Uncertainty relation for the position of an electron in a uniform magnetic field from quantum estimation theory
UUncertainty relation for estimating the position of an electron in a uniform magneticfield from quantum estimation theory
Shin Funada and Jun Suzuki
Graduate School of Informatics and Engineering, The University of Electro-Communications,1-5-1 Chofugaoka, Chofu-shi, Tokyo, 182-8585 Japan (Dated: July 16, 2020)We investigate the uncertainty relation for estimating the position of one electron in a uniformmagnetic field in the framework of the quantum estimation theory. Two kinds of momenta, canonicalone and mechanical one, are used to generate a shift in the position of the electron. We firstconsider pure state models whose wave function is in the ground state with zero angular momentum.The model generated by the two-commuting canonical momenta becomes the quasi-classical model,in which the symmetric logarithmic derivative quantum Cram´er-Rao bound is achievable. Themodel generated by the two non-commuting mechanical momenta, on the other hand, turns outto be a Gaussian model, where the generalized right logarithmic derivative quantum Cram´er-Raobound is achievable. We next consider mixed-state models by taking into account the effects ofthermal noise. The model with the canonical momenta now becomes genuine quantum mechanical,although its generators commute with each other. The derived uncertainty relationship is in generaldetermined by two different quantum Cram´er-Rao bounds in a non-trivial manner. The model withthe mechanical momenta is identified with the well-known Gaussian shift model, and the uncertaintyrelation is governed by the right logarithmic derivative Cram´er-Rao bound.
I. INTRODUCTION
The uncertainty relation based on the quantum estimation theory was investigated by many authors, see forexample [1–8]. It is known that one-parameter unitary model with a pure reference state, the Heisenberg-Robertsontype uncertainty relation and the uncertainty relation by the parameter estimation have the same form. Further,this type of approach is more general than the traditional one, since one can derive the uncertainty relation for non-observables. The celebrated energy-time uncertainty relation [9] is a well-defined relation for time and energy whentreated by the quantum estimation theory. In literature, many authors discussed similarity between two-differenttypes of uncertainty relations. In Ref. [6], they showed that the uncertainty relation for a generic full parameter quditmodel can be different when derived from the quantum parameter estimation theory. Usually, when the uncertaintyrelation is discussed, the uncertainty relation of two non-commuting observables is discussed, see for example [10–12].The aim of this paper is to investigate the uncertainty relation between two communing observables based on themulti-parameter quantum estimation theory [2, 13–16]. At first sight, one might expect that there cannot be sucha trade-off relation. However, as demonstrated in this paper, the quantum estimation theory enables us to derivea non-trivial trade-off relation for estimating the expectation values of two commuting observables. In the presentwork, we set up a specific physical model, a model of one electron in a uniform magnetic field and investigate theuncertainty relation regarding the position of the electron by the parameter estimation problem of two-parameterunitary model. In this model, the Heisenberg-Robertson type uncertainty relation [17, 18] of the position operators
X, Y of an electron only yields the following trivial inequality.(∆ X )(∆ Y ) ≥ |(cid:104) [ X, Y ] (cid:105) ρ | = 0 . (1)This is because two position operators X and Y commute, i.e., [ X, Y ] = 0. In the relation above, ∆ X denotes the(quantum) standard deviation about X with respect to a state ρ , which is defined by(∆ X ) = tr [ ρ ( X − (cid:104) X (cid:105) ρ ) ] = (cid:104) X (cid:105) ρ − (cid:104) X (cid:105) ρ , (2)with (cid:104) X (cid:105) ρ = tr [ ρX ] the expectation value of X . ∆ Y is defined similarly.In order to derive the uncertainty relation between X and Y , we need to introduce a parametric model describingthe position measurement of the electron. We use the unitary transformation generated by the canonical momenta p x and p y with the parameter θ = ( θ , θ ). The state ρ pθ generated by this transformation from the reference state ρ , which is known in advance, is defined asModel 1: ρ pθ = e − i θ p x e − i θ p y ρ e i θ p y e i θ p x . (3) a r X i v : . [ qu a n t - ph ] J u l Using the generators p x and p y , the expectation values of the position operators are (cid:104) X (cid:105) θ = (cid:104) X (cid:105) + θ , (4) (cid:104) Y (cid:105) θ = (cid:104) Y (cid:105) + θ . (5)where (cid:104) X (cid:105) θ = tr [ ρ pθ X ] and (cid:104) X (cid:105) = tr [ ρ X ]. We define (cid:104) Y (cid:105) θ and (cid:104) Y (cid:105) similarly. From Eqs. (4, 5), we see thatestimating the parameters θ and θ amounts to the measurement of the position X and Y . It is worth noting thatthe generators of Model 1 also commute, i.e., [ p x , p y ] = 0.As the main contribution of this paper, we derive an uncertainty relation, or a trade-off relation between thecomponents of mean square error (MSE) matrix by using two different types of the quantum Cram´er-Rao (C-R)inequalities for Model 1. In particular, we find a structure change in the uncertainty relation and derive a conditionfor this transition analytically.We can generate another shift model of the position density probability that gives the same relation as Eqs. (4, 5).That is Model 2: ρ πθ = e − i θ π x e − i θ π y ρ e i θ π y e i θ π x , (6)where (cid:126)π = (cid:126)p + e (cid:126)A . The vector potential for the uniform field (cid:126)B is denoted by (cid:126)A . The charge of an electron is − e ( e > H of this system has an equivalent form of the harmonic oscillator with respect to the generators π x , π y [19]. Both Model 1 and Model 2, therefore, make a shift in the position of the position probability densitywhich is defined by the product of the wave function and its complex conjugate. However, there is a significantdifference between these two models. The generators of Model 2 do not commute, [ π x , π y ] = − i eB unlike those ofModel 1. Furthermore, Model 2 ρ πθ defined by (6) turns out to be a displaced Gaussian state when ρ is a thermalstate.The outline and the summary of this paper are as follows. In Sec. II A, the Hamiltonian of the system is givenin terms of the creation and annihilation operators. In Sec. II C, we explain how the position measurement of theelectron can be set up as a two-parameter estimation problem. In Sec. II D, we derive the uncertainty relation for theMSE matrix for arbitrary two-parameter estimation problem from the quantum C-R inequality.In Sec. III, we investigate the trade-off relation by estimating the position of electron with respect to the referencestate which is a pure state. As the reference state, we choose the lowest energy state with zero angular momentum, orthe lowest Landau level (LLL) [20]. The position probability density of the LLL is known to be a Gaussian function ∝ exp[ − ( x + y ) /λ ] where λ = (cid:112) eB ) − has the dimension of length. We obtain the uncertainty relation fromthe symmetric logarithmic derivative (SLD) C-R inequality for the MSE matrix, which cannot be less than λ / (cid:104) L (cid:105) is fixed. Under this circumstance, for Model 1with the canonical momenta, p x and p y as generators, we see the following three results which are our main claims ofthis paper. (1) We see a trade-off relation, or an uncertainty relation for joint estimation of the expectation values oftwo commuting observables. (2) The trade-off relation, or the uncertainty relation is determined by both of the RLDand the SLD C-R bounds. Therefore the bound has a non-trivial structure. (3) The RLD and SLD C-R bounds havewhether no or two intersections depending on the fixed expectation value of the angular momentum, (cid:104) L (cid:105) . Therefore,a transition occurs in the shape of the uncertainty relation depending on the value of (cid:104) L (cid:105) .For Model 2 with the mechanical (kinetic) momenta, π x and π y as genrators, we see another trade-off relationalthough the observables commute. In contrast, Model 2 is turned out to be a simple Gaussian shift model whichwas well-studied [2, 24]. Therefore, the model is D-invariant and the RLD C-R bound is an achievable bound. In thecase of the thermal state as the reference state also, Model 2 potentially gives a more precise position measurementby estimating the parameters shift generated by the mechanical (kinetic) momentum. The supplement and thecalculations are given in A and B, respectively.Throughout the paper, we use the natural units, where we set c = 1 (the speed of light), (cid:126) = 1 (the Plank constant),and k B = 1 (the Boltzmann constant) unless otherwise stated. II. PRELIMINARIESA. Hamiltonian
The Hamiltonian H for an electron motion in a uniform magnetic field is H = 12 m ( (cid:126)p + e (cid:126)A ) . (7)where − e and m are the charge of an electron ( e > (cid:126)A is a vectorpotential. In the following discussion, we use the coordinate representation of operators. The canonical observablesdescribing this systems are p x , x, p y , and y . We will investigate the uncertainty relation of an electron motion in auniform magnetic field (cid:126)B = (0 , , B ) , B >
0. We use the symmetric gauge. Hence the vector potential is written as (cid:126)A = B ( − y/ , x/ , x − y plane only, because z component solution is a plane wave. With a new vectoroperator, (cid:126)π = (cid:126)p + e (cid:126)A , our Hamiltonian becomes [19] H = 12 m ( π x + π y ) . (8)Here we remark that these mechanical (kinetic) momenta satisfy the canonical commutation relation up to a constantfactor: [ π x , π y ] = − ieB [19]. They together with the guiding center operators are the fundamental observables inthe study of electrons in strong magnetic fields, see for example [25].It is known that the operators x, y and p x , p y are equally described by the two sets of the creation and annihilationoperators, acting on the different Fock spaces, a, a † and b, b † such that [ a, a † ] = [ b, b † ] = 1 with all other commutationrelations vanishing [26].The canonical momenta p x , p y and the position x, y in Eq. (7) are expressed as p x = i2 λ (cid:2) ( a † − a ) + ( b † − b ) (cid:3) , p y = 12 λ (cid:2) ( a † + a ) − ( b † + b ) (cid:3) , (9) x = λ (cid:2) ( a † + a ) + ( b † + b ) (cid:3) , y = − i λ (cid:2) ( a † − a ) − ( b † − b ) (cid:3) . (10)The mechanical momenta π x , π y in Eq. (8) are expressed as π x = i λ ( a † − a ) , π y = 1 λ ( a † + a ) . (11)(12)where λ = (cid:112) eB ) − has the dimension of length. As shown in Eq. (17) below, λ corresponds to the spread of theprobability density of the electron in the LLL.The Hamiltonian H and z component of the angular momentum L are expressed in terms of the two harmonicoscillators as H = ω ( a † a + 12 ) , (13) L = xp y − yp x = a † a − b † b, (14)where ω = eB/m is the cyclotron frequency. B. States
As the states on which the operators a, a † and b, b † act, the number states | n (cid:105) a and | n (cid:105) b that satisfy a † a | n (cid:105) a = n | n (cid:105) a , b † b | n (cid:105) b = n | n (cid:105) b , (15)are often used. The number states | (cid:105) a and | (cid:105) b are the vacuum states of the harmonic oscillators.Since the Hamiltonian H does not include b, b † , its energy eigenstate consists of infinite number of the angularmomentum eigenstates Eq. (14), i.e, the energy eigenstate is degenerated. We choose the state with the energy ω/ | , (cid:105) := | (cid:105) a | (cid:105) b from Eqs. (13, 14).The wave function of this state is known as the LLL, ψ ( x, y ), which is expressed as ψ ( x, y ) = (cid:104) x, y | , (cid:105) = C e − x y λ . (16)where C is the normalization factor. Then, the position probability density | ψ ( x, y ) | is | ψ ( x, y ) | ∝ e − x y λ . (17)This is a Gaussian distribution with its spread λ and with its peak at ( x, y ) = (0 , C. Estimation of the position
The unitary transformations of Model 1 and Model 2 make a shift in the position probability density of the electronby θ = ( θ , θ ). From Eqs. (4, 5), we have θ = (cid:104) x (cid:105) θ − (cid:104) x (cid:105) and θ = (cid:104) y (cid:105) θ − (cid:104) y (cid:105) . Then, the shifted state from thereference state has a sharp peak at ( x, y ) = ( θ , θ ). Therefore, estimating (cid:104) x (cid:105) θ and (cid:104) y (cid:105) θ is equivalent to infer theshift parameters θ = ( θ , θ ). (Under the assumption that we know in advance the expectation value of the positionoperators with respect to the reference state ρ .) We estimate the unknown parameters θ and θ by making arbitrarymeasurement, which is unbiased. We then infer the two parameters from the measurement result. We shall use theMSE matrix to measure the estimation accuracy of the position of the electron. D. Uncertainty relation by quantum C-R inequality
In order to derive the uncertainty relation from the MSE matrix for inferring the position of the electron basedon the quantum estimation theory, the following two factors are essential to formulate the problem: i) Choice of thereference state and ii) generators for the shift in the position of the election. In this paper, we first consider a purereference state, which is the vacuum state of the two harmonic oscillator. Physically, this state is the energy groundstate with zero angular momentum. We then consider a mixed reference state affected by the thermal noise. For thegenerators of unitary transformations, the most natural choice is the canonical momenta p x , p y . We call a parametricfamily of the states generated by them as Model 1 [Eq. (3)]. The other choice of the generator is the mechanicalmomenta π x , π y , and we call this family as Model 2 [Eq. (6)].We next derive the uncertainty relation from the quantum C-R inequality. Consider a general two-parameter modelof which quantum Fisher information matrix is G θ . The quantum C-R inequality then, bounds the MSE matrix V θ = [ V ij ] as V θ ≥ ( G θ ) − . In A 1, we derive the following inequalities: V − g θ ≥ , V − g θ ≥ , (18)( V − g θ )( V − g θ ) ≥ | Im g θ | . (19)where ( G θ ) − = [ g θij ]. We regard these inequalities as a trade-off relation, or an uncertainty relation for estimatingthe two parameters θ = ( θ , θ ). In contrast to the Heisenberg-Robertson type uncertainty relation, the commutationrelationship between two observables do not appear explicitly in the expression above. This is why we can derivea non-trivial uncertainty relation for estimating the position of the electron in our models with the commutingobservables.Note that when the imaginary part of the quantum Fisher information matrix vanishes, i.e., Im g θ = 0, theuncertainty relation is given by Eq. (18) only. In this case, we do not have any trade-off relation between V and V .In the remaining of the paper, we consider the SLD and the RLD quantum Fisher information matrices. But ourformulation can be extended to any quantum Fisher information matrices. E. RLD and SLD Fisher information matrices, generalized RLD information matrix, Z matrix
1. RLD L R , i ( θ ) and RLD Fisher information matrix: G R ( θ ) RLD L R , i ( θ ) is given as a solution of the equation below if one exists. ∂ρ θ ∂θ i = ρ θ L R , i ( θ ) . The RLD Fisher information matrix G R ( θ ) = [ g R , ij ( θ )] is defined by g R , ij ( θ ) = tr [ ρ θ L R , j ( θ ) L † R , i ( θ )] . (20)
2. SLD L S , i ( θ ) and SLD Fisher information matrix: G S ( θ ) SLD, L S , i ( θ ) is also given as a solution of the equation below if one exists. ∂ρ θ ∂θ i = 12 [ ρ θ L S , i ( θ ) + L S , i ( θ ) ρ θ ] . (21)SLD Fisher information matrix G S ( θ ) = [ g S , ij ( θ )] is defined by g S , ij ( θ ) = Re tr [ ρ θ L S , j ( θ ) L S , i ( θ )] .
3. Generalized RLD
In general, the RLD does not exist when a pure state is the reference state [21]. We can show that this holds forModel 1 and Model 2. Instead of the RLD Fisher information matrix, we are able to obtain the generalized RLDFisher information matrix by the method introduced by [21]. Let the generalized RLD Fisher information matrix ˜ G R be ˜ G R = [˜ g R , ij ] , Then, ˜ g R , ij = 4( (cid:104) ∂ i ψ | ∂ j ψ (cid:105) + (cid:104) ψ | ∂ i ψ (cid:105) (cid:104) ψ | ∂ j ψ (cid:105) ) (22)
4. Z matrix L S i ( θ ) is defined by L S i ( θ ) = (cid:88) j g S ji ( θ ) L S , j ( θ ) . where G − ( θ ) = [ g S ij ( θ )]. Then, Z matrix, Z ( θ ) = [ z ij ( θ )] is defined by z ij ( θ ) = tr [ ρ θ L j S ( θ ) L i † S ( θ )] . It is worth noting the relationship between Z matrix and the expectation value of the commutator of SLD’s,tr( ρ [ L S , i ( θ ) , L S , j ( θ )]) [27]. By using the ( i, j ) component of the Z matrix, z ij , we can write the expectation valueof the commutator [ L i S ( θ ) , L j S ( θ )] astr( ρ [ L S ,i , L S ,j ]) = (cid:88) k,(cid:96) g S ,ki (cid:0) z k(cid:96) − ( z k(cid:96) ) ∗ (cid:1) g S ,(cid:96)j = 2i (cid:88) k,(cid:96) g S ,ki Im( z k(cid:96) ) g S ,(cid:96)j . (23)In particular, the expectation value of the commutator is proportional to the imaginary part of Z matrix when theSLD Fisher information matrix is diagonal. III. PURE STATE MODEL
In this section, we first consider an ideal situation, where the reference states are given by a pure state. The deriveduncertainty relation for Model 1 is understood intuitively, since the model is two-independent unitary model. Model2, which is generated by the two non-commuting generators, gives a non-trivial uncertainty relation.
1. Reference state
Since the energy eigenstate of Hamiltonian (8) is infinitely degenerated, we choose the tensor product of thevacuum states as the reference state ρ which is denoted by ρ = | (cid:105) a a (cid:104) | ⊗ | (cid:105) b b (cid:104) | = | , (cid:105) (cid:104) , | . (24)
2. Unitary transformations
We introduce two kinds of unitary transformations, e − i θ p x e − i θ p y and e − i θ π x e − i θ π y . We consider that we havethem act on the LLL, ψ ( x, y ). Since we havee − i θ p x e − i θ p y ψ ( x, y ) = ψ ( x − θ , y − θ ) , (25)this unitary transformation e − i θ p x e − i θ p y makes a shift in x − y coordinate of ψ ( x, y ) from ( x, y ) to ( x − θ , y − θ ).We also have e − i θ π x e − i θ π y ψ ( x, y ) = e i θ θ λ e − i xθ λ e i yθ λ ψ ( x − θ , y − θ ) , (26)where we use the standard Baker-Campbell-Hausdorff formula [28]. Because the difference between Eqs. (25, 26) isonly in the phase shift of the wave function, the unitary transformations e − i θ π x e − i θ π y and e − i θ p x e − i θ p y give thesame effect to the probability density, | ψ ( x, y ) | , i.e., both of them make the shift as follows. | e − i θ p x e − i θ p y ψ ( x, y ) | = | ψ ( x − θ , y − θ ) | , | e − i θ π x e − i θ π y ψ ( x, y ) | = | ψ ( x − θ , y − θ ) | . (27)and thus, we have in both cases, | ψ ( x − θ , y − θ ) | ∝ exp (cid:20) − ( x − θ ) + ( y − θ ) λ (cid:21) . (28)That is, the position probability density is now centered at ( θ , θ ) with its spread λ . A. Uncertainty relation
It is known that the RLD does not exist in general when the reference state is a pure state. In Ref. [29], theauthors showed that the SLD can be defined by taking equivalent classes of the inner product, thereby the SLDFisher information matrix exists uniquely. In later work [21], they also showed that the generalized RLD Fisherinformation matrix exists for a special class of pure-state models, called a coherent model. In our case, Model 1 turnsout to be a quasi-classical model meaning that the SLD Fisher information matrix plays the same role as the classicalFisher information matrix. Whereas Model 2 is shown to be a coherent model, and hence we can derive the quantumC-R inequality based on the generalized RLD Fisher information matrix.
1. Model 1: Unitary model generated by p x and p y In Model 1, the generators, p x and p y commute. We can show that the SLD’s commute on the support of the statesand that the SLD C-R bound is achievable [30].The SLD and the generalized RLD Fisher information matrices are calculated by the way given in [29]. Since theFisher information matrices of the unitary models do not depend on θ , we omit θ for simplicity. The SLD Fisherinformation matrix with respect to the reference state ρ is denoted by G p S . Then, its inverse is calculated in B 2 a as( G p S ) − = λ (cid:18) (cid:19) , From Eq. (18), we obtain V ≥ λ , V ≥ λ . We next calculate the the generalized RLD Fisher information matrix ˜ G p R to find out that the off-diagonal compo-nents of ˜ G p R are zero and that G p S = ˜ G p R for Model 1. Then, ( G p S ) − = ( ˜ G p R ) − = Z holds even though this modelis not coherent. As is explained in the following section, we thus use the generalized RLD for Model 2 only. Thecalculation is shown in B 2 a. Figure 1 shows the SLD C-R bound (dotted lines). λ /
2. Model 2: Unitary model generated by π x and π y Let G π S denote the SLD Fisher information matrix of Model 2 with respect to the reference state ρ . Then, theinverse of SLD Fisher information matrix ( G π S ) − is calculated in B 2 b as( G π S ) − = λ (cid:18) (cid:19) . Notably, the relation ( G p S ) − = 2( G π S ) − holds. This difference, a factor of two results from the difference in thecoefficients in Eqs. (9, 11).Let ˜ G π R denote the generalized RLD Fisher information matrix. The generalized RLD C-R bound [21] is given by V θ ≥ ( ˜ G π R ) − , where ( ˜ G π R ) − = λ (cid:18) − i 1 (cid:19) . (29)The derivation of (29) is given in B 2. By using Eq. (19), we obtain the following inequality,( V − λ V − λ ≥ λ . (30)Figure 1 shows the SLD C-R bound (dashed line) and the generalized RLD C-R bound (solid line). Since thegenerators, π x and π y consist of a, a † only [Eq. (11)], this is a Gaussian shift model and the generalized RLD bound isachievable [21]. Unlike the result of Model 1, the uncertainty relation for Model 2 exhibits a trade-off relation between V and V . This comes from the nature of Model 2 which is purely quantum mechanical. B. Discussion
There are two significant differences between the C-R bounds given by Model 1 and Model 2 even though theunitary transformations of Model 1 and Model 2 make the same shift in the position of the probability density asshown in Eq. (27).First, the SLD and the generalized RLD C-R bounds of Model 2 is lower than the SLD C-R bound of Model1. In particular, the SLD C-R bound of Model 2 is a half of that of Model 1. At first sight, this difference inestimation accuracy might puzzle us, since two models displace the same amount in the position. However, there isno inconsistency in our models, and the simple answer is given as follows. Note that the generators for Model 2 shifttwice of Model 1 as in Eqs. (9, 11) in the parameter space. This results in the larger quantum Fisher information ofModel 2. The measurement accuracy appears to be better in Model 2 only because Model 2 shifts more.Second, Eq. (30) gives the achievable bound of Model 2 [21]. The relation between V and V in the right handside of Eq. (30) is not just a product of V and V unlike the Heisenberg-Robertson type uncertainty relation. Thedifference in the bound between Model 1 and Model 2 is caused by the phase shift of Model 2 in Eq. (26). Althoughthis phase shift in Eq. (26) makes no change in the position probability density, it does make a change in the quantumFisher information matrices. Model 1 SLD Model 2 SLD Model 2 generalized RLD λ - V λ - V FIG. 1: The uncertainty relation of Model 1 and Model 2 given by the inequalities Eqs. (18, 19). The allowed region of Model2 for the MSE matrix components ( V , V ) is above the solid line, the dark gray region and the blue region given by theinequality (30) which is derived from the generalized RLD Cram´er-Rao inequality. The allowed region of Model 2 by the SLDC-R inequality consists of the light gray, dark gray and blue regions. The allowed region of Model 2 by the SLD Cram´er-Raoinequality is the blue region. IV. MIXED STATE MODEL: EFFECT OF THERMAL NOISE
Next, we use a mixed state as the reference state to see how the noise affects the measurement accuracy of theelectron position. For this purpose, as the mixed state, we choose the thermal state. However, in the current systemwe are considering, there is no unique thermal state, because the energy eigenstate is degenerated. Then, the thermalstate of this system is not uniquely specified by the temperature only. To resolve this degeneracy problem, we imposea condition that the expectation value of the angular momentum (cid:104) L (cid:105) is fixed. This is done by introducing a chemicalpotential. A. Reference state
Given (cid:104) L (cid:105) is fixed at a constant, the reference state ρ β, µ is denoted by ρ β, µ = Z − β, µ e − βH + µL , (31)where β = T − is the inverse temperature and Z β, µ = tr [exp( − βH + µL )] is the partition function. The parameter µ is the chemical potential, which will be determined later. The role of the chemical potential µ is to keep (cid:104) L (cid:105) constantto avoid complications by the degeneracy of angular momentum. The use of the chemical potential here is the sameidea as seen in the grand canonical ensemble of statistical physics where the chemical potential is used to keep theexpectation value of the number of particles constant.From Eqs. (13, 14), ρ β, µ = Z − β, µ e − βω e − ( βω − µ ) a † a − µb † b . (32)By using the Gaussian states which are defined by a | z (cid:105) a = z | z (cid:105) a , b | z (cid:105) b = z | z (cid:105) b , (33)the reference state ρ β, µ is expressed as ρ β, µ = ρ , a ⊗ ρ , b , (34)where ρ , a and ρ , b are the thermal states with different temperatures. Explicitly, they are ρ , a = 12 πκ (cid:90) e − | z | κ | z (cid:105) a a (cid:104) z | d z,ρ , b = 12 πκ (cid:90) e − | z | κ | z (cid:105) b b (cid:104) z | d z, βω βω βω -2(cid:0) (cid:1)
10 0 10 (cid:2)(cid:3) (cid:4)3 L > μ FIG. 2: The chemical potential µ as a function of the expectation value of the angular momentum (cid:104) L (cid:105) at three differenttemperature parameters βω = 0 . ,
1, and 5. At lower βω i.e., higher temperature, µ becomes closer to zero, no preference forthe angular momentum. with 2 κ = (e βω − µ − − , κ = (e µ − − . (35)The derivation of Eqs. (34, 35) is given in A 2. It is straightforward to calculate the expectation value (cid:104) L (cid:105) as (cid:104) L (cid:105) = tr [ L ρ β, µ ] = 2 κ − κ . (36)From Eqs. (35, 36), we obtain( (cid:104) L (cid:105) + 1)e µ − (cid:104) L (cid:105) (e βω + 1)e µ + ( (cid:104) L (cid:105) − βω = 0 . (37)When βω and (cid:104) L (cid:105) are given, µ is the variable of Eq. (37). If (cid:104) L (cid:105) = − (cid:104) L (cid:105) (cid:54) = −
1. However, one of them is shown to be unphysical giving a negative temperaturestate in the later case. Then, the chemical potential µ as a function of (cid:104) L (cid:105) and βω is found to bee µ = βω e βω + 1 ( (cid:104) L (cid:105) = − (cid:104) L (cid:105) + 1) (cid:104) (cid:104) L (cid:105) (e βω + 1) + (cid:113) (cid:104) L (cid:105) (e βω − + 4e βω (cid:105) ( (cid:104) L (cid:105) (cid:54) = − . (38)Although the solution of Eq. (37) has a singular point at (cid:104) L (cid:105) = − (cid:104) L (cid:105) (cid:54) = − (cid:104) L (cid:105) = −
1. We can also show that the firstderivative is continuous at (cid:104) L (cid:105) = − µ as a function of (cid:104) L (cid:105) at βω = 0 . , , and 5 from top to bottom. The chemical potential µ asa function of (cid:104) L (cid:105) diverges for (cid:104) L (cid:105) ≥ βω goes to infinity, i.e., the zero temperature limit. At a special case, (cid:104) L (cid:105) = 0, we see µ = βω/ β →∞ µ = ∞ ( (cid:104) L (cid:105) ≥ (cid:104) (cid:104) L (cid:105) − (cid:104) L (cid:105) (cid:105) ( (cid:104) L (cid:105) < . (39)For Model 2, the two-parameter family of the states ρ πθ is expressed as ρ πθ = e − i θ π x e − i θ π y ρ , a ⊗ ρ , b e i θ π y e i θ π x , (40)from Eqs. (6, 34). Since π x and π y consist of a and a † only, ρ π is described as ρ πθ = e − i θ π x e − i θ π y ρ , a e i θ π y e i θ π x ⊗ ρ , b . (41)0Therefore, ρ , b gives no effect to the quantum Fisher information matrices. The reference state ρ β, µ for Model 2, weonly need to use ρ , a . By construction, the family of the states: ρ θ, a = e ξa † − ξ ∗ a ρ , a e ξ ∗ a − ξa † , with ξ = (2 λ ) − ( θ − i θ ), is a Gaussian shift model [2, 24]. It is then known that the RLD C-R bound provides theachievable bound [2, 24]. B. Uncertainty relation
For the mixed-state model, we can calculate the SLD and the RLD C-R bounds. They then provide the uncertaintyrelation for the MSE matrix. The calculations of SLDs and RLDs and their quantum Fisher information matrices aregiven in B 1.
1. Model 1: Unitary model generated by p x and p y Let G p thermalR and G p thermalS be the RLD and the SLD Fisher information matrices with respect to ρ β, µ , respectively.We introduce g R ij and g S ij such that ( G p thermalR ) − = [ g R ij ] , (42)( G p thermalS ) − = [ g S ij ] . (43)The inverse of G p thermalR is calculated as( G p thermalR ) − = λ κ + 2 κ (cid:18) κ + 2 κ + 8 κ κ i (2 κ − κ ) − i (2 κ − κ ) 2 κ + 2 κ + 8 κ κ (cid:19) . From Eq. (19), we have the following inequality,( V − g R11 )( V − g R11 ) ≥ λ (cid:18) κ − κ κ + 2 κ (cid:19) . (44)Next, the calculation of the inverse of G p thermalS reveals that ( G p thermalS ) − is a diagonal matrix and that g S11 is equalto g S22 . ( G p thermalS ) − is written as ( G p thermalS ) − = (cid:18) g S11 g S22 (cid:19) , (45)where g S11 = g S22 = λ + 2 κ + 2 κ + 8 κ κ κ + 2 κ . From Eq. (18), we have V ≥ g S11 , V ≥ g S11 . (46)There are two cases regarding the ordering between the inverse of RLD and SLD Fisher matrices in terms of thematrix inequality.Case i). When |(cid:104) L (cid:105) | ≤ /
2, the SLD C-R bound defines a tighter lower bound. This is because the matrixinequality ( G p thermalS ) − − ( G p thermalR ) − = ∆ g (cid:18) − (cid:104) L (cid:105) (cid:104) L (cid:105) (cid:19) ≥ , holds if and only if |(cid:104) L (cid:105) | ≤ / g is defined by∆ g := g S11 − g R11 = λ κ + 2 κ > . (47)1 Model 1
SLD
Model 1
RLD
Model 2
RLD
Model 2 S LD λ - V λ - V FIG. 3: Uncertainty relation of Model 1 and Model 2 given by the quantum Cram´er-Rao inequalites. The temperatureparameters used are κ = 1 , κ = 1 / , (cid:104) L (cid:105) = 1, i.e., (cid:104) L (cid:105) > /
2. The allowed region of Model 1 for the MSE matrixcomponents ( V , V ) is the blue region. The allowed region of Model 1 is given by the SLD Cram´er-Rao bound (blue dottedlines) and the RLD C-R bound (blue solid line). The allowed region of Model 2 for the MSE matrix components ( V , V ) iscovered by the gray and blue region. The RLD Cram´er-Rao bound (black solid line) is achievable. ∆ V R − S [Eq. (48)] is thedistance between the blue square in the figure and the intersection of SLD Cram´er-Rao bounds (blue dotted lines). βω βω βω -4 (cid:0)2 (cid:1) (cid:2)(cid:3) L > (cid:4) Δ V R - S FIG. 4: ∆ V R − S [Eq. (48)] as a function of (cid:104) L (cid:105) . By the definition of ∆ V R − S , when ∆ V R − S is negative, the bound is determinedby the SLD Cram´er-Rao bound only. Case ii). In the other case, |(cid:104) L (cid:105) | > /
2, however, there is no matrix ordering between the RLD and the SLDFisher information matrices. This means that both inequalities (44) and (46) contribute to the uncertainty relation.Figure 3 shows an example of the bound given by the current analysis with |(cid:104) L (cid:105) | > /
2. The parameters used are κ = 1 , κ = 1 /
2, and thus |(cid:104) L (cid:105) | = 1 > / V , V ). The RLD and the SLD C-R bounds have twointersection points in this case. Let the position of one of the intersection points be ( V R − S11 , g
S11 ) which is markedas the dot in Fig. 3. The RLD C-R bound defines the bound in the region where g S11 < V < V R − S11 . The SLDC-R bound defines in the region where V > V R − S11 and V > V R − S11 . We define ∆ V R − S by ∆ V R − S = V R − S11 − g S11 .Then, ∆ V R − S is ∆ V R − S = ∆ g (4 (cid:104) L (cid:105) − . (48)Figure 4 shows ∆ V R − S as a function of (cid:104) L (cid:105) at three different βω ’s which are the same as Fig. 2.When |(cid:104) L (cid:105) | ≤ /
2, ∆ V R − S is negative as shown in Eq. (48), the RLD C-R bound stays always below the SLD C-Rbound. This is consistent with ( G S ) − ≥ ( G R ) − when |(cid:104) L (cid:105) | ≤ /
2. At larger βω (lower temperature), the possibleranges of V R − S11 and V R − S22 given by the RLD C-R bound becomes larger at the same (cid:104) L (cid:105) .Finally, we briefly discuss achievability of the uncertainty relation above. It is known that the RLD C-R bound is(asymptotically) achievable, if and only when the model is D-invariant [27]. This condition is checked by comparing2two matrices, the inverse of the RLD Fisher information matrix and the Z matrix. As given in B, ( G p thermalR ) − andthe Z matrix Z p thermal are different. Hence, the RLD C-R bound is not tight. We next examine if the SLD C-Rbound is achievable or not. In Refs. [31, 32], the necessary and sufficient conditions are derived for asymptoticallyachievability of the SLD C-R bound. The simplest condition is that the imaginary part of the Z matrix is zero. Inour model, this is equivalent to (cid:104) L (cid:105) = 0 which is also equivalent to κ a = κ b [Eq. (36)]. When (cid:104) L (cid:105) (cid:54) = 0, neither theRLD C-R bound nor SLD C-R bound is even asymptotically achievable. Therefore, the uncertainty relation is nottight, except for the special choice of the parameter, (cid:104) L (cid:105) = 0.
2. Model 2 : Unitary model generated by π x and π y The SLD and the RLD Fisher information matrices of Model 2 are denoted by G π thermalS and G π thermalR , respectively.Their inverse matrices ( G π thermalS ) − and ( G π thermalR ) − are( G π thermalS ) − = λ (cid:18) κ
00 1 + 4 κ (cid:19) , (49)( G π thermalR ) − = λ (cid:18) κ i − i 1 + 4 κ (cid:19) . (50)Since this model is a Gaussian shift model [2, 24], the RLD C-R bound is achievable. By using Eq. (19), the RLDC-R inequality gives the following inequality (cid:20) V − λ κ ) (cid:21) (cid:20) V − λ κ ) (cid:21) ≥ λ . (51)From Eq. (18), we obtain the SLD C-R bound as follows. V ≥ λ κ ) , V ≥ λ κ ) . Figure 3 shows the RLD C-R bound and the SLD C-R bound above for the temperature parameter κ = 1 as well.The gray region is the uncertainty relation given by the RLD C-R bound. The SLD and RLD C-R bounds move awayfrom the origin= (0 ,
0) as κ increases. This makes sense, because the increase in κ means the decrease in β because2 κ = (e βω − µ − − . The C-R bounds of Model 2 stays lower than that of Model 1. C. Discussion
1. Mixed state model
It turns out that in the case of the thermal state as the reference state, Model 2 is a simple Gaussian shift modelwhich is known to be the RLD C-R inequality giving an achievable bound [2, 24]. However, the bound for Model 1has a complicated structure as shown in Fig. 3, although the bound for the pure state is simple. As given in A 3, thetwo-parameter unitary transformation for Model 1, e − i θ p x e − i θ p y can be written ase − i θ p x e − i θ p y = e ξ a † − ξ ∗ a e ξ ∗ b † − ξ b , (52)where ξ = ξ ∗ and ξ = (2 λ ) − ( θ − i θ ). According to Eq. (52), Model 1 is a two-parameter sub-model with aconstraint ξ = ξ ∗ embedded in the four-parameter model. We attribute this dependency between ξ and ξ to thecomplicated bound though it is not clear why the change in the bound occurs at (cid:104) L (cid:105) = 1 /
2. Effects of thermal noise
We next compare the results between the pure state and the thermal state as the reference state as follows. First,Model 1 with the pure state as the reference state, the C-R bound has a quasi-classical feature. The SLD C-R bound3is determined by the constant which is the spread of the position probability density of LLL. With the thermal stateas the reference state, Model 1 is not D-invariant and the uncertainty relation of Model 1 is complicated. The shapeof the C-R bound depends on the expectation value of angular momentum (cid:104) L (cid:105) . The SLD C-R bound becomesachievable only at (cid:104) L (cid:105) = 0. Unless this special condition is satisfied, the mixed state model with the thermal noisehas discontinuity from zero temperature to finite temperature in the quantum C-R bounds. And hence, we cannotsimply take the zero temperature limit from the thermal state in our model.Next, Model 2 with the pure state as the reference state, the generalized RLD exists. The C-R bound given bythe generalized RLD is achievable. The MSE matrix components V and V has a correlation as shown in Eq. (30).With the thermal state as the reference state, Model 2 is a simple Gaussian shift model. Unlike the case of Model 1,Model 2 with the pure state as the reference state is genuine quantum. For Model 2, there exists a limit when thetemperature goes to zero, or equivalently, β → ∞ [21]. This limit yields the result for the pure-state case studied inSec. III A 2.
3. Optical implementation
The models studied in this paper can naturally be realized in the two-dimensional electron gas at low temperature.However, the optimal measurement to attain the quantum C-R bound may not be feasible in such a system. Alter-natively, one can realize our models in a linear optical system with two modes by tuning parameters properly. In thisconnection, we should not forget to mention related works on parameter estimation problems in two mode Gaussianstates Refs. [33–39]. In Ref. [39], authors discussed an optimal encoding and measurement scheme for estimating twoparameters in the pure-state reference state. The optimal state found there also comprises of classical correlation ofthe phase conjugation as in Eq. (52) for Model 1. The optimal POVM for Model 2 requires measuring non-canonicalvariables. π x and π y [40]. Therefore, we can only state that Model 2 potentially gives more precise measurement. Weexpect that our result in the thermal state as a reference state should also relevant to finding the optimal scheme inthe presence of noise. V. CONCLUSION
We have investigated the uncertainty relation for estimating x and y components of the position of one electronin a uniform magnetic field. In the present study, the uncertainty relation upon estimating the expectation valuesof the two commuting observables, ( x, y ) was derived in the framework of the quantum estimation theory. As thegenerators of the unitary transformation, two different sets of generators are used. One is a set of canonical momenta, p x and p y (Model 1) and the other is a set of mechanical momenta, π x and π y (Model 2). Based on the analysis bythe quantum estimation theory, in both cases, we got non-trivial bounds that give the trade-off relations between thetwo commuting observables, x and y , unlike the result of Heisenberg-Robertson type uncertainty relation.Although both Model 1 and Model 2 give the same effect to the position probability density defined by the productof the wave function and its complex conjugate, the C-R bounds of Model 1 and Model 2 are different for pure stateand mixed state (thermal state) as the reference state.With the pure state as the reference state, the C-R bound is quasi-classical for Model 1 and it is quantum mechanicalfor Model 2. With the thermal state as the reference state, the uncertainty relation given by the C-R bounds iscomplicated and the shape of the bounds changes when the expectation value of the angular momentum (cid:104) L (cid:105) is equalto 1 / (cid:104) L (cid:105) = 0. A possible extension might be an analysis by minimizinga weighted trace of the mean square error matrix [8]. However, this method gives asymptotically achievable boundonly. Second, for the thermal state with the (cid:104) L (cid:105) constraint, we see the change in the bound shape depending on (cid:104) L (cid:105) . We have no clue as to why the bound shape changes at (cid:104) L (cid:105) = 1 / Acknowledgment
The work is partly supported by JSPS KAKENHI grant number JP17K05571. We would like to thank Prof. HiroshiNagaoka for the invaluable discussion and suggestion. We would also like to thank anonymous referees for constructivediscussions to improve the manuscript.4
Appendix A: Supplement1. Uncertainty relation by quantum C-R inequality
The quantum C-R inequality for the MSE matrix V θ is V θ ≥ ( G θ ) − , (A1)where G θ is an arbitrary quantum Fisher information matrix. Let ( G θ ) − be( G θ ) − = [ g θij ] , (A2)The RLD C-R inequality (A1) holds iff tr [ V θ − ( G θ ) − ] ≥ V θ − ( G θ ) − ] ≥
0. Thus, we have V − g θ ≥ , V − g θ ≥ , and det (cid:18) V − g θ V − g θ V − ( g θ ) ∗ V − g θ (cid:19) ≥ , where g θ = ( g θ ) ∗ is used. The inequality above gives the following inequality.( V − g θ )( V − g θ ) ≥ | V − g θ | . The right hand side of the inequality above is written as follows. | V − g θ | = | V − Re g θ − i Im g θ | = | V − Re g θ | + | Im g θ | ≥ | Im g θ | . By applying this inequality, we obtain the following inequalities, V − g θ ≥ , V − g θ ≥ , (A3)( V − g θ )( V − g θ ) ≥ | Im g θ | . (A4)When Im g θ = 0, the uncertainty relation is given by Eq. (A3) only.
2. Thermal state and Gaussian state
The thermal state for a single harmonic oscillator, ρ β is described as ρ β = Z − β e − βH , (A5)where Z β = tr [e − βH ] and β = T − . T is temperature.By using Hamiltonian H = ω ( a † a + 1 /
2) and a † a | n (cid:105) = n | n (cid:105) , e − βH ise − βH = ∞ (cid:88) n =0 e − βH | n (cid:105) (cid:104) n | = e − βω ∞ (cid:88) n =0 γ n | n (cid:105) (cid:104) n | , where γ = e − βω . Z β is Z β = tr [e − βH ] = e − βω − γ . We obtain ρ β = Z − β e − βH = (1 − γ ) (cid:88) n γ n | n (cid:105) (cid:104) n | . ρ β by the basis as the Gaussian state, (cid:104) z | ρ β | z (cid:105) . Next, we make the samematrix element of the Gaussian state to see if they match. (cid:104) z | ρ β | z (cid:105) is (cid:104) z | ρ β | z (cid:105) = (1 − γ ) (cid:88) n γ n (cid:104) z | n (cid:105) (cid:104) n | z (cid:105) = (1 − γ )e − | z | − | z | + γz ∗ z . (A6)The Gaussian state S κ is defined by S κ = 12 πκ (cid:90) e − | z | κ | z (cid:105) (cid:104) z | d z. Then its matrix element (cid:104) z | S κ | z (cid:105) is (cid:104) z | S κ | z (cid:105) = 12 πκ (cid:90) e − | z | κ (cid:104) z | z (cid:105) (cid:104) z | z (cid:105) d z = 12 πκ (cid:90) e − ( κ +1) | z | + z ∗ z + z z ∗ d z e − | z | − | z | . By using (cid:90) e − α | z | + βz + γz ∗ d z = πα e βγα , we obtain (cid:104) z | ρ β | z (cid:105) = 12 κ + 1 e − | z | − | z | + z ∗ z κ . (A7)From (A6) and (A7), (cid:104) z | S κ | z (cid:105) = (cid:104) z | ρ β | z (cid:105) holds iff2 κ = γ − γ = 1e βω − . (A8)Therefore, we obtain ρ β = 12 πκ (cid:90) e − | z (cid:48)| κ | z (cid:105) (cid:104) z | d z. (A9)where 2 κ is given by Eq. (A8).
3. Model 1 unitary transformationin terms of the creation and annihilation operators
The unitary transformations of the Model 1, e − i θ p x e − i θ p y ise − i θ p x e − i θ p y = e λ { ( a † − a )+( b † − b ) } θ e − i2 λ { ( a † + a ) − ( b † + b ) } θ . Since p x and p y commute,e − i θ p x e − i θ p y = e − i θ p x − i θ p y = e λ { ( a † − a )+( b † − b ) } θ }− i2 λ { ( a † + a ) − ( b † + b ) } θ . Therefore, e − i θ p x e − i θ p y = e ξa † − ξ ∗ a e ξ ∗ b † − ξb , (A10)where ξ = (2 λ ) − ( θ − i θ ).6 Appendix B: Calculation1. SLD and RLD: The thermal state as the reference state
First, we briefly explain that SLD and RLD Fisher information matrices for the mixed state are independent of theparameters θ = ( θ , θ ) in the unitary transformation U ( θ , θ ).Let Model 1 SLD and Model 1 RLD of Model 1 be L (1)S , j ( θ ) and L (1) R, j ( θ ), respectively. With using the unitarytransformation U ( θ , θ ) = e − i θ p x e − i θ p y , L (1)S , j (0) and L (1)R , j (0) are written as L (1)S , j ( θ ) = U ( θ , θ ) L (1)S , j (0) U † ( θ , θ ) ,L (1)R , j ( θ ) = U ( θ , θ ) L (1)R , j (0) U † ( θ , θ ) . For the RLD Fisher information G (1)R ( θ ) = [ g (1)R , ij ( θ )], we can derive the relation below if the transformation is unitary. g (1)R , ij ( θ ) = tr [ ρ θ L (1)R , j ( θ ) L (1) † R , i ( θ )] = tr [ ρ L R , j (0) L (1) † R , i (0)] . If the transformation is unitary, the RLD Fisher information G R ( θ ) does not depend on the parameters θ and θ .Then, we can write G R ( θ ) as G R = [ g R , ij ]. From Eqs. (20, 22), we can show that the same holds for the SLDFisher information G S ( θ ). Therefore, if we have L (1)S , i (0) and L (1)R , i (0), it is enough to obtain the SLD and RLD Fisherinformation matrices. The same is true for Model 2. a. Model 1 SLD: L (1) S, (0) , L (1) S, (0) , Z matrix Z p thermal L (1)S , (0) = 1 λ (1 + 4 κ ) ( a + a † ) + 1 λ (1 + 4 κ ) ( b + b † ) ,L (1)S , (0) = i λ (1 + 4 κ ) ( a − a † ) − i λ (1 + 4 κ ) ( b − b † ) . With using p x , p y and x, y , L (1)S , (0) = ( 11 + 4 κ + 11 + 4 κ ) p y + 1 λ ( 11 + 4 κ −
11 + 4 κ ) x,L (1)S , (0) = − ( 11 + 4 κ −
11 + 4 κ ) p x + 1 λ ( 11 + 4 κ + 11 + 4 κ ) y. The SLD Fisher information matrix G p thermalS is calculated as G p thermalS = 1 λ ( 11 + 4 κ a + 11 + 4 κ b ) (cid:18) (cid:19) . Z matrix Z p thermal is calculated as Z p thermal = λ κ + 2 κ (cid:18) + 2 κ + 2 κ + 8 κ κ i (2 κ − κ ) − i (2 κ − κ ) + 2 κ + 2 κ + 8 κ κ (cid:19) . From this expression, we have Z p thermal = ( G p thermalR ) − + ∆ g (cid:18) (cid:19) . Since ∆ g (cid:54) = 0, we see that Z p thermal (cid:54) = ( G p thermalR ) − . This implies the model is not D-invariant [27].7 b. Model 2 SLD: L (2)S , (0) , L (2)S , (0) , Z matrix Z π thermal L (2)S , (0) = 2 λ (1 + 4 κ ) ( a + a † ) ,L (2)S , (0) = 2i λ (1 + 4 κ ) ( a − a † ) . With using p x , p y and x, y , L (2)S , (0) = 21 + 4 κ ( p y + 1 λ x ) ,L (2)S , (0) = 21 + 4 κ ( − p x + 1 λ y ) . The inverse of the SLD Fisher information matrix G π thermalS is G π thermalS = 4 λ (1 + 4 κ ) (cid:18) (cid:19) . Z matrix Z π thermal is Z π thermal = λ (cid:18) κ i − i 1 + 4 κ (cid:19) . c. Model 1 RLD: L (1)R , (0) , L (1)R , (0) L (1)R , (0) = 12 λ ( 11 + 2 κ a + 12 κ a † ) + 12 λ ( 11 + 2 κ b + 12 κ b † ) ,L (1)R , (0) = − i2 λ ( −
11 + 2 κ a + 12 κ a † ) + i2 λ ( −
11 + 2 κ b + 12 κ b † ) . The RLD Fisher information matrix G p thermalR is calculated as G p thermalR = 14 λ (cid:32) κ a + κ a + κ b + κ b − i [ κ a (1+2 κ a ) − κ b (1+2 κ b ) ]i [ κ a (1+2 κ a ) − κ b (1+2 κ b ) ] κ a + κ a + κ b + κ b (cid:33) . d. Model 2 RLD: L (2)R , (0) , L (2)R , (0) L (2)R , (0) = 1 λ ( 11 + 2 κ a + 12 κ a † ) ,L (2)R , (0) = i λ ( 11 + 2 κ a − κ a † ) . The creation annihilation operators, a, a † and b, b † are written as follows. a = λ p x + p y ) + 1 λ ( x − i y )] ,a † = λ − i p x + p y ) + 1 λ ( x + i y )] ,b = λ p x − p y ) + 1 λ ( x + i y )] ,b † = λ − i p x − p y ) + 1 λ ( x − i y )] . G π thermalR is G π thermalR = 1 λ κ (1 + 2 κ ) (cid:18) κ − ii 1 + 4 κ (cid:19) .
2. SLD, Generalized RLD: Pure state as the reference state
Let the SLD of a pure state ρ θ = | ψ θ (cid:105) (cid:104) ψ θ | be L S, i . Then, L S , i is expressed as [21], L S , i = 2 ∂ i ρ θ = 2 ∂ i ( | ψ θ (cid:105) (cid:104) ψ θ | ) . (B1) a. Model 1 SLD: L (1)S , ( θ ) , L (1)S , ( θ ) We set the reference state ρ as ρ = | , (cid:105) (cid:104) , | . From Eq. (6), ρ pθ is expressed as ρ pθ = e − i θ p x e − i θ p y | , (cid:105) (cid:104) , | e i θ p y e i θ p x (B2)= U ( θ ) | , (cid:105) (cid:104) , | U † ( θ ) . (B3)where U ( θ ) = e − i θ p x e − i θ p y . From Eq. (B1), the SLD’s of Model 1 are expressed as L (1)S , (0) = − p x , ρ ] ,L (1)S , (0) = − p y , ρ ] . where L (1)S , j ( θ ) = U ( θ ) L (1)S , j (0) U † ( θ ) , ( j = 1 , L (1) S, (0) and L (1)S , (0) are also written as L (1)S , (0) = 1 λ [( a † − a ) + ( b † − b ) , ρ ] ,L (1)S , (0) = − i λ [( a † + a ) − ( b † + b ) , ρ ] . With these SLD’s, the Fisher information matrix G p S is calculated as G p S = 2 λ (cid:18) (cid:19) . From the direct calculation of Eq. (22), we can show that G p S = ˜ G p R . b. Model 2 SLD: L (2)S , ( θ ) , L (2)S , ( θ ) From Eq. (6), ρ πθ is expressed as ρ πθ = e − i θ π x e − i θ π y | , (cid:105) (cid:104) , | e i θ π y e i θ π x . (B4)The unitary transformation e − i θ π x e − i θ π y is calculated as follows [28].e − i θ π x e − i θ π y = e λ θ θ e − i θ π x − i θ π y . (B5)By substituting Eq. (B5) in Eq (B4), we obtain ρ πθ = U ( θ ) | , (cid:105) (cid:104) , | U † ( θ ) . (B6)where U ( θ ) = e − i θ π x − i θ π y .9From Eq. (B1), the SLD’s of Model 2 are expressed as L (2)S , (0) = − π x , ρ ] ,L (2)S , (0) = − π y , ρ ] . where L (2)S , j ( θ ) = U ( θ ) L (2) S, j (0) U † ( θ ) , ( j = 1 , L (2)S , (0) = − λ [ a † + a, ρ ] ,L (2)S , (0) = 2 λ [ a † − a, ρ ] . Thus, the SLD Fisher information G π S is G π S = 4 λ (cid:18) (cid:19) . From Eq. (B1), the generalized RLD Fisher information ˜ G π R is˜ G π R = 4 λ (cid:18) − i 1 (cid:19) . [1] C. W. Helstrom Quantum Detection and Estimation Theory , Academic, New York (1976).[2] A. S. Holevo,
Probabilistic and Statistical Aspects of Quantum Theory , Edizioni della Normale, Pisa, 2nd ed (2011).[3] S.L. Braunstein, C. M. Caves, G. J. Milburn, Annals of Physics, 247(1), 135-173 (1996).https://doi.org/10.1006/aphy.1996.0040[4] H. Nagaoka, in
Surikagaku , no. 508, p. 26–34, Saiense-Sha (2005). (in Japanese)[5] P. Gibilisco, H. Hiai, D. Petz, IEEE Trans. Information Theory. A, Vol. , 439 (2009).https://doi.org/10.1109/TIT.2008.2008142[6] Y. Watanabe, T. Sagawa, M. Ueda, Phys. Rev. A, Vol. , 042121 (2011). https://doi.org/10.1103/PhysRevA.84.042121[7] W. Guo, W. Zhong, X-X.Jing, L-B. Fu, and X. Wang, Phys. Rev. A, Vol. , 042115 (2016).https://link.aps.org/doi/10.1103/PhysRevA.93.042115[8] I. Kull, P. Allard Gu´erin, F. Verstraete, Journal of Physics A: Mathematical and Theoretical, (2020) (forthcoming).https://doi.org/10.1088/1751-8121/ab7f67[9] L. I. Mandel’shtam and I. E. Tamm, Izv. AN SSSR ser. fiz. 9, 122 (1945). I. Tamm, J. Phys. (U.S.S.R.) 9, 249 (1945).[10] M. Ozawa, Phys. Rev. A, Vol. , 042105 (2003). https://doi.org/10.1103/PhysRevA.67.042105[11] M. Ozawa, Phys. Lett. A, Vol. , 367 (2004). https://doi.org/10.1016/j.physleta.2003.12.001[12] S. Wehner and A. Winter J. Math. Phys., Vol. , 062105 (2008). https://doi.org/10.1063/1.2943685[13] M.G.A Paris, Int. J. Quantum Inf. , 125 (2009).[14] F. Albarelli, M. Barbieri, M.G. Genoni, I Gianani. Physics Letters A, 384(12), 126311 (2020).https://doi.org/10.1016/j.physleta.2020.126311[15] J. S. Sidhu, P. Kok, AVS Quantum Science, 2(1), 014701, (2020). https://doi.org/10.1116/1.5119961[16] R. Demkowicz-Dobrzan, W. Gorecki, M. Guta, (2020). arXiv:2001.11742. http://arxiv.org/abs/2001.11742[17] W. Heisenberg, Zeitschr. Phys. Vol. , 172 (1927). http://dx.doi.org/10.1007/BF01397280[18] H. P. Robertson, Phys. Rev. Vol. , 163 (1929). https://doi.org/10.1103/PhysRev.34.163[19] M. Johnson and B. Lippmann, Phys. Rev, Vol. , 828 (1949). https://doi.org/10.1103/PhysRev.76.828[20] L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Nonrelativistic Theory) , Pergamon, New York, (1977).[21] A. Fujiwara, H. Nagaoka J. Math. Phys., Vol. , 4227 (1999). https://doi.org/10.1063/1.532962[22] A. Fujiwara Multiparameter pure state estimation based on the RLD METR 94-9 (1994)[23] A. Fujiwara Multiparameter pure state estimation based on the SLD METR 94-11 (1994)[24] H. Yuen and M. Lax, IEEE Trans. Information Theory, Vol. IT19 , 740 (1973). https://doi.org/10.1109/TIT.1973.1055103[25] R. Kubo, S. J. Miyake, and N. Hashitsume, Solid State Phys., , 269 (1965). https://doi.org/10.1016/S0081-1947(08)60413-0[26] I. A. Malkin and V. I. Man’ko, Soviet Physics JETP., Vol. , 527 (1969).[27] J. Suzuki, J. Math. Phys., Vol. , 042201 (2016). https://doi.org/10.1063/1.4945086[28] J. R. Klauder and E. C. G. Sudarshan, Fundametals of quantum optics , Benjamin, New York (1968).[29] A. Fujiwara, H. Nagaoka Phys. Lett. A, Vol. , 119 (1995). https://doi.org/10.1016/0375-9601(95)00269-9 [30] K. Matsumoto J. Phys. A: Math. Gen., Vol. , 3111 (2002). https://doi.org/10.1088%2F0305-4470%2F35%2F13%2F307[31] S. Ragy, M. Jarzyna, and R. Demkowicz-Dobrza´nski, Phys. Rev. A, Vol. , 052108 (2016).https://doi.org/10.1103/PhysRevA.94.052108[32] J. Suzuki, Entropy, Vol. , 703 (2019). https://doi.org/10.3390/e21070703[33] N. J. Cerf and S. Iblisdir, Phys. Rev. A, Vol. , 032307 (2001). https://doi.org/10.1103/PhysRevA.64.032307[34] M. G. Genoni, M. G. A. Paris, G. Adesso, H. Nha, P. L. Knight, and M. S. Kim, Phys. Rev. A, Vol. , 012107 (2013).https://doi.org/10.1103/PhysRevA.87.012107[35] Y. Gao and H. Lee, Eur. Phys. J. D, Vol. , 347 (2014). https://doi.org/10.1140/epjd/e2014-50560-1[36] T. Baumgratz and A. Datta, Phys. Rev. Lett., Vol. , 030801 (2016). https://doi.org/10.1103/PhysRevLett.116.030801[37] M. Bradshaw, S. M. Assad and P. K. Lam, Phys. Lett. A, Vol. , 2598 (2017).https://doi.org/10.1016/j.physleta.2017.06.024[38] M. Bradshaw, P. K. Lam, and S. M. Assad, Phys. Rev. A Vol. , 012106 (2018).https://link.aps.org/doi/10.1103/PhysRevA.97.012106[39] M. Arnhem, E. Karpov, and N. J. Cerf, Appl. Sci. Vol. , 4264 (2019), https://doi.org/10.3390/app9204264[40] Y. Aharonov and J. Safko, Ann. Phys. Vol.91