity sets a bound (for these type of proto- is completely uncorrelated with Charlie’s (or any other cols) on how well Alice can control the outcome (regard- party’s) phase reference, but it nevertheless provides a less of any restrictions on her), we see that some security bounded version of a shared phase reference for Alice against Alice will be possible. and Bob. It is useful to view this state as the elementary 50

unit of a shared phase reference between Alice and Bob, perselection induced variance V (φ) of this state is defined uncorrelated with any other. to be the variance in the local photon number Because of the dual purpose of this state – either as V (φ) 4 φ Nˆ 2 I φ φ Nˆ I φ 2 . (6.24) as an elementary unit of a shared phase reference, or as ≡ h | A ⊗ B | i − h | A ⊗ B | i a state from which entanglement can be activated with
the use of a shared phase reference – it has been named This SIV satisfies the following properties: (1) it is addi- tive, meaning V (φ φ′)= V (φ)+ V (φ′) for any φ , φ′ ; and categorized in many different ways depending on its ⊗ | i | i intended use. So which way should it be viewed? The (2) it is symmetric under exchange of A and B; and (3) answer is that this state can serve as a resource for both it is a bipartite monotone, in that it is non-increasing entanglement and a shared reference frame, and that one under LOCC operations that can be performed by Alice must trade off its usefulness for one purpose against the and Bob without a shared phase reference. The state ( 1 0 + 0 1 )/√2, which we identified above as other. In fact, a wide variety of tradeoffs between refbits, | iA| iB | iA| iB ebits, cbits, and other resources can be derived (van Enk, an elementary unit of shared phase reference, has an SIV 2005a, 2006), which emphasizes the utility of thinking of of 1. reference frames as yet another form of resource. Two measures – the entanglement and the SIV – com- pletely quantify the nonlocal resources of a bipartite Now that we have identified “standard” elementary state. To prove this result, the general idea is to show unit(s) of a shared reference frame, we can quantify how that Alice and Bob can, through LOCC restricted by well a given quantum state serves as a shared reference the superselection rule, reversibly convert an asymptotic frame by the state’s asymptotic interconvertibility to this number of copies of the bipartite state into a number of standard form, using local operations and classical com- 24 states with only the first type of resource (entanglement) munication. A remarkable property of entanglement and none of the second (SIV), and a number of states of pure bipartite states is that, by observing the prop- with only the second and none of the first. Let Alice erties of asymptotically reversible transformations using and Bob share N copies of a bipartite state φ , which LOCC, entanglement can be quantified by a single ad- | i has entanglement ESSR(φ) and SIV V (φ). In addition, ditive measure: the reversible conversion efficiency to a let Alice and Bob each have in their possession an arbi- standard form of entanglement, the ebit. For an Abelian trary number of quantum registers – quantum systems superselection rule, the resource of a quantum shared that are not restricted by any superselection rule, such reference frame can be quantified by a single additive as were discussed in Sec. III.C.3; these registers are ini- measure in a similar fashion. Thus, the nonlocal prop- tiated in an arbitrary unentangled state 0˜ 0˜ . Then the erties of pure bipartite quantum states in the presence transformation | i| i of an Abelian superselection rule are completely charac-

terized by two additive measures: the entanglement (for N ESSR(φ)N φ ⊗ 0˜ 0˜ ⊗ which we can use an operational measure such as ESSR, | i ⊗ | i| i E (φ)N the entanglement in the presence of a SSR, discussed in c
n N n ψ˜− ⊗ SSR , (6.25) → n| i| − i ⊗ | i Sec. III.C.3) and another measure quantifying the state’s n ability to serve as a shared RF. In the following, we in-  X  vestigate one such measure for the latter, the superselec- is asymptotically reversible, and Alice and Bob can per- tion induced variance (SIV) (Schuch et al., 2004a,b). We form this transformation with LOCC restricted by the note that measures for quantifying quantum shared RFs SSR, where the coefficients cn are Gaussian-distributed in the presence of non-Abelian superselection rules have with variance NV (φ)/4, and ψ˜− is a maximally- not been explored to date. entangled Bell state of a pair of| qubitsi of Alice’s and Let Alice and Bob each have in their possession a Bob’s quantum registers. number of optical modes, and consider a situation as in Let’s analyze the two states on the right side of Eq. (6.25). The first state, c n N n , has SIV of Sec. III.C.1 where they do not share a phase reference. n n| i| − i Thus, Alice and Bob are restricted by an Abelian local NV (φ). Such a state serves as a good “standard” shared RF for large N (Vaccaro et al.P, 2003). Also, although it is superselection rule for photon number. Let A ( B ) be the local Hilbert space for Alice’s (Bob’s)H modes,H and a non-separable pure state, the entanglement in the pres- ˆ ˆ ence of a SSR, ESSR, of this state is zero. In contrast, the NA (NB) be the total local photon number operator for E (φ)N state of the unrestricted registers ψ˜− ⊗ SSR clearly these modes. Consider a bipartite quantum state φ on | i | i contains an amount of entanglement equal to ESSR(φ)N A B that is an eigenstate of total photon number H ⊗ H standard ebits. As this system is a quantum register, and NˆA + NˆB. (This condition ensures that the state is not correlated with another party’s phase reference.) The su- not a system with a phase degree of freedom, it clearly has no function as a shared phase reference; thus, the SIV of this state is zero. The two states on the right side of Eq. (6.25), then, represent standard forms for each type 24 For an alternate measure of how well a quantum state can serve of nonlocal resource – superselection induced variance, as a shared reference frame, based on entropic properties, see and entanglement in the presence of a SSR – and contain Vaccaro et al. (2005). none of the other type. 51

A proof that the transformation (6.25) is asymptot- Bob’s. If their phase references differ by a phase shift ically reversible with LOCC restricted by the SSR can θBA, then these two operations are related by XB = iθBAZ/2 +iθBAZ/2 be found in Schuch et al. (2004a). In their proof, they e− XAe ; see Sec. V.G. Thus, the state used the entropy of entanglement E rather than ESSR; to which they are purifying in this instance is we note that in the asymptotic limit for an Abelian SSR, N N ESSR(φ⊗ ) E(φ⊗ ) for any pure state φ (Wiseman iθBA → | i ( 1 A 0 B + e− 0 A 1 B)/√2 . (6.27) and Vaccaro, 2003). Thus, their proof applies directly | i | i | i | i to the above statement. This result can be extended to apply to mixed states (Schuch et al., 2004b). Finally, If Alice’s and Bob’s local phase references are uncorre- we note that explicit protocols for activation – creating lated, as we assumed, then θBA is completely unknown, states with E = 0 using states with E = 0 us- and the protocol does not yield a state with higher fi- SSR 6 SSR ing a quantum shared reference frame state – have been delity with ( 1 A 0 B + 0 A 1 B)/√2. | i | i | i | i developed (Bartlett et al., 2006a; Vaccaro et al., 2003).

E. Purification of bounded shared reference frames? F. Treating bounded reference frames as decoherence

As noted in the previous section, the state ( 1 0 + | iA| iB We conclude this section with a discussion of a promis- 0 1 )/√2 can be viewed as the elementary unit of a | iA| iB ing approach to describing the effect of using bounded shared phase reference between Alice and Bob, uncorre- RFs. As demonstrated above, bounded RFs limit one’s lated with any other. This state has the appearance of ability to prepare states and to perform quantum opera- a maximally-entangled Bell state (see Sec. III.C), and so tions and measurements on a system, and the nature of a natural question is to ask whether a number of imper- these limitations is similar in many ways to that of de- fect (noisy) states can be purified to a smaller number of coherence. One is led, then, to ask whether it is possible superior states of this form. Of course, for such a pro- to treat bounded RFs externally rather than internally cess to be of any use, it would need to be implementable (in the sense of Sec. IV.A.2) by positing an unavoidable without the use of some other, unbounded shared RF. decoherence. In other words, if such a description ex- An affirmative answer would mean that shared RFs are isted, then the bounded size of the RF could be said to a resource that can be purified, just like entanglement. effectively reduce the purity and/or coherence of systems Unfortunately, however, such a task does not appear to described with respect to it. be possible, as we now demonstrate, following Preskill (2000). While no completely general description of treating Consider a noisy shared RF state that is a mixture bounded RFs in this manner has yet been developed, of the state ( 1 0 + 0 1 )/√2 with probability specific examples of such decoherence mechanisms and | iA| iB | iA| iB their consequences have been discussed in various rela- p > 1/2 and the state ( 1 0 0 1 )/√2 with A B A B tional approaches to quantum theory (Gambini et al., probability 1 p. Let Alice| i and| i Bob−| sharei | i two copies of 2004a,b; Milburn and Poulin, 2006; Page and Wootters, this mixed state.− With these states, they attempt to per- 1983; Poulin, 2006). These discussions have primarily fo- form the following simple entanglement purification pro- cussed on the tricky issue of internalizing time in quan- tocol (Bennett et al., 1996): they each apply a CNOT on tum theory. Unsurprisingly, given the interpretation of the two qubits in their possession, and then perform an certain types of phase references as clocks, these rela- X measurement on the target qubit. Each party obtains tional formulations generally follow along the lines of the a measurement outcome 1, which they communicate procedures we have already reviewed. One begins by with each other classically,± and compare whether the re- treating all systems which can serve as a clock as inter- sults are the same or different. Effectively, though this nal, constructs (pure or mixed) states that are invariant process, they have measured the joint non-local operator under global time shifts, identifies relational spaces in

(XAXB)1 (XAXB)2 . (6.26) the decoherence-free subsystems, re-factorizes the Hilbert · space in terms of the induced tensor product, and finally In the standard entanglement purification protocol, if Al- interprets the new formulation as the ‘true’ dynamical de- ice and Bob keep only those states where they obtain the scription. As expected, a form of decoherence in this new same measurement results, the resulting states will have description is found whenever the size of the internalized higher fidelity with the state ( 1 A 0 B + 0 A 1 B)/√2. reference system(s) is bounded. It is an interesting open Note, however, that this protocol| i | i requires| i | operationsi problem to identify the appropriate decoherence maps (if which are not U(1)-invariant. For example, a mea- they exist) that describe the dynamics of a system rela- surement of X must be performed relative to that tive to a bounded (particularly non-Abelian) RF. While party’s local phase reference. Let Alice and Bob make one can debate the appropriateness of these approaches use of unbounded local phase references in this proto- from a foundational perspective, such an approach would col. Note that the operator XA is defined with re- certainly be useful for addressing questions in the field of spect to Alice’s phase reference, and XB with respect to quantum information. 52

VII. OUTLOOK key (Sec. III.D), the private communication of unspeak- able information (Sec. V.J), and secret sharing of un- The study of reference frames and superselection rules speakable information. Many more analogies of this sort in the context of quantum information theory is an unfin- could be considered. Indeed, for almost any information- ished task. In this section, we provide an overview of the theoretic task of interest today, it is interesting to muse topics we have discussed together with some open prob- about possible analogues of it for unspeakable informa- lems and research directions, while outlining the practical tion. (A particularly intriguing question to consider is and foundational significance of this sort of investigation. whether there is such an analogue for computation.) On It is useful to divide the practical applications into two the experimental side, the implementation of quantum broad categories corresponding to whether their purpose protocols for even the best-studied of these sorts of tasks, is the manipulation of speakable information or of un- the alignment of reference frames, has, with the exception speakable information, that is, corresponding to the na- of phase estimation, only just begun to be investigated. ture of their inputs and outputs. As emphasized in the introduction, imposing a restric- The first category contains the standard problems of tion on operations generically leads to the identification interest in quantum information theory, both those that of a novel resource to overcome this restriction, and we use quantum systems to manipulate classical informa- are then compelled to develop a theory for how that re- tion, and those whose inputs and outputs are themselves source may be manipulated. For instance, under the re- quantum information. Even though these ultimately pro- striction of LOCC, entanglement becomes a resource, and cess speakable information (whether classical or quan- the theory of how this resource can be manipulated — tum), protocols for such tasks must always encode this the theory of entanglement — has been the subject of a information using some degree of freedom, which re- significant amount of work in recent years. Others have quires some form of RF. Thus we are led to ask how considered the theory of communication under natural much the absence of a particular RF or of a shared RF restrictions such as local operations and public communi- among separated parties decreases the efficiency of vari- cation (Collins and Popescu, 2002) or restrictions to only ous information-processing tasks, or increases the practi- Gaussian quantum-optical states and operations (Eisert cal difficulty of implementing them. What is the answer and Plenio, 2003). A superselection rule (either local and to such questions in the case where one has an imprecise global) is another sort of natural restriction, and under RF or two parties share RFs that are only partially cor- this restriction, any quantum state that acts as an RF related? Such questions have been considered here for a becomes a resource. The theory of such resources might variety of tasks, such as quantum and classical communi- aptly be called the theory of quantum reference frames cation (Sec. III.A), quantum key distribution (Sec. III.B), or the theory of unspeakable quantum information. It en- and implementing quantum gates (Sec. VI.B.1). There deavors to answer questions such as how this resource is are many more tasks that could be considered. Also, depleted with use, transformed from one form to another, most of the communication and cryptographic problems shared among several parties, etcetera. Such a theory considered to date have determined the efficiency only in has only begun to be developed. The limited results on the case where one demands perfect fidelity encoding and bounded quantum reference frames, described in Sec. VI decoding and perfect security. Furthermore, these sorts (see also Sec. III.C) are evidence of this. Moreover, in a of questions have been scarcely addressed for the case of sense there is a family of theories to be developed, be- shared RF that are partially correlated. Finally, although cause we obtain different results depending on the group there have been a few experiments demonstrating the vi- G with which the superselection rule is associated. Many ability of some of these schemes, such as relational encod- investigations to date have applied only to the cases of ings (Sec. III.A.3), the development of realistic physical the group U(1) and/or the group SU(2). Ultimately, one implementations remains as much a source of experimen- would like to have a generic theory of unspeakable in- tal challenges as any other quantum technology. formation that applies to any group; in particular, non- The second category of applications consists of tasks compact groups (such as the Lorentz or Poincar´egroups) that explicitly involve the manipulation of unspeakable may require more general mathematical tools than those information, such as clock synchronization or the align- discussed here. ment of Cartesian frames. Quantum considerations be- It is worth noting that this research is not necessarily come important to achieve the optimal precision and driven by applications. In this sense, it is similar to the it is the tools of quantum information theory that are study of entanglement in quantum information theory, best suited to a treatment of the problem. We may which although initially motivated by its practical appli- describe the alignment of remote reference frames as cations, has increasingly become an interesting subject in the communication of unspeakable information, and as its own right. Of course, just as the development of the soon as one starts describing and thinking about such theory of entanglement has led to many unforeseen prac- tasks in the language of information theory, many new tical dividends, we may expect this of a general theory tasks suggest themselves. Examples mentioned in this of unspeakable information as well. review are: dense coding of unspeakable information Finally, applying the tools of quantum information the- (Sec. V.K), using private shared RFs as a cryptographic ory to the study of RFs and SSRs can shed light on foun- 53 dational issues in quantum theory. Examples from this Bagan, E., S. Iblisdir, and R. Munoz-Tapia, 2006a, Phys. 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