# Minimax estimation of qubit states with Bures risk

Anirudh Acharya and Mădălin Guţă
School of Mathematical Sciences, University of Nottingham,
University Park, NG7 2RD Nottingham, UK
###### Abstract

The central problem of quantum statistics is to devise measurement schemes for the estimation of an unknown state, given an ensemble of independent identically prepared systems. For locally quadratic loss functions, the risk of standard procedures has the usual scaling of . However, it has been noticed that for fidelity based metrics such as the Bures distance, the risk of conventional (non-adaptive) qubit tomography schemes scales as for states close to the boundary of the Bloch sphere. Several proposed estimators appear to improve this scaling, and our goal is to analyse the problem from the perspective of the maximum risk over all states.

We propose qubit estimation strategies based on separate and adaptive measurements, that achieve scalings for the maximum Bures risk. The estimator involving local measurements uses a fixed fraction of the available resource to estimate the Bloch vector direction; the length of the Bloch vector is then estimated from the remaining copies by measuring in the estimator eigenbasis. The estimator based on collective measurements uses local asymptotic normality techniques which allows us derive upper and lower bounds to its maximum Bures risk. We also discuss how to construct a minimax optimal estimator in this setup. Finally, we consider quantum relative entropy and show that the risk of the estimator based on collective measurements achieves a rate under this loss function. Furthermore, we show that no estimator can achieve faster rates, in particular the ‘standard’ rate .

## 1 Introduction

Quantum state tomography plays in an important role in quantum information processing tasks, as a method of estimating and validating experimentally prepared quantum states [MIT]. The aim of state tomography is the estimation of an unknown density matrix from the outcomes of measurements performed on identical copies of the state available as a resource. The quality of the resulting estimate is quantified in terms of its average error, or risk. Given a measurement design , and the corresponding set of outcomes , the risk of the measurement-estimator pair is

 R(ρ,^ρ):=E[D(^ρ(X),ρ)], (1)

where the expectation is taken with respect to the measurement outcomes , given the unknown state . The risk depends on the choice of the error, or loss function , which is a measure of the deviation of the estimated state from the true state . Examples of commonly used loss functions are squared Frobenius (norm-two) distance, infidelity, the trace-norm distance, and the Bures distance. The risk is a function of the resource size , and one is interested in its behaviour in the limit of large . Typically, for a ‘good’ estimator and particular choices of loss functions, (e.g. locally quadratic functions) the optimal risk exhibits a rate of uniformly over all states .

However, for certain loss functions (e.g. squared Bures distance, or infidelity) , the risk is known to behave differently for states of unknown purity [MahlerRozema, AcharyaGuta]. This is readily illustrated in the qubit case, where the fidelity between the state and its estimate is defined as and can be expressed as

 F(ρ,^ρ):=12(1+√1−|r|√1−|^r|+r⋅^r),

where are the Bloch vectors of the two states. For within the Bloch sphere, the fidelity is locally quadratic in the components , with . However, for states close to the boundary of the Bloch sphere, the fidelity becomes linear in . Standard tomographic estimation of the Bloch vector components by measuring the spin operators gives an accuracy of the order of in estimating . This implies that for a loss function such as the infidelity , or the square Bures distance , the risk scales as for states within the Bloch sphere, but only as for nearly pure states.

This poor scaling for nearly pure states is significant as the preparation and estimation of pure states is ubiquitous in quantum information processing tasks. Although many papers discuss the issues surrounding quantum tomography for Bures risk, we consider the problem in the context of minimax estimation (see also [FerrieBK2, FerrieBK]), i.e. where the figure of merit is the maximum risk over all states

 Rmax(^ρn):=supρR(ρ,^ρn)=supρE[DB(ρ,^ρn)2] (2)

Our aim is to show that adaptive, separate measurement strategies can achieve the scaling of the maximum risk. We also consider collective measurements, and derive an upper bound to the asymptotic constant of the maximum risk. Our analysis shows that the problem of finding minimax estimators, i.e. estimators with smallest possible asymptotic constant, reduces to that of finding minimax estimator for the ‘classical’ problem of estimating the coin toss probability with respect to the square Hellinger distance risk.

Several estimation methods have been proposed in the literature, involving both global and local measurement strategies, with the aim of improving the poor scaling of fidelity based risks for nearly pure states. However, to the best of our knowledge a scaling of of the maximum risk has not been demonstrated for any of these estimators. Two-steps adaptive quantum tomography [GillMassar, RehacekEnglert, BaganBallesterGill, MahlerRozema, HouZhu] involves using a fraction of the available resource to obtain a preliminary estimate of the eigenbasis of , and then performing measurements along the estimated eigenbasis on the remaining copies of the state. In [BaganBallesterGill] it is shown that using a vanishing fraction , with for the preliminary estimate gives a rate of for the average infidelity with respect to certain distributions over states. However, it has been pointed out in [MahlerRozema], that for certain states a vanishing fraction is insufficient, and that for almost pure states the worst case infidelity scales as . Numerical results in [MahlerRozema] suggest that using a fixed fraction instead gives the scaling for nearly pure states. The two step adaptive protocols have been experimentally implemented [MahlerRozema, HouZhu], showing a quadratic improvement in scaling for nearly pure states. The extension of the two-step adaptive protocol to a fully adaptive one has been considered in [RehacekEnglert, OkamotoLefuji, Fujiwara, Ferrie], where the measurement basis is aligned according to a current estimate after every measurement step. In the Bayesian framework, ‘self-learning’ measurement protocols have been considered in [FischerKienle, GranadeFerrie, HannemannReiss, KravtsovStraupe]. A detailed review of various adaptive protocols and experimental results is found in [Straupe].

Protocols considering collective (or joint rather than separate) measurements have also been considered [BaganBallester, BaganBaig, GutaJanssens, GutaKahn]. It is known that joint measurements perform better than separate measurements in the case of mixed states [BaganBaig]. In a Bayesian framework, [BaganBallester] showed that with certain optimal joint measurements, the asymptotic infidelity averaged over a prior distribution achieved a value of for mixed qubit states, where is the mean purity over the prior distribution. Work in [GutaJanssens] proposes a two-step adaptive estimation strategy that is shown to be locally optimal, achieving an infidelity risk of for mixed qubit states. However these theoretical results cannot be directly used to derive the scaling of the minimax risk.

We propose two different estimators, one based on adaptive local measurements similar to [BaganBallesterGill, MahlerRozema], and a second based on global collective measurements and LAN as in [GutaJanssens, GutaKahn].

In terms of local measurements, we consider a two step adaptive strategy much in line with already proposed estimators [GillMassar, RehacekEnglert, BaganBallesterGill, MahlerRozema, HouZhu]. A fixed fraction of the total sample size is used to obtain a preliminary estimate of the Bloch vector of the state, by performing standard tomographic measurements of the spin observables. The remaining copies of the state are used to estimate the eigenvalues of the state by performing measurements along the estimated direction . The final estimate of the state is then constructed from the estimated eigenvalues in the adaptive step and the preliminary estimate. For this estimator, we upper-bound the maximum Bures risk and demonstrate a scaling of .

The estimator based on global collective measurements uses established LAN results for qubit states [GutaJanssens, GutaKahn]. The measurement strategy involves two stages. The first stage involves the standard tomographic measurements of the spin components on a vanishing number of copies of the state . A preliminary estimate is constructed from the outcomes. The second measurement stage depends on this preliminary estimate, and for technical purposes related to the asymptotic analysis we consider the following two cases: and for some small constant .

When is close to the fully mixed state, the standard tomographic measurements are performed on the remaining copies of the state. When the preliminary estimate is away from the fully mixed state, a joint measurement is performed on the remaining copies of the state. The joint state has a block-diagonal form following the Weyl decomposition of the underlying space . Information about the eigenvalue parameter is encoded in a probability distribution over the different blocks of the decomposition, while information about the local parameters is encoded in the block states. We consider a parameterisation of states , with , where parametrises the smallest eigenvalue of the states and are certain local rotation parameters around a fixed state . The parameter is then estimated from the outcomes of a “which-block” measurement, while the local parameters are optimally estimated by exploiting the local asymptotic normality of the block states. The LAN results in [GutaJanssens] establish that in the limit , the block states converge to Gaussian states , with displacement proportional to parameters (Theorem 1). The optimal estimator of is then the optimal estimator of the displacement of a Gaussian state , which is known to be the heterodyne measurement. We derive minimax upper and lower bounds for the risk (2), and demonstrate that the maximum Bures risk for the estimator scales as , with being a constant. We obtain lower bound of for this constant, and an upper bound of .

An important element in the derivation of the upper-bounds for the maximum risk of both estimators is the fact that Bures distance can locally approximated as a sum of contributions from the eigenvalue parameter and the ‘rotation’ parameters. More explicitly, we have that

 DB(ρ,^ρn)2≈DH(λ,^λ)2+14(1−2λ)(1−2^λ)√(1−λ)(1−^λ)+√λ^λΦ2,

where with is the classical Hellinger distance, and is the angle between the Bloch vectors of the two states. The optimal rate of estimation of the ‘rotation’ parameters is easily shown to be for all states. The problem of establishing minimax results for the Bures distance therefore converts a problem of establishing minimax results for the Hellinger risk of estimating the eigenvalue parameter. To the best of our knowledge such minimax results for the Hellinger risk are not known. Instead, we upper-bound this risk by the Kullback-Leibler risk, and use known results about the minimax estimator in this case. However, in section 5 we propose that a minimax optimal estimator of the classical parameter under the Hellinger risk gives a minimax optimal estimator for qubit states.

In section 6, we consider the quantum relative entropy (QRE) , and bound the maximum risk under this loss function. As in the case with the Bures distance, an important element is a local decomposition of the QRE risk into contributions from the Kullback-Leibler risk and a term involving the ‘rotation parameters’. Namely, we have that for qubit states

where is the Kullback-Leibler distance between the two distributions and . Using this local decomposition we show that the global estimator we propose achieves a rate of . Furthermore we make the case that no estimator can achieve faster rates, and show that the minimax QRE risk scales as .

The paper is organised as follows. In section 2 we consider an estimator based on local measurements and detail a two step adaptive measurement strategy. We demonstrate that the proposed estimator achieves a minimax rate of . In section 3 we propose a second estimator based on global collective measurements. We begin in section 3.1 by describing the preliminary measurement stage and introduce our parametrisation of states. In section 3.2, we describe the second measurement stage, and overview the block decomposition of the joint state and results of LAN. The minimax bounds for this estimator are derived in section 4, and in section 5 we discuss and state the proposition that a minimax estimator for the Hellinger loss function implies a minimax optimal estimator for qubit states. Finally in section 6 we consider the quantum relative entropy and establish that the minimax rate under this loss function scales as .

## 2 Estimator based on local adaptive measurements

We let be an arbitrary density matrix associated with a single qubit state. Given identical copies of the state as a resource, we wish to construct an estimator of the state. As briefly discussed in the introduction, in this section we propose a two-step adaptive measurement strategy based on local measurements. While the idea of an adaptive local measurement strategy is not new and has been treated in various instances in the literature [GillMassar, RehacekEnglert, BaganBallesterGill, MahlerRozema, HouZhu], we are interested in analysing the performance of the proposed estimator in terms of the maximum risk with respect to the Bures distance defined in equation (2) and its asymptotic rescaled version

 rmax(^ρ)=limsupn→∞supρnE[DB(ρ,^ρ)2]=limsupn→∞nRmax(^ρn).

We will derive an upper bound for the latter risk, thereby demonstrating a scaling for maximum risk over all states . Since the maximum risk of any estimation procedure cannot scale faster than , this implies that the existence of a non-trivial scaling constant for the minimax risk given by

 Rminmax:=limsupn→∞inf^ρnnRmax(^ρn).

Finding the value of the minimax constant remains an open problem. We will come back to this problem in section 5 where it is shown that the minimax qubit estimation problem reduces to that of minimax estimation of a coin probability with respect to the square Hellinger distance risk.

The estimator we propose is constructed as follows. The first stage is a preliminary localisation step involving standard projective measurements of Pauli observables on a fixed fraction of the total number of qubits . An estimate of the direction vector is constructed from the outcomes of these measurements. The following lemma shows that with high probability the estimated directional vector is within an angle of of the true vector, where is a fixed (small) positive constant.

###### Lemma 1.

Let be the outcomes of measurements of performed on independent qubits in state with Bloch vector , where . Let be the estimate of the Bloch vector, where each Bloch vector component is obtained by averaging the outcome results, e.g . Then we have that for ,

 P(∥r−~r∥22>6n−1+2ϵ11)≤6exp(−2n2ϵ113). (3)

The proof of this lemma follows directly from the Hoeffding’s inequality applied to the binomial distribution corresponding to each of the Bloch vector components. The concentration inequality implies that when is bounded away from zero, the magnitude of the angle between the directional vectors and is of the order with high probability.

The second adaptive stage involves preforming measurements along this estimated direction. That is, projective measurements of the observable are performed on the remaining copies of the state. Let be the total number of outcomes from these measurements. It is easy to see that is distributed binomially with binomial parameter . We estimate this parameter as from the measurement outcomes using the ‘add-beta’ estimator [BraessDette, BraessSauer] defined as follows,

 ^pn2=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩1/2n2+5/4,k=0,2n2+7/4,k=1,k+3/4n2+3/2,k=2,…,n2−2,n2−1/4n2+7/4,k=n2−1,n2+3/4n2+5/4,k=n2. (4)

While this estimator is not in any sense optimal, it is known to be the minimax estimator for the Kulback-Leibler risk, which will be used below in deriving the upper bound for qubit tomography.

The final estimate of the state puts together the estimate and the estimated Bloch vector as follows

 ^ρn=12(I+2^pn2−1|~r|~r⋅→σ). (5)

It is easy to see that by construction, where is the eigenvalue of the estimate .

### 2.1 An n−1 scaling upper bound scaling

We now look at deriving an upper bound for the Bures risk of this estimator, and demonstrate that the maximum over all states scales as . Recall that the Bures distance between the final estimate and the true state is defined as , where is the fidelity, expressed in terms of the Bloch vectors and as

 F(ρ,^ρn) =12(1+√1−|r|2√1−|^r|2+r⋅^r) =12(1+√1−|r|2√1−|^r|2+|r||^r|cosΦ), (6)

where is the angle between the Bloch vectors, or equivalently the angle between the vectors and by construction. From Lemma 1, this angle is known to be small and of the order with high probability. This implies that the cosine term in (2.1) can be expanded to leading order in . In this case, the Bures distance is expressed as

 DB(ρ,^ρn)2=DH(λ,^λ)2+14(1−2λ)(1−2^λn)√(1−λ)(1−^λn)+√λ^λnΦ2+O(Φ4) (7)

where are the smallest eigenvalues of and respectively , and

 DH(λ,^λ)2:=(√λ−√^λ)2+(√1−λ−√1−^λ)2

is the square Hellinger distance between the probability distributions and . The proof of this approximation can be found in Appendix A.2.

Identifying , we upper bound the Bures distance as

 DB(ρ,^ρn)2 ≤DH(λ,^pn2)2+14Φ2+O(Φ4) ≤2DH(λ,p)2+2DH(p,^pn2)2+14Φ2+O(Φ4)

where the second inequality is established using the fact that the Hellinger distance satisfies the triangle inequality. Using the inequality and we further upper bound the risk as

 DB(ρ,^ρn)2≤2DH(p,^pn2)2+(14+|r|)Φ2+O(Φ4).

Taking expectation with respect to the measurement outcomes given the true state , we have

 E[DB(ρ,^ρn)2] ≤2E[2DH(p,^pn2)2]+54E[Φ2]+O(n−2+4ϵ11)

The maximum risk of the estimator is therefore bounded from above as

 supρE[DB(ρ,^ρn)2] ≤ supρ2E[DH(p,^pn2)2]+O(n−11)+O(n−2+4ϵ11) (8) ≤ supρ2E[DKL(p,^pn2)]+O(n−11)+O(n−2+4ϵ11)

In the first inequality we upped bounded as ; this follows from the concentration inequality of Lemma 1. The second step employs the inequality between the squared Hellinger distance and the Kullback-Leibler (KL) distance, defined as . The reason for employing this inequality is that to the best of the authors’ knowledge the asymptotic minimax optimal Hellinger risk of estimating the binomial parameter is not known in the literature. A discussion related to the difficulty in obtaining the asymptotic minimax Hellinger risk is left to section 5. However, the minimax optimal rate for the KL loss function under the binomial distribution is known. The minimax optimal estimator is precisely the ‘add beta’ estimator defined in (4) , and is known to achieve the asymptotic rate [BraessDette, BraessSauer]. Choosing to be constant fraction of the total number of samples establishes an overall rate of , for example by choosing .

## 3 Irreducible representations, collective measurements and local asymptotic normality

In this section we propose a two stage estimator of the qubit state, which uses collective rather than separate measurements. The first stage, much like in the pervious section, is a preliminary localisation stage. However, this stage does not use a fixed fraction of the total number of copies of the state, but a vanishing fraction of the overall ensemble of identical qubit . The second measurement stage involves performing joint measurements of the remaining copies of the state, and is based on the techniques of Local Asymptotic Normality (LAN) established in [GutaJanssens, GutaKahn]. This section is structured as follows. In subsection 3.1 we describe the preliminary measurement stage, and define a choice of parameterisation of states. In subsection 3.2 we describe the joint measurement strategy. In subsection 3.3 we provide a brief review of established LAN results and techniques which can be used to derive asymptotic estimation bounds. Minimax results for the proposed estimator are then established in section 4.

### 3.1 Preliminary localisation and parametrisation

The first stage is a preliminary localisation step that involves performing standard projective measurements of the Pauli observables on a vanishing fraction of the overall ensemble of identically prepared qubits. An estimate of the state is constructed from the outcomes of these measurements. The following lemma shows that with high probability the true state lies within a ball of radius of this estimate . This allows us to restrict our attention to a local neighbourhood of the preliminary estimator in the second stage of the estimation.

###### Lemma 2.

Let be independent outcomes of measurements of performed on independent qubits in state with Bloch vector , where . Let be the estimator with Bloch vector obtained by averaging the outcome results, e.g. .

In order to obtain a physical state, the final estimate of the state is constructed as

 ~ρ:=argminτ∈S2∥τ−(1+~r⋅σ)/2∥21 (9)

where the minimisation is over all the space of all density matrices . For this estimator , we have that for all ,

 P(∥~ρ−ρ∥21>3n2ϵ2−1)≤6exp(−2~nn2ϵ2−13),      for all ρ∈S2. (10)

The proof of this lemma follows from an application of Hoeffding’s inequality, and can be found in Appendix A.1. Setting , with , the probability of failure is exponentially small. The preliminary measurement stage therefore places the estimate in a local neighbourhood around the true state. Since , the angle between the two normalised Bloch vectors and is of the order with high probability.

For technical reasons related to local asymptotic normality theory and the derivation of certain error bounds, the subsequent measurement depends on , and we distinguish the following two cases.

i) If is within a fixed but small ball of radius around the fully mixed state (i.e, ), the secondary measurement stage consists of the standard tomographic measurements in the , bases. For each , measurements of are performed on identical copies of the state. The final estimate of the state is constructed from the outcomes of these measurements, and is detailed in section 4.

ii) If is away from the fully mixed state, we can apply the tools of LAN. The remaining copies of the state available for the second stage are rotated such that the estimated Bloch vector is pointing along the -axis. From Lemma 2, the angle between the directional vectors and is known to be of the order with high probability. This allows us to consider a restricted parametrisation of states which we describe now for an arbitrary but fixed state (which plays the role of ) with its Bloch vector along the -axis

 ρ0=(1−λ000λ0) (11)

with . We consider a parametrisation of states obtained by small unitary rotations of , and different choices of the eigenvalue. We choose the parameter vector , where corresponds to the small unitary rotations of the eigenvectors, and is the smallest eigenvalue. That is, any state described by is of the form

 ρθ:=U(w√n)(ccc1−λ00λ)U(w√n)∗ (12)

where the unitary is given by

 U(w√n):=exp(i√n(uσx+vσy))=(cos|w|/√n−exp(−iφ)sin|w|/√nexp(iφ)sin|w|/√ncos|w|/√n)

with . Note that in this parametrisation we have with . The aim of the second measurement stage is then to estimate the unknown parameter vector corresponding to the true state .

### 3.2 The ‘which block’ measurement stage

The second measurement stage involves a joint measurement on the remaining copies of the state. We therefore consider the joint states on identical qubits, with the parametrisation around the preliminary estimator described above. It is known that the states have a block-diagonal form with respect to the decomposition of the underlying space in irreducible representations of the groups and [GutaJanssens, BaganBallester, CiracEkert]. The representation of is given by for any , and the representation of the symmetric group is given by the permutation of factors

 ~π(n)(τ):v1⊗…⊗vn→vτ−1(1)⊗…⊗vτ−1(n),       τ∈S(n) (13)

According to Weil’s Theorem, the following decomposition holds

 (C2)⊗n=n/2⨁j=0,1/2Hj⊗Hjn (14)

where the lower limit in the direct sum is for even and for odd . The two group representations decompose into direct sums of irreducible representations as and where is the irreducible representation of with total angular momentum which acts on , and is the irreducible representation of the symmetric group acting on with

 nj=(nn/2−j)−(nn/2−j−1),          j≠n/2

and . The density matrix is invariant under permutations and can be decomposed as

 ρθn=n/2⨁j=0,1/2pn,λ(j)ρwj,n⊗1nj

where the probability distribution is given by [GutaJanssens, GutaKahn, BaganBallester]

 pn,λ(j):=nj1−2λλn/2−j(1−λ)n/2+j+1(1−p2j+1) (15)

with . The above distribution can be written in the form

 pn,λ(j):=Bn,λ(n/2−j)×K(j,n,λ), (16)

where is the binomial distribution and the term is given by

 K(j,n,λ):=(1−p2j+1)n+(2(j−jn)+1)/(1−2λ)n+(j−jn+1)/(1−λ),     jn:=n(1/2−λ).

The binomially distributed variable concentrates around its mean value of with high probability

 P[nλ−n1/2+ϵ3≤n/2−j≤nλ+n1/2+ϵ3]≥1−2exp(−2n2ϵ3),

where is an arbitrary constant. This follows from a straightforward application of Hoeffding’s inequality (50) to the binomial distribution. The mass of the distribution therefore concentrates over values of in the interval

 Jn:={j| jn−n1/2+ϵ3≤j≤jn+n1/2+ϵ3}. (17)

For all , the factor provided that is bounded away from , which is one of the reasons we chose to treat the two cases above separately. Additionally we note that the factor remains bounded over all values of as long as . From the concentration of the binomial distribution over values of , and the value of in this interval it follows that

 pn,λ(Jn)=1−O(n−1/2+ϵ3). (18)

A “which block” measurement corresponds to an output of a particular value of from the distribution (16), and an associated posterior state . This value of lies in the set with high probability. The eigenvalue parameter is estimated form this value of . As in the case of the local adaptive estimator in section 2, in order to derive a minimax upper bound, we shall define as the ‘add-beta’ estimator, identifying with in (4). However a discussion regarding this choice for the estimator is discussed later in section 5. We note that a possible physical implementation of such a measurement is detailed in [GutaJanssens], and involves coupling the joint states to different bosonic field and performing a homodyne measurement.

Information about the ‘rotation’ parameters are contained in the block state . These parameters are estimated using established LAN results [GutaJanssens] which we recall in section 3.3 below.

### 3.3 Local asymptotic normality

The block state encodes information about the rotation parameters . The optimal estimation strategy for these parameters has been established using results about the LAN of qubit states [GutaJanssens]. This shows that for large , the block states approach a Gaussian state of a one-mode continuous variables system uniformly over all and . The rotation parameters are encoded linearly into the mean of the Gaussian state . So the problem of the optimal estimation of these parameters for the block state can be translated into one of estimating the displacement of . These ideas have been treated in detail in [GutaJanssens], and we only include a brief overview here. The block states depend on the parameters in the following way

 ρwj,n=Uj(w√n)ρ0j,nUj(w√n)∗ (19)

where the unitaries are defined as , with being the generators of rotations in the irreducible representation of SU(2). The state is expressed as

 ρ0j,n=1−p1−p2j+1j∑m=−jpj−m|j,m⟩⟨j,m| (20)

with as before. The set is an orthonormal basis on such that . It has been demonstrated [GutaJanssens] that the family of states is asymptotically Gaussian. This mean that as the family of states “converges” to a family of Gaussian states of a one-mode continuous variables system, for all and . In order to make this convergence more precise, we let

 ϕ0:=(1−p)∑k=0pk|k⟩⟨k| (21)

be a centred Gaussian state of a one mode continuous variables system, with denoting the Fock basis. The states are defined as

 ϕw:=D(√1−2λαw)ϕ0D(−√1−2λαw) (22)

where . The operator is the displacement operator that for every maps the vacuum vector to the coherent state , with being the creation and annihilation operators satisfying . The convergence of the block state to the Gaussian state is formalised in the following Theorem.

###### Theorem 1.

Let be the isometry

 Vj:|j,m⟩→|j−m⟩ (23)

that maps the orthonormal basis of into the Fock basis of . Then for the family of block states defined by (19), and the family of Gaussian states defined by (22), the following convergence holds for any and

 sup∥w∥≤nηmaxj∈Jn∥Vjρwj,nV∗j−ϕw∥1=O(n−1/4+η+ϵ3) (24)

over the set . The convergence is uniform over for an arbitrary fixed .

The interpretation is that the block state can be mapped by means of physical transformations (in this case an isometric embedding) into the Gaussian state with vanishing norm-one error, uniformly over the unknown parameter and over the block index . A possible physical implementation is detailed in [GutaJanssens]; the ensemble of qubits is coupled with a Bosonic field such that the state is transferred to the field after some time.

In order to estimate the rotation parameters , one first maps the qubit state via the isometry , and then performs a heterodyne measurement, which is optimal for estimating displacement. In the next section we discuss the Bures risk of the estimation procedure described above.

## 4 Upper and lower bounds for collective measurements

The overall measurement procedure we propose can be briefly summarised as consisting of a preliminary localisation stage, where a vanishing number of copies of the state is used to localise the state . This estimate informs the choice of measurements in the second stage. When is within a fixed ball of radius around the fully mixed state, standard tomographic measurements are performed on the remaining copies of the state. However, if lies outside this ball, the measurements performed in second stage uses techniques based on the principle of LAN to estimate the parameter vector . In this section we look at the Bures risk of the measurement estimator pair As we will show below, the risk of a good estimators scales as , and we would like to find asymptotic upper and lower bounds for the rescaled maximum risk

 Rmax(^ρ)=limsupn→∞ supρ∈S2 nR(ρ,^ρn).

### 4.1 The upper bound

We now make concrete our final estimator for the state . The first stage involves using a vanishing number of copies (with ) to get a rough estimate . This estimate informs the second measurement stage. The subsequent measurement stage differs depending on whether the state is estimated to be close to the fully mixed state.

If the estimate lies in a small ball of radius around the fully mixed state, then measurements in the standard , basis are performed on copies of the state. The outcomes of each measurement , and the associated probabilities are , where the projectors are defined via . The total number of outcomes obtained by measurements of is binomially distributed . The final estimate of the state is constructed as the maximum likelihood (ML) estimate from these measurement outcomes

 ^ρn=argmaxτ∈S2∑i=x,y,znilogTr(τP+1i)+(n/3−ni)logTr(τP−1i) (25)

where the maximisation is over the space of all density matrices .

On the other hand, if the preliminary estimate lies away from the fully mixed state, we perform the following measurements to estimate the parameter vector . A ‘which block’ measurement outputs a value of from which the eigenvalue is estimated, cf. section 3.2. Similarly to the separate measurements strategy, we consider the following ‘add-beta’ estimator for the eigenvalue [BraessDette, BraessSauer]

 ^λn=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩1/2n+5/4,n2−j=0,2n+7/4,n2−j=1,n/2−j+3/4n+3/2,n2−j=2,…,n−2,n−1/4n+7/4,n2−j=n−1,n+3/4n+5/4,n2−j=n (26)

The range of possible values of is , and therefore only some of the rules of the estimator described above are used. However, we describe the estimator over the range as this will be used shortly to upper bound the minimax risk.

Conditional on , we are left with the block state . Using Theorem 1 we can isometrically map this state onto the Fock space, close to the Gaussian state . In order to estimate the displacement parameter we perform a heterodyne measurement with outcome . The final estimate of our the state is constructed from the estimated parameter vector as

 (27)

We now state precisely the an upper bound for the minimax risk of the Bures distance for the measurement strategy described above.

###### Theorem 2.

Let be the estimator described above. The asymptotic rescaled maximum risk is bounded from above as

 limsupn→∞ supρ nR(ρ,^ρn)≤32. (28)

The proof of this theorem is detailed in the appendix (A.3), and here we only provide an outline for the arguments employed. The choice of measurements in the second stage depend on whether the preliminary estimate lies inside or outside a small ball of radius around the fully mixed state. In keeping with this, let us therefore denote as the estimator (25) chosen when , and let be the LAN based estimator (27), when . The minimax risk can then be bounded from above as follows

 limsupn→∞ supρ nR(ρ,^ρn)≤max{limsupn→∞supρ∈B1nE[DB(ρ,^ρ1n)2∣∣ |~r|≤δ],limsupn→∞supρ∉B2nE[DB(ρ,^ρ2n)2∣∣ |~r|>δ]} (29)

where and are balls of radius and respectively. The two terms are evaluated explicitly in section (A.3) of the appendix. The term corresponding to the estimator is straightforward to evaluate as the Bures distance is locally quadratic for states in . From this quadratic expansion and the efficiency of the maximum likelihood estimator in the asymptotic regime, the risk can be expressed as

 E[DB(ρ,^ρ1n)2]≈1nTr(I(ρ)−1G) (30)

where is the weight matrix reconstructing the quadratic approximation of the Bures distance, and is the Fisher information matrix. From the explicit form of and , the asymptotic risk can be bounded as

 limsupn→∞ supρ∈B1nE[DB(ρ,^ρ1n)2]≤34(1+δ1−δ). (31)

The other term in (29) corresponding to the estimator uses the local parameterisation of states , and the approximation of the Bures distance used in section 2, and detailed in the appendix (A.2). We therefore get that the risk can be bounded as

 E[DB(ρ,^ρ2n)2] ≤E[DH(λ,^λn)2+14Φ2]+O(
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