GHZ states as near-optimal states forreference frame alignment

Mear M. R. Koochakie∗, Vahid Jannesary†, andVahid Karimipour‡

Department of Physics, Sharif University of Technology,Tehran 14588, Iran

Abstract

Let two coordinate systems, in possession of Alice and Bob, berelated to each other by an unknown rotation R ∈ SO(3). Alice isto send identical states |ψ0〉 to Bob who will make measurements onthe received state and will determine the rotation R. The task ofBob is to estimate these parameters of the rotation R by the bestpossible measurements. Based on the Quantum Fisher Information,we show that Greenberger-Horne-Zeilinger (GHZ) states are near op-timal states for this task. Compared to the optimal states proposedbefore, the advantage of GHZ states are that they can be more eas-ily prepared experimentally, and more importantly, we show concretemeasurements which will allow Bob to determine the rotation R. Wealso study the robustness of these states in keeping their encoded in-formation, against common sources of noises.

1 Introduction

A class of quantum estimation theory problem can be stated as follows: Aunitary operator U(θ) acts on a quantum state |ψ0〉 turning it into |ψ(θ)〉 =

∗email: m.koochakie@gmail.com†email: vahid.jannesary@gmail.com‡email: vahid@sharif.edu

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U(θ)|ψ0〉. The parameter θ is for example related to the coupling constant ofa certain Hamiltonian, i.e. the strength of a magnetic field whose direction isknown to us and our task is to estimate its strength by performing the bestmeasurements on the state |ψ(θ)〉. An important question is which referencestate encodes this information in an optimal way, where ”optimality” canbe defined in various ways. Intuitively one can say the optimal state shouldbe the one which is the most sensitive to variations of θ. For example, ifthe direction of the magnetic field is known to be in the z− direction and θencodes only the strength of this magnetic field, then the optimal states arethose which are in the xy plane, i.e. the so-called equatorial states, and themore these states are tilted toward the z-axis, the less effective they are incarrying out the information. In passing, we note that it has been shown thatin this simple setting, entanglement can enhance the precision of parameterestimation [1, 2].

Suppose however that we do not even know the direction of the magneticfield and only a probability distribution of the directions is at hand. Thenthe above intuitive argument is no longer valid and we have to resort toa quantitative measure to answer this question. Besides, using tools frommulti-parametric estimation theory, we should also do an averaging over alldirections or parameters in order to find the optimal state. Such an optimalstate will certainly be the optimal state for a certain direction but not forthe others, but on the average, it will be optimal when the probability dis-tribution is taken into account. For a review see [3, 4].

In a more general setting the problem can be stated as follows: Let G bea group of transformations and |ψ〉 a reference state. The information of agroup element g = g(θ1, · · · θn) ∈ G is encoded into a unitary representationU(g) acting on the state |ψ〉 in the form |ψ(g)〉 = U(g)|ψ0〉 and the parame-ters (θ1, · · · , θn) of g are to be estimated by making suitable measurementson this state. Given that the elements of the group are chosen according to ameasure P (g)dg with

∫GP (g)dg = 1, our task is to find the optimal reference

state |ψ〉 for this task.

In a particular setting, G can be the group of rotations SO(3) and theparameters (θ1, θ2, θ3) can be the Euler angles. In this setting this is relevantamong other things to the problem of reference frame alignment, i.e. whentwo previously aligned reference frames are distorted due to noise. While

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classical means are always available for reference frame alignment, it hasbecome apparent in recent years that quantum resources like entanglement,can lead to the improved Heisenberg scaling in the achievable precision. [1, 4,2]. Moreover there are other applications closely related to this problem, likeestimating the direction and magnitude of a magnetic field, or the amountof distortion of photon polarization sent through an optical fiber.

1.1 Two main approaches to the problem of sharedreference frames

Before proceeding we should discriminate between two approaches which hasbeen followed in the past years for dealing with the problem of referenceframes. In one approach which is the subject of this paper, it is tried to alignthe two reference frames by quantum mechanical means before doing anyquantum information task. By quantum mechanical means we means thatone party sends an optimal reference to the other which is to be measuredand based on the results of the measurement the other reference frame shouldbe properly rotated to be aligned with the first one. This is the approachwhich was motivated by [5, 6] and initiated in [7, 8]. This approach wasthen developed in several works including [9, 10, 11, 12, 13]. There is how-ever another approach which seeks to do quantum information task withoutaligning the reference frames. In this approach it is either tried to encodeclassical or quantum information into particular states which are insensitiveto alignment [14, 15, 16, 17] or else try to model the effect of this misalign-ment as a noisy channel and assess their effect on the classical or quantuminformation which is being communicated [18, 19, 20]. A general theory ofreference frames and its relation with asymmetry and resource theory hasalso been developed in [21, 22, 23].

1.2 A brief history of the first approach

The approach that we follow here is the first one where the two parties try toalign their reference frame by estimation of optimal states sent transferredbetween the frames. This approach has a long history for itself which isquite different from the second one. First Gisin and Popescu showed in [5]that sending two anti-parallel spin one-half particles is better than two par-allel spins for sharing a direction, then Massar and Popescu [6] showed that

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the fidelity F of transmission of a direction with N parallel spins scales as1− F ∼ 1

Nwhen the receiver (Bob) is allowed to make collective muti-qubit

measurements. Later, Peres and Scudo [7] and Bagan et al [10] showed in-dependently that the fidelity can increase quadratically (i.e. 1 − F ∼ 1

N2 )with the number of particles if the state is encoded into an entangled stateof N spin one-half particles, namely a state which is an eigenstate of totalspin operator in the desired direction. Alleviating the need for multi-qubitmeasurements, it was shown in [9] that one can do single qubit measurementson singlet states shared between the two parties and use the correlations inmeasurements to set up a common direction with the same fidelity as inMassar and Popescu [6]. The figure of merit used in all these works was thefidelity between the actual direction sent by Alice and the direction guessedby Bob, averaged over all sent directions.

These studies naturally were extended to sending copies of a specific statewhich encodes the information of a whole Cartesian frame [8, 10, 11, 12, 13].The idea is that Alice takes a particular state |ψ0〉 in her own Caterzian frame(x, y, z) and sends copies of it to Bob whose frame (x′, y′, z′) is not alignedwith that of Alice. This state appears in the frame of Bob as

|ψ(θ)〉 = U(θ)|ψ0〉, (1)

where θ = (θ1, θ2, θ3) are the three parameters which designates the rotationoperator U(θ) aligning Bob’s frame with that of Alice. It is important tonote that |ψ0〉 itself does not have any intrinsic dependence on θ. Bob thenperforms optimal measurements on this state in his own frame to estimatethe values of the parameter θ. The aim of optimal measurements of Bob isto estimate the values of these parameters from the statistics of his measure-ments. In [8], it was shown that despite the spherically symmetric potentialof a Hydrogen atom a particular Rydberg state in the n−th level of a Hydro-gen like atom, |ψ0〉 =

∑n−1j=0

∑jm=−j ajm|j,m〉, can indeed optimally encode

this information. Here the figure of merit used was the average fidelity overall orientations of frames and the explicit form of the states, i.e. the valuesof aj,m, were found numerically.

Besides states of Rydberg atoms, collective spin states of a groups of Nspin 1/2 particles have also been studied in a number of works and this is theapproach that we will also follow in our work. That is the state we consider

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is of the type

|ψ0〉 =∑i1,···iN

ψi1,i2,···in|i1, i2, · · · iN〉 (2)

where |ik〉 ∈ {|0〉, |1〉} are the basis spin states of one particle. In [12],the multi-parameter Cramer Rao bound was considered and it was shownthat when the frames are slightly mis-aligned (an approach called the localapproach in [12]) simpler states can encode the information of Alice’s frame.The basic premise in the local approach of [12] is that for small misalignment,the rotation operator is linearized

U(θ) ≈ I + iθ · S, (3)

which leads to a drastic simplification of the Quantum Fisher Informationmatrix. Here S = (Sx, Sy, Sz) is the total spin operator acting on the spins,i.e.

Sa =1

2

∑k

σa,k. (4)

The optimal states are thus found to be the so-called anti-coherent states ofspin j− systems, i.e. those which satisfy the following conditions

〈ψ0|Sa|ψ0〉 = 0 〈ψ0|S2a|ψ0〉 =

j(j + 1)

3a = x, y, z. (5)

Therefore the authors of [12] found analytical forms for the optimal stateswhich turned out to be certain specific states of N particles, studied in [25]and named anti-coherent states. When represented in terms of Majoranarepresentation [26, 25], they correspond to Platonic solids: Any symmetricstate can be decomposed as superposition of product states, i.e.

|ψ〉 = N∑σ∈SN

|~nσ(1)〉 ⊗ |~nσ(2)〉 ⊗ · · · |~nσ(N)〉, (6)

where N is a normalization constant, the summation is performed over allpermutations and the states |~nk〉 are pure states on the Bloch sphere. Thereis a one-to-one correspondence between any such state and the points on thesurface of Bloch sphere [25].

Quite recently [13] the problem has been studied in the so called globalapproach, i.e. where the two frames are misaligned by arbitrary finite rota-tions. To calculate the Fisher information and find the optimal state in this

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approach, it is necessary to parameterize the rotation operator by the threeEuler angles, as

U(α, β, γ) = e−iαSze−iβSye−iγSz . (7)

Intriguingly and despite the asymmetry of the three axes, what the authorsof [13] obtain are still regular polyhedrons. They also obtained new formsof symmetric states corresponding to Archimedean solids which contain thestates of [12] as a subset, figure (2).

The basic premise in the work of Goldberg and James [13] which leadsto this symmetry of optimal states, despite the asymmetry of their parame-terization is that they use a result of [12], namely (5) which is valid only forsymmetric parameterization (3) of the rotation operator.

Remark: It is crucially important to note that in all of these works the op-timal state cannot be prepared experimentally due to its complicated form,whether it be calculated analytically or numerically, whether in the local orglobal approach.

1.3 Our main results

In this article we start from scratch, use Euler parameterization for rotationand show that the optimal state is in fact a GHZ state of the form

|ψ0〉 =1√2

(|N2,N

2〉+ |N

2,−N

2〉) =

1√2

(| ↑, ↑, · · · ↑〉+ | ↓, ↓, · · · ↓〉), (8)

where the states | ↑〉 and | ↓〉 are the spin up and down in Alice’s z− directionand δ is an arbitrary phase. (In fact the two terms can also have a phase dif-ference eiδ without altering any of the results.) The basic point is that whena rotation operator is decomposed as in Eq. (7), the sensitivity of the sentstate to variations of the three parameters is certainly not equal and thereis an obvious difference between the z-direction and the other directions.Therefore it is understandable that the optimal state may not correspond toa regular polyhedron. As we will see, the optimal state that we will obtain isstill a symmetric state with a nice Majorana representation shown in figure(2). Had the Euler decomposition been chosen in a different form, i.e. withdifferent orders of rotation, the GHZ state would have been different. In any

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case, the important point is that such states can be created quite easily inthe laboratory with a quantum circuit. Our analysis shows in yet anotherquantum information task, the usefulness of GHZ states as superposition ofmacroscopically distinct states.

The figure of merit that we use is the related to Quantum Fisher Infor-mation of the encoding state uniformly averaged over all rotations. Basedon the this figure of merit, we show that Greenberger-Horne-Zeilinger statesare optimal states for this task. The advantage of these states are that, com-pared to the optimal states which have been proposed so far, they can beexperimentally prepared [27, 28, 29, 30] . We also study the robustness ofthese states in keeping their encoded information, against common sourcesof noises.

The structure of this paper is as follows: In section (2) we set up theproblem and explain in detail the figure of merit which leads to the optimalstate. Then we will show that the GHZ states are near optimal in a sensewhich will be explained. Finally we compare our final results with those ofreference [13]. We conclude the paper with a conclusion.

2 Cramer-Rao bound for frame alignment

As explained in the introduction, Alice takes a particular state |ψ0〉 in herown Caterzian frame (x, y, z) and sends copies of it to Bob whose frame(x′, y, z′) is not aligned with that of Alice. This state appears in the frameof Bob as |ψ(θ)〉 = U(θ)|ψ0〉 where θ = (θ1, θ2, θ3) are the three parameterswhich designates the rotation operator U(θ) aligning Bob’s frame with that ofAlice. The state |ψ0〉 itself does not have any intrinsic dependence on θ. Letus denote the estimated values by θe = (θ1,e, θ2,e, θ3,e). Here optimality of thewhole process of estimation depends on the state |ψ0〉 chosen by Alice and theclever measurements of Bob. In all the works which have taken the QuantumFisher Information as their figure of merit, as also in the present work, theemphasis is to find the optimal state |ψ0,opt〉 which Alice should send, ratherthan on the optimal measurement performed by Bob and the estimationmethod he may use. These may be quite complicated and varied. Whatreally allows these two optimal choices to be made separately is the celebrated

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mutli-parameter Cramer Rao bound, which states that the variance betweenthe real values of the parameters θi and their estimated values θi,e satisfiesthe following matrix inequality [31, 32, 33, 34]

∆(θe,θ) ≥ F−1(ψ(θ)), (9)

where∆(θe,θ)m,n = 〈(θm,e − θm)(θn,e − θn)〉, (10)

and

Fmn(ψ(θ)) = 2〈∂mψ(θ)|∂nψ(θ)〉+ 2〈∂nψ(θ)|∂mψ(θ)〉+ 4〈ψ(θ)|∂mψ(θ)〉〈ψ(θ)|∂nψ(θ)〉. (11)

Here |ψ(θ)〉 is the state |ψ0〉 as appeared in the frame of Bob, that is|ψ(θ)〉 = U(θ)|ψ0〉. The important point about (9) is that the right handside is independent of the measurement process and any estimation procedurethat Bob may use. It sets a theoretical lower bound on the variance of thevalues of estimated parameters compared with their real values. Note alsothat we have used the form of Quantum Fisher Information for pure states,since we have used the fact that F is a convex function of the parameterizedquantum states, and hence takes its optimum value at the extreme points ofthe space of states, that is the pure states.

An important question is whether this lower bound can be saturated andit has been shown in [35] that this is indeed the case if

Im〈∂mψ(θ)|∂nψ(θ)〉 = 0. (12)

One usually defines a cost function as the sum of variances of all three pa-rameters, i.e.

C(θe,θ) :=∑m

〈(θm,e − θm)2〉 = Tr∆(θe,θ), (13)

which is bounded below as

C(θe,θ) ≥ Tr[F−1(ψ(θ))

]= Tr

[F−1(U(θ)ψ0)

]. (14)

Therefore for communicating a ”known” parameter θ, the optimal state|ψ0,opt〉 is the state which minimizes the right hand side. Naturally this

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state depends on the parameter θ, therefore we can write ψ0,opt = ψ0,opt(θ).

While these relations are valid for any process in which a set of ”known”parameters are encoded into quantum states and communicated to someother point, in the problem of frame alignment, we are faced with a situationin which the parameter is not known to the sender. Therefore in this casethe optimal state cannot have any dependence on θ even after optimization.Instead one has to define a cost function averaged over all values of theparameter θ. A natural cost function is defined as follows

C :=

∫dθTr(∆(θe,θ)), (15)

where dθ is a suitable measure over the space of parameters. Thus C dependsonly on the estimation procedure and the initial state |ψ0〉, both of which aresuppressed in the notation for C for simplicity. Then equation (9) leads to

C ≥∫dθ Tr(F−1(U(θ)ψ0) =: Tr(F−1). (16)

One can obtain a still lower bound [36, 37] by adopting the so called van Treesinequality and the Baeysian-Cramer Rao bound, however we suffice here tothis slightly looser bound. In view of the convexity of the quantum Fishermatrix [38] and the inequality Tr(X−1)Tr(X) ≥ n2 for any n dimensionalpositive matrix, one can also write

C ≥ 9∫dθ Tr(F (U(θ)ψ0)

=:9

Tr(F ). (17)

Remark: While inequality (16) can be saturated if for each θ, one insertsψ0,opt(θ) inside the integral on the right hand side, the inequality in (16), isnot normally saturated, that is, the quantum Fisher information, as used in(16) and (17) gives a slightly less tight lower bound compared to the casewhere a known parameter is to be communicated by encoding it into a quan-tum state. Therefore no matter how the right hand side of (16) is minimized,the Cramer-Rao bound cannot be saturated for frame alignment and all theoptimal states found in different works, including the present one, shouldactually be called near-optimal states.

What we will do is to calculate the quantum Fisher information matrixand show that the GHZ state which almost maximizes the quantity TrF . The

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large benefit which comes from this approach is the simplicity of creating aGHZ state in the laboratory compared with other states.

3 The Quantum Fisher Information for frame

alignment

The state |ψ0〉 that Alice sends appears in Bob frame as

|ψ(θ)〉 = U(θ)|ψ0〉, (18)

where U(θ) is the rotation which align the two coordinate systems. Thereare a multitude of ways for parameterization of a rotation operator. Theparameterization U(θ) = eiθ·S, which has many good properties is suitableonly for small rotations, when it comes to calculating the Fisher matrix.Instead we use the Euler angle parameterization which allows us to determinethe Fisher matrix for finite rotations. Thus we use

U(θ) = U(α, ,¯γ) = e−iαSze−iβSye−iγSz ≡ Z(α)Y (β)Z(γ), (19)

where α, β and γ are the Euler angles which transform Alice’s frame to thatof Bob. It should be noted that there are many variants of the Euler anglesand there is no preference among them, since they all lead to valid lowerbounds for precision. In other words, had we decomposed the rotation ina different form, we would have ended up again with an optimal state as aGHZ state in a different direction, but this is only a matter of labeling qubitstates. The fact that Alice should send a GHZ state remains unchanged.More importantly, when Bob does measurements, the way this rotation hasbeen parameterized is not relevant, see section (3.1).

We can now use Eq. (11) and calculate QFI matrix of |ψ(α, β, γ)〉 = U(α, ,¯γ)|ψ0〉.

For a pure state, the components of the Quantum Fisher Information (QFI)are given by [40, 39]

Fmn = 4cov (Hm, Hn) , (20)

where Hm = i(∂mU†)U, m = α, β, γ and

cov(Hm, Hn) :=1

2〈ψ0|{Hm, Hn}|ψ0〉 − 〈ψ0|Hm|ψ0〉〈ψ0|Hn|ψ0〉. (21)

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It is straightforward to check that

Hα = − cos βSz + sin β(cos γSx − sin γSy), (22)

Hβ = − cos γSy − sin γSx, (23)

Hγ = −Sz. (24)

In particular we have

Fm,m = 4〈ψ0|H2m|ψ0〉 − 4〈ψ0|Hm|ψ0〉2, m = α, β, γ. (25)

To calculate TrF , we need a prior distribution for the parameters α, β andγ. We take a uniform distribution (SO(3) Haar measure)

z(α, β, γ) dαdβdγ =1

8π2sin β dαdβdγ. (26)

Here we have assumed that the distribution of the z axis is uniform over asphere in Bob’s frame. This gives the factor 1

4πsin β dαdβ in (26). Once the

z or the x-y plane is fixed, the x axis can be uniformly distributed in thisplace, with a distribution 1

2πdγ, hence Eq. (26).

Using the above distribution, we show that (see the appendix)

Tr(F ) =16

3(∆Sz)

2 +10

3(∆Sx)

2 +10

3(∆Sy)

2

=16

3(〈S2

z 〉 − 〈Sz〉2) +10

3(〈S2

x〉 − 〈Sx〉2) +10

3(〈S2

y〉 − 〈Sy〉2)

=10

3〈S · S〉+ 2〈S2

z 〉 − 〈Sx〉2 − 〈Sy〉2 − 〈Sz〉2. (27)

This expression clearly shows the special role which the z− axis plays com-pared with the other two axes. Note however that had we decomposed therotation in a different form, we would have ended up again with an optimalstate as a GHZ state in a different direction, but this is only a matter oflabeling qubit states. The fact that Alice should send a GHZ state remainsunchanged.

To find the state |ψ0〉, which maximizes this quantity, we take an arbitrarystate of N spin 1/2 particles,

|ψ0〉 =

N2∑

j=0, 12

j∑m=−j

aj,m|j,m〉, (28)

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where j runs from 0 or 1/2 for even and odd N respectively and |j,m〉 isthe familiar angular momentum notation. It is then obvious that Tr(F ) ismaximized when j takes its largest value and 〈Sx〉 = 〈Sy〉 = 〈Sz〉 = 0. Sucha state is nothing but a GHZ state, i.e. a superposition of macroscopicallydistinct states

|ψ0〉 = |GHZ〉 =|N2, N

2〉+ eiδ|N

2,−N

2〉

√2

, (29)

where δ is an arbitrary phase and |N2, N

2〉 = | ↑, ↑, · · · ↑〉 and |N

2,−N

2〉 = | ↓

, ↓, · · · ↓〉 in which the spin up and spin down are along the z directions of

Alice frame. These states maximize Tr(F ) to the value N(4N+5)3

. It turns outthat the phase δ has no effect on the QFI and hence we set δ = 0 hereafter.

3.1 Measurement

In general it is not easy to determine the optimal measurement which satu-rates the Cramer-Rao bound. To the best of our knowledge, simple measure-ments have not been proposed for anti-coherent symmetric states proposedin [12] and [13] or for the Rydberg states proposed in [8]. Nevertheless it isa highly legitimate question as to how after all , Bob determines the coor-dinate system of Alice by doing measurements on a single type state thatAlice sends him. Here we propose a measurement procedure that he cando for this task, without claiming that it is the optimal one. Note that ifthe two frames are related by the rotation U(θ)†, the state |ψ0〉 will ap-pear in Bob frame as U(θ)|ψ0〉, where U(θ) is given in (19) . Thereforethe GHZ state |ψ0〉 = 1√

2(|z〉⊗N + |(−z)〉⊗N) appears in Bob’s frame as

|ψ′0〉 = 1√2(|n〉⊗N + e−iγN | − n〉⊗N), where n is the unit vector with polar

coordinates (α, β) as shown in figure (1). Bob’s task is to do measurementson this state and discern α, β and γ. Knowing that |n〉 and | − n〉 are spin

states given by |n〉 =

(cos β

2

sin β2eiα

)and | − n〉 =

(− sin β

2

cos β2eiα

)respectively,

Bob concludes that 〈ψ′0|Sa|ψ′0〉 = 0 ∀ a = x, y, z . This means that he cannotdetermine α and β simply by the expectation values of total spin component.However he can determine these angles by the variance of spin measurements,since a simple calculation reveals to him that

〈ψ0|S2x|ψ0〉 =

1

4(N +N(N − 1) sin2 β cos2 α)

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〈ψ0|S2y |ψ0〉 =

1

4(N +N(N − 1) sin2 β sin2 α)

〈ψ0|S2z |ψ0〉 =

1

4(N +N(N − 1) cos2 β). (30)

This implies that by measuring the variance of any two components he candetermine the two angles α and β. It remains for him to determine the angleγ. To this end he uses a second bunch of states |ψ′0〉 and rotate them to alignthem along his z axis in which case their form will be given by

|ψ′0〉 =1√2

(|0〉⊗N + e−iγN |1〉⊗N) (31)

where for convenience, we have used the computational notation, and hemeasures the operator Π = σx,1σx,2 · · · σx,N . He simply finds

〈ψ′0|Π|ψ′0〉 = cos γN (32)

from which he determines γ.

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Figure 1: The GHZ state |ψ0〉 = 1√2(|z〉⊗N + |− z〉⊗N) sent by Alice, appears

as |ψ′0〉 = 1√2(|n〉⊗N+eiγN|−n〉⊗N) in frame of Bob. Bob can then determine

the angles α , β and γ by suitable measurements as detailed in section (3.1).

13

3.2 Comparison

As can be seen, in Tr(F ) the role of z direction differs from other directions.This is due to the way Euler rotations are parameterized and is also reflectedin the Majorana representation of the state (29), shown in figure (2) whichis a superposition of equally spaced product states in the equatorial plane ofAlice, namely

|ψ0〉 = N∑σ

|zσ(1)〉|zσ(2)〉 · · · |zσ(N)〉, (33)

where |zk〉 = 1√2(e

iδN |0〉+ωk|1〉) in which ω is the N−th root of unity, ωN = 1

and |0〉 and |1〉 are the σz eigenstates in Alice’s frame.

z

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Figure 2: Left: Majorana representation of an optimal states correspondingto an Archimedean solid, [13]. Each vertex corresponds to a pure state on theBloch sphere and the full state is a symmetric superposition of these states.Right: Majorana representation of the GHZ state (29) for communicating aframe, when the rotation is parameterized in terms of Euler angles.

14

This work: Maximizing Tr FGoldberg and James (2018)

Tr(F�1( 0))

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Figure 3: (Color online) Tr[F−1(U(α, ,

¯γ)|ψ0)

]for the case where |ψ0〉 is

the GHZ state (29) in our approach versus the case where |ψ0〉 is the anti-coherent symmetric state of [13]. Both depend only on the angle β and it isseen that they differ only slightly for almost all values of the rotation angleβ. See the remark (1) for more details.

Remark 1 Although by minimizing 1Tr(F )

, (averaged over all rotations), we

have shown that the GHZ state (29) is near optimal, we can calculate Tr(F−1(ψ0))for this near optimal state for any specific rotation U(α, β, γ), to see how thisstate works for an arbitrary rotation between the frames. Note that while op-timizing Tr(F−1), averaged over all rotations, is intractable analytically, itis possible to calculate Tr(F−1(|GHZ〉) for arbitrary rotations.

To this end, we insert the optimal state (29), into the expressions (11) andafter straightforward calculations obtain the F matrix

15

F (ψ(α, ,¯γ)) =

N2 cos2 β +N sin2 β 0 N2 cos β0 N 0

N2 cos β 0 N2

, (34)

with

Tr(F ) = N(1 + sin2 β) +N2(1 + cos2 β). (35)

Then the inverse of the F matrix becomes

F−1(ψ(α, ,¯γ)) :=

1

N sin2 β

1 0 − cos β0 sin2 β 0

− cos β 0 cos2 β + 1N

sin2 β

. (36)

Therefore we find for the GHZ state with

Tr[F−1(U(α, ,

¯γ)|GHZ〉)

]=

1

N2+

2

N sin2 β. (37)

This is to be compared with the result of [13] which in our notation reads

Tr[F−1(U(α, ,

¯γ)|Φ〉)

]=

3

N(N + 1)

(1 +

2

sin2 β

), (38)

where |Φ〉 is the anti-coherent symmetric state of [13]. It is seen that bothquantities depends only on the parameter β of the rotation, which is theangle between the axes z and z′ and is insensitive (symmetric) to the othertwo rotations. The comparison is done in figure (3). The vertical axis showsTr[F−1(ψ(α, ,

¯γ))]

for our near optimal state ψ0 (blue color) and for the anti-coherent polyhedron symmetric state of [13](pink color). This figure showsthat the lower bound for both these states are very close for all values of Nand β. We should emphasize however that none of these states are the realoptimal state which should be found by rigorously minimizing the right hand

side of Cramer Rao inequality (16), i.e. minimizing Tr(F−1

). In our case,the shortcoming is that we really maximize Tr(F ). In case of Goldberg andJames, the shortcoming is that for the Euler parameterization of rotations,the basic condition (5) which is true for infinitesimal rotations and hencesymmetric parameterization is no longer valid. At a fundamental level, aswe have explained in the remark after equation (17), for frame alignment inthe global approach, when we integrate over all angles of rotations, no statecan be absolutely optimal in the sense of saturating the lower bound of theCramer-Rao bound. What we have obtained with our near-optimal state isthe simplicity in experimental realization.

16

4 The effect of noise

It is quite conceivable that the optimal state which has been taken to be apure state up to now, is affected by noise either by imperfect preparation inthe first lab or during transmission between the two labs. The optimal statethen changes to a mixed state and it is then desirable to see how the QuantumFisher Information (QFI) of this optimal state is affected and whether or notit is still useful for estimation of parameters encoded into it. This is theproblem which we now study. To this end, we should resort to a formulafor Quantum Fisher Information which is appropriate for mixed states andis a generalization of (20). According to [39, 40], QFI of a mixed state withspectral decomposition

∑k λk|φk〉〈φk| (evolving unitarily) is

Fm,n =∑k

4λkcovk (Hm, Hn)−∑k 6=l

8λkλlλk + λl

Re (〈φk|Hm|φl〉〈φl|Hn|φk〉) . (39)

For a pure state, this will reduce to (20). We use the above result anddetermine the effect of two very important and common sources of noise,namely the dephasing and the depolarizing noise, on the QFI of the optimalstate that we already found (29).

4.1 Dephasing noise

Here we study the effect of a very common noise on the optimal state andits viability as a carrier of reference frame information. We consider randomphase kicks on all the qubits. This noise changes the optimal state |ψ0〉 to amixed state in the form

|ψ0〉 −→ ρ =

∫dU U |ψ0〉〈ψ0|U †, (40)

whereU = eiθ1σz ⊗ · · · ⊗ eiθNσz ,

and dU = P (θ1, · · · θN)dθ1 · · · dθN , where we have not made any assumptionon the probability distribution P (θ1, · · · θN). The random kicks can be cor-related or not and can be different on various qubits, the net effect is thatthey transform the optimal state to

ρdephased = p|ψ0〉〈ψ0|+1− p

2

(| ↑ ·· ↑〉〈↑ ·· ↑ |+ | ↓ ·· ↓〉〈↓ ·· ↓ |

), (41)

17

in which

p =

∫dθ1 · · · dθNe2i(θ1+···+θN )P (θ1, · · · , θN). (42)

This means that the optimal state |ψ0〉 may decohere to product states withprobability 1 − p. Note that when there is no phase kick, p = 1 and weget back our original pure optimal state and when the kicks are completelyrandom, p = 0, the state becomes a statistical mixture of all-up and all-downspins. Interestingly this state can be written as [42]

ρdephased =1 + p

2|ψ0〉〈ψ0|+

1− p2|ψ′0〉〈ψ′0|, (43)

where

|ψ′0〉 =|N2, N

2〉 − |N

2,−N

2〉

√2

, (44)

The noisy optimal state (43) is already in the spectral decomposed form.Inserting this in equation (39) we find after simplification

Fm,m = 2(1 + p)(〈ψ0|H2m|ψ0〉 − 〈ψ0|Hm|ψ0〉2)

+ 2(1− p)(〈ψ′0|H2m|ψ′0〉 − 〈ψ′0|Hm|ψ′0〉2) (45)

− 4(1− p2)(〈ψ′0|Hm|ψ0〉〈ψ0|Hm|ψ′0〉). (46)

It turns out that the first two lines of (45) are equal. This is in fact anexample of a result mentioned before that the relative phase between thetwo terms in the optimal state (29) is not relevant. For the cross terms inthe last line we need

〈ψ′0|Ha|ψ0〉 = 〈ψ′0| − cos βSz + sin β(cos γSx − sin γSy)|ψ0〉

= cos β〈ψ′0|Sz|ψ0〉 =−N

2cos β. (47)

Similarly we find

〈ψ′0|Hβ|ψ0〉 = 〈ψ′0| − cos γSy − sin γSx|ψ0〉 = 0, (48)

where for definiteness we have taken N > 1 and

〈ψ′0|Hγ|ψ0〉 = 〈ψ′0|Sz|ψ0〉 =N

2. (49)

Combining the above, we find

Tr(F )(ρdephased) = Tr(F )(ψ0)−N2(1− p2)(1 + cos2 β). (50)

18

To compare it with the case for optimal pure state, it is instructive to writetheir explicit form here, from (35)

Tr(F )(|ψ0〉) = N(1 + sin2 β) +N2(1 + cos2 β),Tr(F )(ρdephased) = N(1 + sin2 β) + p2N2(1 + cos2 β). (51)

Thus the effect is second order in p but increases also quadratically with thenumber of particles N .

4.2 Depolarizing noise

Another common noise is when the GHZ state undergoes a global depolariz-ing noise (say in the production process) where the optimal state is changedas

|ψ0〉 −→ ρdepol = p|ψ0〉〈ψ0|+1− p2N

I. (52)

Incidentally this simple form still holds when each of the qubits locally un-dergo depolarizing noise when N = 2. This is also already in the form ofspectral decomposition, with eigenvalues and eigenvectors{

|ψ0〉 ξ := p+ 1−p2N,

|ψ⊥k 〉 η := (1−p)2N

,(53)

where |ψ⊥k 〉 are 2N − 1 states which are perpendicular to |ψ0〉. The simpleform of the spectrum and the high degeneracy of the eigenvalue η, allows usto write (39) in a compact and simple form. First we rewrite the diagonalelements of F from (39) as

Fm,m =∑k

4λk〈φk|H2m|φk〉 −

∑k,l

8λkλlλk + λl

(〈φk|Hm|φl〉〈φl|Hm|φk〉) . (54)

Then we use the completeness relation |ψ0〉〈ψ0|+∑

k |ψ⊥k 〉〈ψ⊥k | = I to writefor any operator A,

∑k〈ψ⊥k |A|ψ⊥k 〉 = Tr(A)− 〈ψ0|A|ψ0〉 and simplify all the

terms, specially the cross terms in (54). After straightforward calculationswe find

Fm,m(ρdepol) = (ξ + η − 4ξη

ξ + η)Fm,m(ψ0), (55)

19

or inserting the values of ξ and η from (53)

Tr(F )(ρdepol) = Λ(p,N) TrF (|ψ0〉), (56)

where

Λ(p,N) := p+O(1− p2N

), (57)

showing that the Tr(F ) decreases linearly with increase of the level of noise.

5 Conclusion

The question we have asked is the following: Given a state |ψ0〉 and a unitaryoperator U(R) representing a finite rotation R(α, ,

¯γ), where α, β, and γ are

the Euler angles, which state is the optimal state if we want to estimate theangles by making the best measurements on the state |ψ(α, ,

¯γ)〉 := U(R)|ψ0〉.

By calculating the Quantum Fisher Information, we have shown that the op-timal state is a |GHZN〉 = 1√

2(| ↑ ·· ↑〉+ | ↓ ·· ↓〉) state and have determined

in closed form the dependence of the QFI on the Euler angles and also onN . We have also determined, again in closed form, the effect of two verycommon sources of noise on the QFI, if they happen to affect the optimalpure state. Our work raises several questions. First of all, our work pro-vides only a near-optimal state which lower bounds a reasonable figure ofmerit for estimation of the rotation parameters. Considerations of QuantumFisher information do not normally yield specific measurement proceduresfor achieving this lower bound. Therefore it will be interesting to find aparticular measurement setting for achieving or coming close to any of theselower bounds. Second, it is very interesting to generalize this to a more gen-eral estimation problem, i.e. when the parameters of a certain group elementg of a continuous group G is to be estimated. In this way connections withthe asymmetry properties of states under general group transformations [24]seems to be relevant. Finally we have here considered the case where the twoplayers have no information about the relative orientation of their frames,hence we have used a Haar measure over the rotation group. In [12], it isassumed that the two frames are slightly misaligned, leading to an optimalstate which is symmetric over the Bloch sphere. It will be interesting to seehow the optimal state depends on the distribution function, i.e. to the widthof a spherical Gaussian distribution of what is known in the literature as avon Mises-Fisher distribution [18].

20

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Appendix: Deriving Tr(F ) of Euler rotation

To calculate the mean trace QFI Tr(F ) for Euler rotation w.r.t. the distri-bution Eq. (26), we should calculate following integral:

Tr(F ) =

∫ 2π

γ=0

∫ π

β=0

∫ 2π

α=0

TrF (α, β, γ) z(α, β, γ) dαdβdγ (58)

=1

8π2

∫ 2π

γ=0

∫ π

β=0

∫ 2π

α=0

(Fαα + Fββ + Fγγ) sin β dαdβdγ, (59)

where the quantum Fisher matrix, F , is defined in Eq. (20). To calculateeach of the three integrals in (58) we use the expression for Fα,α from (22)and calculate

24

∫ 2π

γ=0

∫ π

β=0

(〈Hα〉 − 〈Hα〉2

)sin β dβdγ =

2π

∫ π

β=0

cos2 β sin β dβ(〈S2

z 〉 − 〈Sz〉2)

+

∫ π

β=0

sin3 β dβ

∫ 2π

γ=0

cos2 γ dγ(〈S2

x〉 − 〈Sx〉2)

+

∫ π

β=0

sin3 β dβ

∫ 2π

γ=0

sin2 γ dγ(〈S2

y〉 − 〈Sy〉2)

=4π

3

(〈S2

x〉 − 〈Sx〉2 + 〈S2y〉 − 〈Sy〉2 + 〈S2

z 〉 − 〈Sz〉2). (60)

Then we use the expression for Fβ,β from (23) and calculate

∫ 2π

γ=0

(〈H2

β〉 − 〈Hβ〉2)

dγ =∫ 2π

γ=0

cos2 γ dγ(〈S2

y〉 − 〈Sy〉2)

+

∫ 2π

γ=0

sin2 γ dγ(〈S2

x〉 − 〈Sx〉2)

= π(〈S2

y〉 − 〈Sy〉2 + 〈S2x〉 − 〈Sx〉2

). (61)

Finally we the expression for Fγ,γ from (24) and calculate∫ 2π

γ=0

(〈H2

γ〉 − 〈Hγ〉2)

dγ = 〈S2z 〉 − 〈Sz〉2. (62)

Putting everything together the final expression for Tr(F ) is simplified to

Tr(F ) =10

3

(〈S2

x〉 − 〈Sx〉2)

+10

3

(〈S2

y〉 − 〈Sy〉2)

+16

3

(〈S2

z 〉 − 〈Sz〉2), (63)

which is the expression used in (27).

25