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A. Overview of main results A. Canonical reference frame states

In section II we introduce the primitive quantum ref- In what follows we shall assume that the QRF is in a pure state QRF . The system spin states are la- erence frames and use the symmetries of G-twirling to | i restrict to a canonical set of states. belled with respect to a defining, background CRF so that J = +1/2 0 and J = 1/2 1 and we In section III we determine the optimal primitive refer- | z i ≡ | i | z − i ≡ | i order the three spins so that = QRF system. ence frame that preserves a single spin state and find that For any single spin unitary UH, andH any composite⊗ H state the best one can ever do is a 26% reduction of the volume 3 3 ρ on we can use [ρ]= [U ⊗ ρ(U †)⊗ ] to deduce that of state space, for orthogonal reference frame spins with H G G

2 2 2 2

½ ½ ½ entropy of entanglement E =0.19. [U ⊗ ½ρ(U †)⊗ ] = [ ⊗ U †ρ ⊗ U](1) G ⊗ ⊗ G ⊗ ⊗ Section IV then deals with the operational criterion of and so a rigid rotation of the QRF is equivalent to an entanglement preservation, in which two distinct cases application of the inverse rotation to the system of in- arise: either Alice lacks a directional frame or both Alice terest, and does not affect the issue of irreversible loss and Bob lack a directional reference frame. We estab- of . This equivalence under the two lish a useful theorem for the negativity of a bipartite different transformations is simply the equivalence be- state and analyse the loss in entanglementN in terms of tween active and passive transformations in physics [1]. a sub-normalized quantum operation on one half of the In particular this means that, without loss of generality, entangled state. Surprisingly, we find that the optimal we may restrict ourselves to a subset of canonical QRF alignment of the product reference frame spins is at an states. angle of 87 degrees, which preserves 23.6% of the nega- We can thus work in an external frame in which the tivity, and by adding some entanglement to the reference Schmidt decomposition of the canonical QRF state is pa- frame this can be raised to 24.4%. We also find that it is rameterized as possible to obtain greater negativity if we use a less than maximally entangled state. For the case of both Alice β β QRF (α,β,δ) = cos α 0 (cos 0 + eiδ sin 1 ) and Bob lacking a reference frame we find that the most | i | i⊗ 2 | i 2 | i entanglement one can preserve with primitive reference β iδ β + sin α 1 (sin 0 e− cos 1 ). (2) frames is = 7%, with identical frames for Alice and | i⊗ 2 | i− 2 | i Bob, butN now with reference frame spins aligned at an angle of 83 degrees. III. OPTIMAL FOR A SINGLE SPIN STATE In section V we consider how increasing the spin from the primitive reference frame scale to the classical limit affects matters. We show how classical vector addition The G-twirling of 3 spins under rotations, results in a emerges sharply in the large-L limit, analyse how well single protected subsystem qubit, and in what follows we the spin-L reference does at preserving entanglement and shall consider the encoding of a single spin state into this give evidence that suggests that 90 degrees is asymptot- protected subsystem. The technical details of G-twirling ically optimal [15]. are contained in appendices A and B, while appendix C describes the spin observables that allow access to the We conclude with section VI and provide some techni- virtual subsystem degrees of freedom. cal details on G-twirling in the appendix. The key property of the G-twirling map on the 3 spin system is that it splits up into a fully decohering map on the 4-dimensional subspace corresponding to a total angular J =3/2, and a partially decohering II. THE PRIMITIVE QUANTUM REFERENCE map on the orthogonal J = 1/2 subspace. The J = FRAME 1/2 subspace can in turn be split into two virtual qubit systems . More explicitly, we have M2 ⊗ N2 We look at the simplest possible quantum reference [ρ] = 1[Π1ρΠ1]+ 2[Π2ρΠ2] (3) frame for specifying spatial directions in 3 dimensions - G D D two non-aligned spin-1/2 particles. We refer to any QRF where Π1 is the projector onto the J =3/2 subspace, and consisting of two spin-1/2 particles simply as a primi- Π2 is the projector onto the J =1/2 subspace. The op- tive quantum reference frame. Our task is to determine eration is fully decohering on the support of Π , while D1 1 the optimal primitive quantum reference frame for both 2 = 2 ½ 2 , which signifies that it decoheres fully state preservation and entanglement preservation. The onD a 2-dimensionalDM ⊗ N virtual subsystem while leaving M2 absence of a CRF is described mathematically by the ac- the 2-dimensional virtual subsystem 2 unaffected (see tion of G-twirling, which involves an averaging of a quan- appendix B for more details). N tum state over the rotation group. The technicalities of We adopt the convention of (α,β,δ) for the QRF pa- G-twirling as they relate to this work are described in rameters as given in (2), and use single spin state param- the appendix. eters (θ, φ), in the state θ, φ = cos θ 0 + eiφ sin θ 1 . | i | i | i 3

The encoding of the spin state into the protected sub- For two aligned spins we have β = 0 and the Bloch 2 2 system defines a quantum channel from the physical sys- vector images get set to (0, 0, 3 sin θ) and so is a very tem into a virtual subsystem. The quality of the QRF poor image of the Bloch sphere.− Similarly for anti- at protecting quantum information is then equivalent to aligned spins, β = π, we get ( 1 cos2θ, 0, 1 ) and so − √3 3 how noisy this resultant quantum channel is, and as such is also a very poor image of the Bloch sphere; both are we do not worry about unitary rotations in the encoding 1-dimensional images of the sphere. of the state. We also note that the reduction in purity As already mentioned, we can view the action of the G- under the encoding can vary greatly with the direction twirling as defining a single qubit quantum channel. The along which the spin is polarized. To eliminate scenar- mapping induced by the G-twirling can then be written ios in which the state is strongly dephased (e.g. ones as an affine map of the state space x A(α,β,δ)x + that preserve only classical information) we should re- b(α,β,δ), composed of a linear transformation→ followed quire that the encoding of the quantum state does not by a global translation along a fixed direction. The vol- decohere strongly along any one direction. ume distortion factor is obtained from the determinant In light of this, we use the volume of the image of of the linear transformation and for a canonical product the state space under the mapping induced by the G- 2 2 state QRF is found to equal detA = 9 sin β, which has twirling as a suitable measure of how well the QRF pre- a peak for βopt = 90 degrees. Thus, the best product serves a general spin state. A large volume for the image QRF for the preservation of the single spin logical state of the Bloch sphere implies a high average purity for all is obtained when we orient the two spins orthogonally, spin polarizations, and conversely an encoding that pre- for which the Bloch sphere is shrunk by about 22%. The serves a high level of purity for all polarizations will result image of the Bloch sphere becomes in a large image volume. Furthermore, since the trace distance coincides with the Euclidean distance between 1 ( cos2θ + cos(δ φ) sin 2θ) 2√3 − − Bloch vectors [10], we will simply use the Euclidean vol- R(θ, φ) = 1 sin 2θ sin(δ φ) . √3 ume element as our measure. Given an image volume V˜  1 −  6 (cos2θ + cos(δ φ) sin 2θ) 3V˜ 1/3 − we can define a characteristic distance r = ( 4π ) 1   for the encoding, which gives a measure for the average≤ Note that the arbitrary phase angle on the QRF acts distinguishability of two orthogonal states under the en- simply as a linear translation of the phase of the encoded coding. state. We first consider a canonical QRF that is in a product For the most general QRF in which we allow the two state spins to be entangled, it is possible to once again calculate the volume distortion of the state space. In this case β β QRF = 0 (cos 0 + eiδ sin 1 ) (4) 2 | i | i⊗ 2 | i 2 | i detA(α, β) = (cos α sin α)4(cos α + sin α)2 sin2 β 9 − with the full state Ψ = QRF θ, φ then subject where for simplicity we have set δ = 0 as this turns out to to G-twirling. Only| thei reduced| i state ⊗ | oni the protected be optimal. We now have a maximum volume for β = virtual subsystem is unaffected, and we may obtain the opt √ Bloch vector in the protected virtual subsystem by sim- 90 degrees and αopt = arctan(2 2 3). This corresponds to an entropy of entanglement [8]− of E[ QRF QRF ]= ply computing the qubit state Tr 2 [Π2 Ψ Ψ Π2]. | ih | The bases for the J =3/2 andMJ =1/| 2ih are| J,s,p , 0.19 for the QRF. and are given explicitly in appendix B. We{| definei} a Thus, the best possible primitive quantum reference frame for the encoding a single spin state shrinks the virtual set of two qubits for the J = 1/2 sector as 64 ¯ ¯ 1 Bloch Ball by a factor 243 26%, or roughly to a radius i j := 2 ,i,j . In this notation, the second of the ≈ two| i ⊗ virtual | i qubits| i is the protected virtual subsystem in of 0.64, and shows that an orthogonal frame is best, but which the quantum state is stored. a little bit of entanglement also helps. To determine the protected qubit state we first project into the J =1/2 sector and then trace out the first virtual IV. OPTIMAL FOR SPIN ENTANGLEMENT qubit. Upon doing this for the uncorrelated QRF (4), we find that the Bloch vector of the protected state is given We now determine the best primitive QRF for the by R = (Rx, Ry, Rz) where preservation of entanglement. As mentioned before we have a global freedom to rigidly rotate the QRF in space. 1 2 β Rx = ( 2cos2θ sin + cos(δ φ) sin β sin 2θ) We consider a composite state Ψ = QRF ϕ 2√3 − 2 − A QRF , where ϕ is a maximally| i entangled| i ⊗state | i⊗ of 1 | Bi | i Ry = sin β sin 2θ sin(δ φ) two spin-1/2 particles, however rotation of the two local −2√3 − QRFs is equivalent under G-twirling to the application of 1 R = (cos β( 2+cos2θ) rotation of the two spins that make up ϕ . Since every z 6 − such pure maximally entangled state is locally| i equivalent +cos2θ + cos(δ φ) sin β sin 2θ). (5) to the singlet state, we can take ϕ = ψ− and work − | i | i 4 with a general QRF, or alternatively we can fix QRF state the projection probabilities are given by | Ai and QRFB to be canonical QRF states and leave ϕ unspecified,| i but maximally entangled. We opt for the| i 1 1 p1,2 = (cos β latter. 2 ± 6 As the measure of bipartite entanglement, we use neg- 2cos α cos β cos δ sin α + cos δ sin 2α). (9) 1 TB − ativity [11], which is defined as [ρAB ] := 2 ( ρAB 1 1) N || || − We wish to maximize the negativity of the full state where M 1 := Tr√M †M and TB denotes partial trans- || || [Ψ] across A and B. Since negativity is unitarily in- pose with respect to B. While the negativity is usually G TB variant we attack the problem in the virtual basis, and convenient to calculate, the spectrum of ρAB does not admit an analytic expression for the most general QRF, establish the following lemma and theorem, which are and so for such cases numerics will be needed. useful for our analysis. There are two distinct situations of interest. The first Lemma IV.1. Given hermitian operators and is where either Alice or Bob does not share a classical Ai , with each pair in having mutually{ orthogo-} reference frame with Eve; the second is where neither Bi Ai {nal} support, we have that{ } A B = A B , Alice nor Bob share a CRF with Eve. Since the latter | i i ⊗ i| i | i| ⊗ | i| where M := √M M for any operator M. is obtained by two independent G-twirlings for Alice and | | † P P Bob (see appendix D), we shall first consider the situation where Alice lacks the appropriate CRF. Moreover, this is Proof. By spectral decomposition, any hermitian opera- tor M can be written as M = M+ M , where M closest to our main aim of determining the best primitive − − ± QRF, since this situation tests the quality of a single are positive operators with orthogonal support, and thus M = M+ + M . In particular, QRF, as opposed to the joint functioning of two such | | − reference frames. Ai Bi = (Ai,+ Bi,+ + Ai, Bi, ) ⊗ ⊗ − ⊗ − (Ai,+ Bi, + Ai, Bi,+). (10) A. Twirling Alice − ⊗ − − ⊗ However, Ai, Bi, is orthogonal to Ak, Bk, for ± ⊗ ± ∓ ⊗ ± We consider the state consisting of a canonical QRF i = k by definition of the decomposition into positive operators, and orthogonal for i = k by assumption on the for Alice and a maximally entangled two spin state. The 6 set Ai . Hence, we have X := Ai Bi = X+ X four spins are in the state { } i ⊗ − − where X+ = i, (Ai, Bi, ) and X = i, (Ai, ± ± ⊗ ± P − ± ∓ ⊗ Ψ = QRF (α,β,δ) ϕ . (6) Bi, ), and so X = Ai Bi = Ai Bi . | i | i ⊗ | i ± P| | | i ⊗ | i | |P ⊗ | | G-twirling Alice involves Ψ Ψ Ψ transforming as Theorem IV.2. P P ≡ | ih | For any bipartite state ρAB = piϕA,i σAB,i with orthogonal states ϕA,i , the neg- Ψ [Ψ] = 1[Π1ΨΠ1]+ 2[Π2ΨΠ2] (7) i → G D D ativity of the⊗ state is given by [ρ ]={ p} [σ ]. N AB i iN AB,i which acts only on the first three spins. P Proof. The negativity is defined as P (ρ ) := Since measures of entanglement are unitarily invariant, N AB 1 ( ρTB 1), where X := Tr X and T denotes we define a unitary U from the physical basis for the three 2 || AB||1 − || ||1 | | B spins to a virtual one, consisting of orthonormal basis partial transpose on B. Since ρTB = p ϕ σTB , AB i i A,i ⊗ AB,i vectors for the two sectors 1 and 2. We then analyse it follows from the previous lemma and basic trace prop- H H 1 P TB 1 the two (unnormalized) states p1ρ1 = U 1[Π1ΨΠ1]U † erties that (ρAB) = 2 i piTr(ϕA,i)Tr σAB,i 2 = and p ρ = U [Π ΨΠ ]U , where we projectD into the N | | − 2 2 2 2 2 † i pi (σAB,i). two sectors ofD the first 3 spins and transform to the vir- N P tual basis. In the virtual basis we use the states 0¯ and PDue to the orthogonality of the different local sectors 1¯ of the first spin to label the two orthogonal sectors.| i | i for A, the above theorem tells us that our task of maxi- The action of the G-twirling on the first three spins mizing the negativity of the full state is reduced to max- with probability p1, results in the first ‘sector spin’ being (2) imizing [ρAB] = p2 [σ ] over the virtual two spin projected into the 0¯ state and spins 2 and 3 in being N N 34 | i output states σ(2). fully depolarized, and with probability p2, the sector spin 34 being projected into 1¯ and spin 2 fully depolarized. This leaves us with | i B. Induced quantum operation on a single spin

1 (1) ½ ¯ ¯ ½ ρ1 = 0 0 2 3 σ4 4| ih |⊗ ⊗ ⊗ From the analysis in the previous section we estab- 1 (2)

¯ ¯ ½ ρ2 = 1 1 2 σ34 (8) lished that it is sufficient to consider the induced selec- 2| ih |⊗ ⊗ tive quantum operation on the one half of the maximally (2) (subscripts on the right-hand side label the qubits) and entangled state ϕ , that takes ϕ p2σ34 . This oper- ρ = [Ψ] = ρ = p ρ +p ρ . We find that for the singlet ation depends on| thei specific QRF→ state QRF (α,β,δ) G 1 1 2 2 | i 5 involved in the G-twirling and can be described by the αopt = 0.15) and the spins aligned at an angle 82 de- linear completely positive map grees! Interestingly,− this QRF is quite similar to the ana- lytic optimal one we obtained for the quite different task A[ ϕ ϕ ] = M1 ϕ ϕ M1† + M2 ϕ ϕ M2† of preserving a logical state. Furthermore, both do ap- E | ih | | ih | | ih | proximately 25% as well as the classical limit, with the where M1,2 act non-trivially only on the first spin. The addition of entanglement providing a similar benefit in operators M1,2 can be found by projecting into the J = each case. 1/2 sector and then tracing out the virtual subsystem It is also of interest to consider partially entangled . For the canonical primitive QRF state (2) they are states ϕ . If we use the entangled state ϕ = cos γ 01 givenM by sin γ 10| ,i instead of a maximally entangled| i state we| findi− that| thei optimal alignment angle obeys 1 (eiδ cos α sin α) sin β 0 √2 2 π M1 = − lim βopt = (14) 1 iδ β 2 β γ 0 2 (e cos α + sin α) sin cos α cos ! → − √6 2 3 2 however, more unusual is that we can preserve more out- 0 1 (eiδ cos α sin α) sin β q M = √2 − 2 . (11) put entanglement by using a less than maximally entan- 2 0 1 (eiδ cos α + sin α) sin β √6 2 ! gled input state. A careful numerical analysis finds that the most entanglement that can be preserved with any We have fixed the QRF state to be canonical, and can QRF and any entangled two spin state is with βopt = 82 now consider a maximally entangled spin state shared degrees, δopt = 0, αopt = 0.2 and γopt = 38 degrees. For between A and B which, in general, can be written as this we get = 25.2%. Here− we note that the negativity N ϕ = ½ U ψ− , for some unitary U. The action of varies slowly for angles between about 80 and 90, while | i ⊗ | i the quantum operation on this state is then A[ ϕ ϕ ]= the addition of entanglement to the QRF contributes to

E | ih |

½ ½ ½ (M ½) ϕ ϕ (M † ) = ( U) [ ψ ψ ]( U ) a much larger variation for . The unusual parameters i i i A − − † N and from⊗ unitary| ih | invariance⊗ of⊗ negativityE | ih we| conclude⊗ for the optimal QRF in each case arise from the finite- thatP all maximally entangled spin states ϕ will suffer sized effects within the full Hilbert space, which occur exactly the same loss in negativity. | i once we impose the constraint that the QRF must be a (2) separate, disentangled, system. For such small systems It is then a simple matter to compute p2σ = 34 the states of the protected virtual subsystem only have A[ ψ− ψ− ] and analyse its negativity. In particular, forE | theih case| of a canonical product state QRF, the state partial overlap with those accessible product states. 2 σ34 is rank 2, and (for 0 β π) has eigenvalues β ≤ ≤ 1 cos 2 λ1,2 = 2 cos β 3 with corresponding eigenstates C. Twirling Alice and Twirling Bob ± − β β e1 = ( √3cot , √3, tan , 1) The tools acquired in the previous section can be di- | i − 4 − − 4 rectly applied to the situation where neither Alice nor β β Bob has access to the reference frame of the prepared e2 = (√3 tan , √3, cot , 1) (12) | i 4 − 2 maximally entangled spin state. As before, we restrict to primitive quantum reference frames QRFA and QRFB in the virtual basis. for Alice and Bob respectively. For β = π/2 the negativity of the bipartite state may The composite system is composed of six spins, in the be found analytically and is [ρ] = 1 23.57%, and N 3√2 ≈ pure state Ψ = QRFA ϕ QRFB . However, the furthermore, to good approximation we have that independent| G-twirlingi | ati ⊗ A | andi ⊗ | B transformsi the state as Ψ [ [Ψ]] [Ψ] where 1 → GB GA ≡ G (β,γ) sin β sin 2γ (13) N ≈ 3√2| | [Ψ] = [ [Π ΨΠ ]] (15) G DB,i DA,j i,j i,j i,j for an entangled state ϕ = cos γ 01 sin γ 10 . Sur- X prisingly however, it turns| i out that| ai− state of| orthogo-i where we again label local sectors as either 1 or 2, and define Π Π Π . nal spins is not the optimal product state QRF. It can i,j ≡ B,i A,j be shown that one can do slightly better, and obtain We transform to the two local virtual bases, and use [ρ]=23.60% for β = 87 degrees. the orthogonality of the spin states labelling the sectors N opt (2,2) As in the previous section, the addition of a small bit of to deduce that [ [Ψ]] = p2,2σ3,4 . For identical prod- N G 1 2 entanglement increases the performance of the QRF. Nu- uct reference frames we have p2,2 = 18 (7 cos β) sin β/2 merics on the full set of quantum reference frames show (2,2) − and σ3,4 is rank 2. We obtain this state more directly that the optimal primitive reference frame for preserv- via an application of the selective operations derived ing entanglement in a maximally entangled spin state above, in other words achieves [ρ] = 24.4% for the QRF in the state with δ = 0, anN entropy of entanglement of E = 0.167 (for p σ(2,2) = [ [ ϕ ϕ ]] (16) 2,2 3,4 EB EA | ih | 6 with corresponding single spin Kraus matrices 0.12 M A,M B given by the local QRFs as in (11). It { i j } is then straightfoward to determine, for example, that 0.1 in the case of product state QRFs for both Alice and Bob, the largest negativity preserved under the two 0.08 G-twirlings is only = 7% for a QRF with spins aligned N 0.06 at βA = βB = 83 degrees. 0.04 Scaled Probability V. THE CLASSICAL LIMIT: SPIN-L 0.02 QUANTUM REFERENCE FRAMES 0 0 0.5 1 1.5 2 We have obtained optimal angles close to 90 degrees J/L for primitive QRFs, and so it is natural to ask what hap- pens to these angles in the classical limit. To do this we FIG. 1: [Color online] Quantum Pythagoras’ theorem: consider a QRF composed of two spin-L particles and getting √2 with spins up to L = 25. The vertical axis is a analyse how β varies with L for a singlet state input. rescaled probability to account for the increase in data points opt as we increase L. The horizontal axis is J/L where J is The sector structure at A now becomes ( 2L 2L 1 H ⊕ H − ⊕ the measured total angular momentum of the two orthogo- 0) 1/2 = 2L+1/2 2 2L 1/2 2 1/2, nal spins. and···⊕H we get⊗ 2HL virtualH protected⊕ H qubits− on⊕···⊕ 2L differentH sectors. As before, we unitarily transform A to a local ‘sector’ spin ‘√2’ is always a rational number, which would have plus ‘junk’ system and a protected qubit system. The pleased the ancient Greeks. Classicality emerges through negativity of the resultant state (assuming singlet be- the distribution over the different sectors, and the value tween A and B) is then given by for the negativity of the twirled state is dominated by states σkc where k labels a sector with total angular k c [ [ QRF (α,β,δ) ψ− ]] = p [σ ] (17) N G | i ⊗ | i kN momentum near to the classical value. Xk For any angle of inclination β for the QRF spins, we can construct the corresponding state where k ranges over the 2L sectors containing protected qubits, σk is the reduced state on the two spin subsystem, QRF = Jz = L Jnˆ(β) = L (18) and pk is the projection probability of obtaining sector | i | i ⊗ | i k. where J = L is a coherent spin state polarized at an We may once again obtain a reference frame depen- | nˆ(β) i k k angle β to the Z axis, and then compute the negativity dent quantum operation A : ρ M ρM † that E 7→ ik ,k ik ik of [ QRF ψ− ]. Intuitively we expect the preserved describes the action of the G-twirling purely on Alice’s negativityG | i to ⊗ increase | i with L for optimal angles at each P k spin and gives the required ensemble terms pkσ = value of L. k k M ψ− ψ− M † for the negativity. ik ik | ih | ik In figure 2 we have computed the optimal angles P βopt(L), for a QRF composed of two spin L particles, that preserve the most negativity in the singlet state. The nu- A. Sectors for the Quantum Reference Frame merical algorithm used is as follows: for a fixed L, we first construct Alice’s unitary transformation UA(L) from the To construct the unitary to go from the physical basis physical basis of the spin system = H HL ⊗ HL ⊗ H1/2 states to the k sector states we make use of the Clebsch- to the virtual basis s,j,p where s is the sector la- H {| i} Gordan coefficients [12, 13] L,L; m1,m2 J, M to couple bel, j labels the ‘junk’ degrees of freedom affected by the h | i the two spin-L particles in the QRF. G-twirling, while p labels the degrees of freedom of the In this setting it is illuminating to see how for large protected virtual subsystems. For any fixed angle β, we

L the state of two orthogonal spins J = L J = construct QRF (β) ψ and apply U (L) ½ | z i ⊗ | x A − AB A B L is distributed over the different sectors. Classically to the full| state. Thei negativity⊗ | i after G-twirling is⊗ then onei expects the sharp vector addition law where the two obtained by projecting onto sectors, tracing out onto the orthogonal vectors add to one of length √2L. Numerics protected qubit systems and using (17) to find the total show that as L increases the distribution of J = L preserved negativity between A and B. | z i⊗ Jx = L is sharply peaked on the sector with total J We find that the function β (L)risesto a peakof 96.5 | i opt value closest to √2L as one would expect (see figure 1). degrees for L = 3 before starting a slow decline. For large For example, with L = 17 it peaks on sector J = 24 values of L we find that the function (β,L = constant) with 24/17 1.412 being a good approximation to √2 flattens out over the interval 0 to 180N degrees and van- 1.414. By increasing≈ L we probabilistically recover the≈ ishes at the two end-points where the QRF becomes de- standard vector addition, although for any finite-sized generate. The maximal preservation of negativity in- 7

98 ysed how well the most primitive such frame can ever perform under certain information preserving criteria, 96 and have found a rough concordance as to the properties of the optimal primitive reference frame. The optimal 94 frames involve spins roughly orthogonal (between 82 and

90 degrees), and with a small degree of entanglement (an

opt 92 β entropy of entanglement of about 0.15). The actual per- 90 formance of a single optimal reference frame in each case is roughly 25% of the infinite-limit classical frame. We 88 also studied the classical limit of such a quantum refer- 1 86 0 5 10 15 L 0.8 FIG. 2: The classical limit: The optimal angle of inclina- tion βopt for the two spins in the QRF as a function of L. A peak occurs of βopt = 96.5 degrees at L = 3. Numeri- 0.6

cal analysis for large L implies that (β) approaches a step Negativity function with zeros at β = 0 and β =N 180 degrees, and that 0.4 asymptotically βopt 90 degrees. →

0.2 creases rapidly with L, reaching 90% already by L =6.5, 0 5 10 15 L see figure 3. Some intuition as to why small values of L have β > opt FIG. 3: Negativity in the classical limit: Preservation 90 degrees can be gained from figure 1. For small values of negativity increases rapidly with increasing spin L. States of L the probability distribution is asymmetric and has σkc from sectors with angular momentum close to the classical a relatively large weight on the sector with total angular value dominate the negativity. momentum 2L. This sector contains no protected sub- system, and so it is clearly beneficial to have an angle of inclination slightly above 90 to reduce the contribu- tion from a sector that can preserve no negativity. Of ence frame and found that the usual vector addition law course there is another competing aspect. Having the emerges gradually, and that a large amount of negativity spins strongly anti-aligned results in virtual qubit states can be preserved even for modest-sized spins. The op- σk with poor negativity. Note that the L = 1/2 case is timal angle of inclination as a function of L displays an in fact the only one with βopt < 90 degrees, and so in unusual peak at L = 3, before slowly decreasing towards this case it is not merely a matter of avoiding the sec- 90 degrees, where finite-size effects are washed out and tor of largest L, but there is also a strong dependence of the reference frame no longeer degrades the spin entan- the negativity for each σk on the QRF spin alignment. glement. In light of these results, it would be of interest in future work to study this emergence of classicality, and to fur- ther analyse the competing mechanisms at work for small systems. Acknowledgments VI. CONCLUSIONS DJ acknowledges the support of EPSRC, and thanks In this work we have considered the fundamental limits Terry Rudolph and Yeong-Cherng Liang for helpful dis- of finite-sized quantum reference frames. We have anal- cussions.

[1] R. M. Wald, General Relativity, The University of [4] S. Bartlett, T. Rudolph, R. Spekkens, and P. Turner, Chicago Press, (1984). New J. Phys. 11 063013 (2009). [2] S. Bartlett, T. Rudolph, and R. Spekkens, Rev. Mod. [5] M. Ahmadi, D. Jennings and T. Rudolph, Phys. Rev. A Phys. 79, 555 (2007). 82, 032320 (2010) [3] F. Costa, N. Harrigan, T. Rudolph, and C. Brukner, New [6] D. Poulin, and J. Yard, New. J. Phys. 9, 156 (2007). J. Phys. 11, 123007 (2009). [7] S. D. Bartlett, T. Rudolph, and R. W. Spekkens, Phys. 8

Rev. A. 70, 032321 (2004). cles in total. We refer to the application ρ [ρ] of this [8] S. Popescu, and D. Rohrlich, Phys. Rev. A 56, R3319- group averaging as ‘G-twirling’. → G R3321 (1997). [9] T. Frankel, The Geometry of Physics, Cambridge Uni- versity Press, 2nd edition (2004). [10] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, (2000). [11] G. Vidal and R. F. Werner, Phys. Rev. A 65 032314 (2002) Appendix B: Protected Subsystems [12] M. E. Rose, Elementary Theory of Angular Momentum, (Wiley, New York, 1957). [13] A. R. Edmonds, Angular Momentum in Quantum Me- The tensor product representation U(Ω) is reducible, chanics, Princeton University, Princeton, New Jersey, and the full Hilbert space = can be split (1957). H H1 ⊗···⊗HN [14] H. Weyl, The Classical Groups. Their Invariants and into irreps of the rotational symmetry group. Specifically, Representations, Princeton University Press, (1939). the Schur-Weyl duality [14] tells us that the full Hilbert space splits into a sum of multiplicity-free irreps of the [15] Although of course in this limit any linearly independent H frame of spins preserves entanglement perfectly. group SU(2) N , where N is the discrete permutation group of N elements.×S In otherS words, we have that = ( ) where is the subspace sectorH ⊕q Mq ⊗Nq Mq ⊗Nq ≡ Hq Appendix A: A lack of a classical reference frame in which SU(2) acts trivially on q and irreducibly on N q, while N acts trivially on q and irreducibly on M. S M Here we describe how G-twirling arises when there is Nq not a shared directional reference frame. We can imag- We shall assume that Eve shares an ordering reference ine an agent Eve, preparing a system of spin particles frame with both Alice and Bob, or at the least, she main- that make up a quantum system in a state Ψ . This | i tains the same ordering of systems when sending multiple pure quantum system is prepared in conjunction with a copies of the same state. ‘classical reference frame’ CRFE which sharply specifies directions in space. We could regard CRFE as a large Since the subsystems undergo a unitary channel, q spin-j system in a highly coherent state, however here are called protected or decoherence free virtual subsys-N we simply take it as an abstract background setting and tems and can hold information that is not erased by ro- place it on the classical side of the Heisenberg cut. tations. Eve then sends the system to Alice, but unfortunately Alice does not share a private reference frame with Eve. It is found that 2 spins can protect 1 classical bit, while Her local axes are related to Eve’s by some unknown ro- we need a minimum of 3 spins to encode a single virtual tation and so Alice must average the state uniformly over spin subsystem. This follows from the angular momen- all spatial rotations. The processed state represents the tum addition = = H H1/2 ⊗ H1/2 ⊗ H1/2 M3/2 ⊗ N3/2 ⊕ updated knowledge of the randomly rotated state, or the 1/2 1/2 with dim( 3/2) =4, dim( 3/2) = 1, and state from Alice’s point of view, where she lacks knowl- Mdim( ⊗ N) = dim( )M = 2. N M1/2 N1/2 edge of how her CRFA is related to Eve’s CRFE . In the case of multiple copies, instead of a single-shot procedure, This decomposition into irreps of SU(2) S3 can be × Eve sends multiple copies of the same state to Alice, but described in an orthonormal basis for of the form H if the systems are sent identically then the same unitary λ,s,p where λ = 3/2, 1/2 is a symmetry class label {| i} rotation is applied to each system then we have a per- and corresponds to the different total angular momentum fect channel, where no averaging takes place. However sectors, while the remaining labels s and p correspond to if the systems are independently and randomly rotated the action of the rotation group and permutation group around, then Alice must once again use an average over respectively. In terms of computational spin bases, we all rotations. These two perspectives correspond to the have for the J =3/2 sector bayesian and frequentist views of quantum states. In both cases the averaged state becomes 3 , 0, 0 = 000 [ Ψ Ψ ] dΩU(Ω) Ψ Ψ U †(Ω) (A1) |2 i | i G | ih | ≡ | ih | 3 1 Z , 1, 0 = ( 001 + 010 + 100 ) where we use the Haar measure over the set of unitaries |2 i √3 | i | i | i U(Ω) = U (Ω) U (Ω) induced by the spatial rota- 3 1 1 ⊗ · · · N , 2, 0 = ( 110 + 101 + 011 ) tion R(Ω) and integrate over all rotations in SO(3). Each |2 i √3 | i | i | i subsystem k in = 1 N transforms under the 3 H H H ⊗···H , 3, 0 = 111 (B1) rotation via an irrep Uk(Ω) of SU(2) - N spin-1/2 parti- |2 i | i 9 while for the J =1/2 sector The observables Ni are rotationally invariant, but not invariant under permutations, as expected. 1 1 , 0, 0 = ( 010 100 ) Using the “order 2 with order 2” operator identity |2 i √2 | i − | i 1 1 , 0, 1 = (2 001 010 100 ) [J J , J J ]= i(J J ) J (C5) |2 i √6 | i − | i − | i a · c b · c a × b · c 1 1 , 1, 0 = ( 011 101 ) |2 i √2 | i − | i together with the “order 2 with order 3” identity 1 1 , 1, 1 = ( 2 110 + 101 + 011 ) (B2) |2 i √6 − | i | i | i i [J J , (J J ) J ] = J (J J ) (C6) b · c a × b · c 2 a · c − b Appendix C: Accessing the protected qubit one can readily verify that N obey the su(2) Lie { i} We can ask which spin observables must Alice manip- algebra relations [Ni,Nj] = iǫijkNk and also satisfy 2 ulate in order to access the virtual qubit subsystem, pro- Ni = ½ 2 = projector onto the j = 1/2 sector, justi- tected from the G-twirling. fying ourH labels of x,y,z for the virtual spin observables. Alice has in her possession three spin-1/2 parti- Furthermore, changes of the order reference of the par- cles, with angular momentum operators J1, J2, J3 , ticles corresponds to the action of the permutation group where for a fixed Cartesian frame with Pauli{ matrices} , which preserves the commutation relations and cor- σx, σy, σz on spin a we will use the compact vector 3 a a a Sresponds to rotations of the virtual Bloch sphere through notation{ J} := (J x,J y,J z) = ( 1 σx, 1 σy, 1 σz). a a a a 2 a 2 a 2 a angles of 120 degrees. She knows that for a given state ρ the relevant G- twirling map takes the form

[ρ] = [Π ρΠ ]+ [Π ρΠ ] (C1) G D1 1 1 D2 2 2 where the j=3/2 and j=1/2 sectors are written 1 and Appendix D: Twirling Alice, Twirling Bob = , with projectors Π and Π , and whereH H2 M2⊗N2 1 2 D1 fully decoheres 1 while 2 fully decoheres the virtual subsystem andH leavesD unaltered. The observable In section IV we are interested in how well local ref- M2 N2 S that has the sectors 1,2 as eigenspaces is given, in erence frames do in the preservation of bipartite entan- terms of local spin observables,H by glement. The initial product state is assumed to take the form Ψ = QRFA ϕ QRFB , where ϕ is 1 | i | i ⊗ | i ⊗ | i | i J J J J J J an entangled two spin state and QRFA and QRFB S = ( 1 2 + 2 3 + 1 3) (C2) | i | i 3 · · · consist of NA and NB spins respectively. In the absence of a shared classical reference frame for both A and B where the inner products in S are defined by this means that both sides are G-twirled independently, x x y y z z Ψ Ψ ρAB = A B [ Ψ Ψ ], or more explicitly J J := (J J + J J + J J ) ½ . (C3) a · b a ⊗ b a ⊗ b a ⊗ b ⊗ c | ih | → G ⊗ G | ih | We see that S is a relational observable, roughly being the average degree of alignment between the three spins, [ Ψ Ψ ]= dΩdΩ′U (Ω, Ω′) Ψ Ψ U † (Ω, Ω′) GA ⊗ GB | ih | AB | ih | AB and is both rotationally invariant and permutationally ZZ invariant, as expected from the Schur-Weyl duality.

The physical observables that Alice must measure to NA+1 NB +1 where U (Ω, Ω′) = U(Ω)⊗ U(Ω′)⊗ is the access the state on the protected virtual qubit system AB ⊗ N2 unitary corresponding to the rigid rotation of all spins at are given in terms of the physical spin observables by A through an angle Ω, and the rigid rotation of all spins 1 at B through an angle Ω′. Nx = (J2 J1) J3 √3 − · A brute-force numerical simulation of this G-twirling 2 rapidly gets difficult, and so one must exploit the struc- Ny = (J1 J2) J3 √3 × · ture of the decoherence-full/free subsystems to determine 1 how much entanglement is lost for a given pair of local N = (J J + J J 2J J ). (C4) reference frames QRF and QRF . z 3 2 · 3 1 · 3 − 1 · 2 A B