A tight Cramér-Rao bound for joint parameter estimation with a pure two-mode squeezed probe
aa r X i v : . [ qu a n t - ph ] J a n A tight Cram´er-Rao bound for joint parameter estimation with apure two-mode squeezed probe
Mark Bradshaw, Syed M. Assad, Ping Koy Lam
Centre for Quantum Computation and CommunicationTechnology, Department of Quantum Science,Research School of Physics and Engineering,Australian National University, Canberra ACT 2601, Australia.
Abstract
We calculate the Holevo Cram´er-Rao bound for estimation of the displacement experienced byone mode of an two-mode squeezed vacuum state with squeezing r and find that it is equal to4 exp( − r ). This equals the sum of the mean squared error obtained from a dual homodynemeasurement, indicating that the bound is tight and that the dual homodyne measurement isoptimal. Keywords.
Quantum physics, Quantum information, Quantum optics, Parameter estimation,Cram´er-Rao bound . INTRODUCTION In quantum mechanics, there is a limit to the precision to which we can simultaneouslymeasure two observables that do not commute. If the observables are complementary vari-ables, such as position and momentum, this limit is described by the Heisenberg uncertaintyprinciple.In continuous variable quantum optics, the bosonic quadrature operators Q and P areanother pair of complementary variables, and obey the canonical commutation relation[ Q, P ] = 2 i , where throughout the paper we use units where ¯ h = 2. The bosonic field is alsodescribed by the annihilation operator a and creation operator a † , which are related to thequadrature operators by Q = a + a † and P = i ( a † − a ). The displacement operator is givenby D ( α ) = exp (cid:0) αa † − α ∗ a (cid:1) , (1)where α = ( q + ip ) / ρ which undergoes adisplacement of D ( α ) resulting in ρ θ = D ( α ) ρ D † ( α ), how well can we estimate the twoparameters θ := q = 2 Re( α ) and θ := p = 2 Im( α )? Our figure of merit is sum of themean square error (MSE), V := E [(ˆ θ − θ ) ]+ E [(ˆ θ − θ ) ] where E is the expectation value,and ˆ θ and ˆ θ are the estimates of θ and θ respectively. We aim to find a lower bound to V . These bounds are called Cram´er-Rao bounds (CR bounds) and will only depend on thestate ρ θ and independent of the measurement performed on it. We calculate the CR boundbased on the work of Holevo [1, 2], which we call Holevo CR bound. First, we calculatethe Holevo CR bound when the probe state ρ is a single mode squeezed state, for whichtight bounds are already known. Next, we calculate the Holevo CR bound when the probeis a two mode squeezed state. We find that it is superior to bounds calculated by previousauthors [3, 4]; and that the bound can be reached by a simple measurement.Our paper is divided up into sections as follows. In section II, we briefly describe theGaussian quantum optics used in our results. In section III, we summarize parameter es-timation theory including CR bounds. In section IV, we summarize the bounds found byother authors and the MSE from a dual homodyne measurement. In section V, we calculatethe Holevo CR bound for one- and two-mode squeezed probes and discuss our results.2 I. GAUSSIAN QUANTUM OPTICS
Consider a state consisting of m bosonic modes. Let the annihilation and creation oper-ators of the k th mode be a k and a † k , respectively, and the quadrature operators be Q k and P k . Define a vector Z to contain all the quadrature operators: ~Z = ( Q , P , ..., Q m , P m ) (2)The mean of the quadrature operators of a state ρ , otherwise known as the displacementvector of the state, is given by M = [ h Z j i ] j , (3)where h A i = tr( ρA ) is the expectation value of operator A . Define the covariance matrix V ,which contains the variances of the quadrature operators, by V = (cid:20) h Z j Z k − Z k Z j i − h Z j i h Z k i (cid:21) jk . (4)A thermal state is given by ρ th ( N ) = 1 N + 1 ∞ X n =0 (cid:18) N N (cid:19) n | n i h n | (5)where | n i are the Fock states, and N is the mean number of photons in the bosonic mode.A thermal state has a zero displacement vector and a covariance matrix of V th = (2 N + 1) I where I is the 2 × S ( r ) = exp (cid:18)
12 ( ra − ra † ) (cid:19) , (6)where r is the squeezing parameter. When acting on the vacuum state, this gives thesqueezed vacuum state | S ( r ) i = S ( r ) | i . The squeezed vacuum state has a zero displacementvector and a covariance matrix of V sq = e − r
00 e r . (7)The two-mode squeezing operator is given by S ( r ) = exp (cid:16) ra a − ra † a † (cid:17) . (8)3hen acting on two vacuum states it gives the two-mode squeezed vacuum state, also knownas the Einstein-Podolski-Rosen (EPR) state. The two-mode squeezed vacuum state has zerodisplacement vector and covariance matrix of V EPR = cosh(2 r ) 0 sinh(2 r ) 00 cosh(2 r ) 0 − sinh(2 r )sinh(2 r ) 0 cosh(2 r ) 00 − sinh(2 r ) 0 cosh(2 r ) . (9)A beam splitter is used to mix two modes. It is described by the unitary transformation B ( φ ) = exp (cid:16) φ ( a † a − a a † ) (cid:17) (10)where τ = cos φ is the transmissivity of the beam splitter. III. PARAMETER ESTIMATION THEORY
Let ρ θ be a family of states parametrized by d parameters θ = ( θ , θ , ...θ d ). The goal ofparameter estimation is to estimate the value of θ based on the outcome of a measurementon ρ θ . In quantum mechanics, a measurement is described by a positive operator-valuedmeasure (POVM) Π = { Π x } . Each measurement outcome x has a corresponding non-negative hermitian operator Π x associated with it, where the probability of measuring x ona state ρ θ is p θ ( x ) = tr(Π x ρ θ ), and the POVM elements sum to Identity: P x Π x = I . Wethen need an estimator ˆ θ ( x ), which maps the observed outcome x to an estimate for θ . Anestimator is called locally unbiased at θ if E [ˆ θ ( x )] = θ at the point θ . An estimator is calledunbiased if and only if it is locally unbiased at every θ .The MSE matrix V θ [ˆ θ ] of the estimator ˆ θ is given by V θ [ˆ θ ] = "X x p θ ( x )( ˆ θ j ( x ) − θ j )( ˆ θ k ( x ) − θ k ) jk . (11)The sum of the MSE V is the trace of the MSE matrix: V = Tr n V θ [ˆ θ ] o . (12)Here, Tr {·} denotes the trace of an estimator matrix. The Cram´er-Rao bound provides alower bound to the MSE matrix for a classical probability distribution p θ ( x ): V θ [ˆ θ ] ≥ ( J θ [ p θ ]) − (13)4here A ≥ B for matrices A and B means A − B is positive semi-definite. Taking the tracewe get a bound for the MSE, V ≥ Tr (cid:8) ( J θ [ p θ ]) − (cid:9) . (14) J θ [ p θ ] is the classical Fisher information matrix is given by J θ [ p θ ] = "X x p θ ( x ) ∂ log p θ ( x ) ∂θ j ∂ log p θ ( x ) ∂θ k jk . (15)This provides a bound to the MSE matrix for a fixed measurement Π. Next we define themost informative quantum Cram´er-Rao bound, by minimizing over all POVMs. C MIθ = min Π Tr { J θ [Π] − } (16)In practice, this minimization is difficult to perform, so lower bounds are used instead. Thefirst one is base of the symmetric log derivative (SLD) operator L θ,j which is defined by ∂ρ θ ∂θ j = 12 ( ρ θ L θ,j + L θ,j ρ θ ) . (17)This is used to calculate the SLD quantum fisher information matrix defined by G θ = [ h L θ,j , L θ,k i ρ θ ] jk , (18)where we use an inner product defined by h X, Y i ρ θ = tr (cid:18) ρ θ
12 (
Y X † + X † Y ) (cid:19) , (19)where tr( · ) denotes trace of a density matrix. This leads to a bound on the sum of the MSE,which we call the SLD CR bound C Sθ [5, 6]. V ≥ C Sθ = Tr (cid:0) G − θ (cid:1) . (20)The next quantum CR bound we consider is based on the right log derivative (RLD) operator˜ L θ,j defined by ∂ρ θ ∂θ j = ρ θ ˜ L θ,j . (21)This is used to calculate the RLD quantum fisher information matrix˜ G θ = [ h ˜ L θ,j , ˜ L θ,k i + ρ θ ] jk , (22)5here we use an inner product defined by h X, Y i + ρ θ = tr (cid:0) ρ θ Y X † (cid:1) . (23)This leads to a bound on the sum of the MSE [7], which we call the RLD CR bound C Rθ ,given by V ≥ C Rθ = Tr { Re ˜ G − θ } + TrAbs { Im ˜ G − θ } , (24)where TrAbs {·} denotes the sum of the absolute values of the eigenvalues of a matrix.While the RLD and SLD CR bounds are easy to compute [8, 9], in general they are notalways achievable. The SLD CR bound corresponds to performing the optimal measurementsfor the estimation of each parameter ignoring the other. But for non-commuting observables,it might not be possible to perform the two optimal measurements simultaneously. Similarly,the RLD CR bound is in general not obtainable by a valid measurement.Holevo derived another bound for the MSE [1, 2], which we shall call the Holevo CRbound. The Holevo CR bound is defined through the following minimisation C Hθ := min ~X ∈X h θ [ ~X ] (25)and X := { ( X , X , ...X d ) } where X j are Hermitian operators satisfying the locally unbiasedconditions tr( ρ θ X j ) = 0 (26)tr (cid:18) ∂ρ θ ∂θ j X k (cid:19) = δ jk (27)and h θ is the function h θ [ ~X ] := Tr n Re Z θ [ ~X ] o + TrAbs n Im Z θ [ ~X ] o . (28) Z θ [ ~X ] is a d × d matrix Z θ [ ~X ] := [tr ( ρ θ X j X k )] j,k . (29)For any ~X satisfying the condition Eq. (27), h θ [ ~X ] ≥ C Sθ and h θ [ ~X ] ≥ C Rθ . At the minimumof h θ , defined above as C Hθ , V ≥ C Hθ . So, the Holevo CR bound is always greater than orequal to the RLD and SLD bounds. See for example [10] for proof of the above statements.The Holevo CR bound involves a minimization over the measurement space, is in gen-eral hard to compute, and can be attained by a collective measurement [11]. When theprobe state has rank one, an individual measurement is sufficient to attain the Holevo CRbound [12]. 6 . . . . . . . . . r s u m o f m e a n s q u a r ee rr o r , V SLD bound, C Sθ RLD bound, C Rθ max (cid:8) C Sθ , C Rθ (cid:9) Dual homodyne, V DH FIG. 1: Bounds for V as a function of squeezing parameter r when the number of thermal pho-tons N = 0 . r . IV. SLD AND RLD CR BOUNDS FOR TWO-MODE SQUEEZED PROBE
Let the probe be a two-mode squeezed thermal state given by ρ = S ( r ) ( ρ th ( N ) ⊗ ρ th ( N )) S † ( r ), where if N = 0 we get the two-mode squeezed vacuum. Thefirst mode of the probe state undergoes a unitary displacement operation D ( θ ) and ends upin the state ρ θ . The SLD CR bound and RLD CR bounds are given by [3, 4]: C Sθ := 2 + 4 N cosh 2 r (30) C Rθ := 8 N (1 + N )(1 + 2 N ) cosh 2 r − . (31)Now let us consider a measurement which we call the dual homodyne measurement. Themeasurement consists of interfering the two modes on a beam splitter with transmissivity τ = 1 /
2, followed by homodyne measurement of the Q quadrature of the first mode anda homodyne measurement of the P quadrature of the second mode. The dual homodynemeasurement gives V = (8 N + 4) exp( − r ) := V DH .We plot V DH and the two bounds in Fig. 1. The dual homodyne measurement MSE does7ot reach the bounds for most values of r . This means that either the measurement is notoptimal or the bounds are not tight. To help determine which is the case, we will calculatethe Holevo CR bound and compare it to V DH . V. RESULTS
Here, we calculate the Holevo CR bound for two cases: a pure single-mode squeezedprobe and a pure two-mode squeezed probe.
A. Calculation of Holevo bound for pure single-mode squeezed probe
Consider a single mode squeezed probe ρ = S ( r ) ρ th ( N ) S ( r ) † . Applying the displace-ment operator D ( θ ), we end up with the state ρ θ = D ( θ ) ρ D † ( θ ). For pure states when N = 0, this case is shown in [12–14] to be coherent and the bound from the RLD is knownto be tight [13–15]. One optimal joint estimation strategy is to alternatively apply the op-timal strategy for each parameter. In general, when ρ is a mixed state, the Cram´er-Raobounds are [3, 4] C Sθ := (2 + 4 N ) cosh 2 r (32) C Rθ := 2 + (2 + 4 N ) cosh 2 r . (33)The RLD bound is always greater than the SLD bound and is hence a more informativelower bound. In fact, the dual homodyne measurement gives V = 2 + (2 + 4 N ) cosh 2 r whichsaturates the RLD bound [3]. As the bound increases with r , squeezed state probes performworse than a coherent state probe ( r = 0).Although we already know what the result will be, as an exercise we compute the Holevobound for the pure single-mode squeezed probe. In this case, ρ θ = | ψ i h ψ | where | ψ i = D ( θ ) | S ( r ) i . The derivatives of the displacement operator D ( θ ) = exp (cid:0) i θ Q − i θ P (cid:1) with respect to θ and θ are ∂∂θ D ( θ ) = (cid:18) − i P + i θ (cid:19) D ( θ ) (34) ∂∂θ D ( θ ) = (cid:18) i Q − i θ (cid:19) D ( θ ) . (35)8ee appendix A 1 for the derivation.To simplify the calculation, we calculate the Holevo CR bound when θ is small, andhence evaluate at θ = 0, and assert that the bound will be the same for all θ . The reason wecan do this is because the Holevo CR bound is asymptotically attainable with an adaptivemeasurement scheme, given a set of n identical states ρ ⊗ nθ with n → ∞ . A rough estimate for θ can be obtained for a small number of measurements using √ n states, Then the remaining n − √ n states can be displaced by D ( − ˜ θ ) where ˜ θ is the rough estimate for θ , resulting instates with a small θ .We compute | ψ i and the derivatives of | ψ i with respect to θ and θ evaluated at θ = 0. | ψ i = | ψ i (cid:12)(cid:12)(cid:12)(cid:12) θ =0 = | S ( r ) i (36) | ψ i = ∂∂θ | ψ i (cid:12)(cid:12)(cid:12)(cid:12) θ =0 = − P | S ( r ) i i | ψ i = ∂∂θ | ψ i (cid:12)(cid:12)(cid:12)(cid:12) θ =0 = Q | S ( r ) i i . (38)The inner products are h ψ | ψ i = 1 (39) h ψ | ψ i = − i h S ( r ) | P | S ( r ) i = 0 (40) h ψ | ψ i = i h S ( r ) | Q | S ( r ) i = 0 (41) h ψ | ψ i = 14 h S ( r ) | P | S ( r ) i = e r h ψ | ψ i = 14 h S ( r ) | Q | S ( r ) i = e − r , (43)9here we have used that the displacement vector of | S ( r ) i is zero, and the covariance matrixgiven by Eq. (7). h ψ | ψ i = − h S ( r ) | P Q | S ( r ) i . (44)From the commutation relation [ Q, P ] = 2 i , we get Im h P Q i = −
1. The covariance of Q and P is given by V QP = 12 h QP + P Q i − h Q i h P i (45)= Re h P Q i − h Q i h P i . (46)From Eq. (7) this should equal zero, and since the displacement vector is also zero,Re {h S ( r ) | P Q | S ( r ) i} = 0, so we have that h ψ | ψ i = i . (47)We introduce a set of orthonormal vectors {| e i , | e i} such that | ψ i = | e i (48) | ψ i = | e i e r | ψ i = | e i i e − r , (50)which satisfies the inner products. With this, the constraint Eq. (26) becomes h e | X | e i = 0 (51) h e | X | e i = 0 , (52)The density matrix for | ψ i and its derivatives at the point θ = 0 are ρ = ρ (cid:12)(cid:12)(cid:12)(cid:12) θ =0 = | ψ i h ψ | (53) ρ = ∂∂θ ρ (cid:12)(cid:12)(cid:12)(cid:12) θ =0 = | ψ i h ψ | + | ψ i h ψ | (54) ρ = ∂∂θ ρ (cid:12)(cid:12)(cid:12)(cid:12) θ =0 = | ψ i h ψ | + | ψ i h ψ | . (55)10he constraint Eq. (27) is therefore h e | X | e i = e − r (56) h e | X | e i = − i e r . (57)Because we are interested in the minimization of Eq. (25), we can set to zero all componentsof X and X not involved in the constraints or not complex conjugates of componentsinvolved in constraints. The minimization is trivial, and the solution occurs when Z θ = e − r i − i e r . (58)The Holevo CR bound for a pure single-mode squeezed state probe is therefore C Hθ = 2 + 2 cosh 2 r , (59)which equals the RLD bound and the variance from a dual homodyne measurement when N = 0 as expected. B. Calculation of Holevo CR bound for pure two-mode squeezed probe
To calculate the Holevo CR bound for a pure two-mode squeezed probe, we follow a similarprocedure as the single-mode case. The two-mode probe state can be transformed into aproduct state of two single-mode squeezed probes by a beam splitter with trasmissivity .The beam splitter is a unitary transformation, which does not affect the Holevo CR bound.Furthermore, when N = 0, ρ has rank one, and this transformed version of ρ can be writtenas U ρU † = | ψ i h ψ | where | ψ i = D ( θ/ √ | S ( r ) i ⊗ D ( − θ/ √ | S ( − r ) i , (60)We compute | ψ i and the derivatives of | ψ i with respect to θ and θ evaluated at θ = 0. | ψ i = | ψ i (cid:12)(cid:12)(cid:12)(cid:12) θ =0 = | S ( r ) i | S ( − r ) i (61) | ψ i = ∂∂θ | ψ i (cid:12)(cid:12)(cid:12)(cid:12) θ =0 − P | S ( r ) i | S ( − r ) i i √ | S ( r ) i P | S ( − r ) i i √ | ψ i = ∂∂θ | ψ i (cid:12)(cid:12)(cid:12)(cid:12) θ =0 = Q | S ( r ) i | S ( − r ) i i √ − | S ( r ) i Q | S ( − r ) i i √ . (63)Of interest are the inner products involving the states | ψ i , | ψ i and | ψ i , so we calculatethem now. h ψ | ψ i = 1 h ψ | ψ i = − i √ h S ( r ) | P | S ( r ) i + i √ h S ( − r ) | P | S ( − r ) i (64)= 0 (65) h ψ | ψ i = i √ h S ( r ) | Q | S ( r ) i − i √ h S ( − r ) | Q | S ( − r ) i (66)= 0 (67) h ψ | ψ i = 18 h S ( r ) | P | S ( r ) i + 18 h S ( − r ) | P | S ( − r ) i = 18 e r + 18 e − r = cosh 2 r h ψ | ψ i = 18 h S ( r ) | Q | S ( r ) i + 18 h S ( − r ) | Q | S ( − r ) i = 18 e − r + 18 e r = cosh 2 r h ψ | ψ i = − h S ( r ) | P Q | S ( r ) i − h S ( − r ) | P Q | S ( − r ) i = i . (70)To satisfy the inner products, we introduce an orthonormal set of states {| e i , | e i , | e i} such that | ψ i = | e i| ψ i = | e i cosh r | e i sinh r | ψ i = | e i i cosh r − | e i i sinh r h e | X | e i = 0 (72) h e | X | e i = 0 , (73)12nd the constraint Eq. (27) becomesRe(cosh r h e | X | e i + sinh r h e | X | e i ) = 1 (74)Re(cosh r h e | X | e i + sinh r h e | X | e i ) = 0 (75)Re( i cosh r h e | X | e i − i sinh r h e | X | e i ) = 0 (76)Re( i cosh r h e | X | e i − i sinh r h e | X | e i ) = 1 . (77)The matrix Z θ in Eq. (29) is given by Z θ = tr( ρ X X ) tr( ρ X X )tr( ρ X X ) tr( ρ X X ) . (78)Because we are interested in the minimization of Eq. (25), we can set to zero all componentsof X and X not involved in the constraints Eq. (72–77) or not complex conjugates of com-ponents involved in constraints. Define the components in terms of their real and imaginaryparts: h e | X | e i = t + ij h e | X | e i = s + ik h e | X | e i = t + ij h e | X | e i = s + ik . (79)And so Z θ = (cid:16) t + j + s + k t t + j j + s s + k k + i ( j t − j t + k s − k s ) t t + j j + s s + k k + i ( j t − j t + k s − k s ) t + j + s + k (cid:17) , (80)The Holevo function from Eq. (28) becomes h = t + t + s + s + j + j + k + k | {z } f +2abs { j t − j t + k s − k s } | {z } g . (81)Using the constraints Eq. (74–77), we can eliminate four variables by making the substitu-tions t = sech r − s tanh rj = k tanh rt = − s tanh r = − sech r + k tanh r . (82)The problem now is to minimize h over the four remaining variables. This is accomplished inappendix A 2. We find that the minimum of h and the Holevo CR bound is C H = 4 exp( − r ).This is equal to the sum of the MSE obtained from the dual homodyne measurement. Hence,the dual homodyne measurement is the optimal measurement and the Holevo CR bound istight. C. Conclusion
We calculated the Holevo CR bound for a pure single-mode squeezed state probe expe-riencing a unknown displacement of D ( θ ). As expected, this equals the RLD CR boundwhich is known to be tight.We calculated the Holevo CR bound for a pure two-mode squeezed state probe to be C H = 4 exp( − r ). This bound is superior to the SLD and RLD CR bound found by [3, 4].The dual homodyne measurement obtains our bound indicating that the bound is tight.Our calculation relied on the probe state being pure, so a natural extension to our workis to find the Holevo CR bound for N >
0, i.e. when the probe is a two mode squeezedthermal state, and to determine whether the dual homodyne measurement is also optimalis this case.
Acknowledgements
This research is supported by the Australian Research Council(ARC) under the Centre of Excellence for Quantum Computation and Communication Tech-nology (CE110001027). We would also like to thank Jing Yan Haw for comments on thepaper.
Appendix A: Calculations required for results1. Derivatives of displacement operator
To calculate the derivatives of the displacement operator, we use the Baker-Campbell-Hausdorff identity e A + B = e A e B e − [ A,B ] , (A1)14here A and B are operators that do not commute, but commute with [ A, B ]. ∂∂θ D ( θ ) = ∂∂θ exp (cid:18) i θ Q − i θ P (cid:19) (A2)= ∂∂θ exp (cid:18) − i θ P (cid:19) exp (cid:18) i θ Q (cid:19) exp (cid:18) − θ θ [ P, Q ] (cid:19) (A3)= (cid:18) − i P − θ [ P, Q ] (cid:19) exp (cid:18) − i θ P (cid:19) exp (cid:18) i θ Q (cid:19) exp (cid:18) − θ θ [ P, Q ] (cid:19) (A4)= (cid:18) − i P − θ [ P, Q ] (cid:19) D ( θ ) (A5)= (cid:18) − i P + i θ (cid:19) D ( θ ) , (A6)where we have used [ Q, P ] = 2 i . Similarly, ∂∂θ D ( θ ) = ∂∂θ exp (cid:18) i θ Q − i θ P (cid:19) (A7)= ∂∂θ exp (cid:18) i θ Q (cid:19) exp (cid:18) − i θ P (cid:19) exp (cid:18) − θ θ [ Q, P ] (cid:19) (A8)= (cid:18) i Q − θ [ Q, P ] (cid:19) exp (cid:18) i θ Q (cid:19) exp (cid:18) − i θ P (cid:19) exp (cid:18) − θ θ [ Q, P ] (cid:19) (A9)= (cid:18) i Q − θ [ Q, P ] (cid:19) D ( θ ) (A10)= (cid:18) i Q − i θ (cid:19) D ( θ ) . (A11)
2. Performing the minimization for the two-mode probe
In order to contend with the absolute value in Eq. (81), we consider two cases, case 1: g is greater or equal to zero, and case 2: g is less than zero. We consider the case when r = 0.When r = 0, the problem reduces to a single-mode probe and is discussed in section V A. case 1: g ≥ Minimize h = f + 2 g (A12)subject to g ≥ . (A13)15rom the Karush-Kuhn-Tucker conditions, necessary (but not sufficient) conditions for theminimum are − ∇ ( f + 2 g ) = − λ ∇ g (A14) g ≥ λ ≥ λg = 0 (A17)where ∇ = ( ∂∂s , ∂∂k , ∂∂k , ∂∂s ). Equation (A14) becomes − r λ − λ − − r r λ −
20 0 λ − r s k k s = − λ sinh r − λ sinh r . (A18)For r = 0, this set of equations has no solutions when λ = 4 cosh r . We thus consider λ = 4 cosh r , and find s = k = λ sinh r r − λ and k = s = 0 . (A19)From Eq. (A17) we have that either λ = 0 or g = 0. Let us consider each case seperately. case 1a: λ = 0 When λ = 0 the solutions (A19) become s = k = k = s = 0 , (A20)which gives g = − sech r < case 1b: g = 0 When g = 0, we solve for λ to get λ = 4e ± r cosh r . Both are valid solutions and give h = 4e ± r . Although h = 4e r satisfies the Karush-Kuhn-Tucker conditions, is not theminimum so we can ignore this solution. 16 ase 2: g < Minimize h = f − g (A21)subject to g < − ∇ ( f − g ) = λ ∇ g (A23) g ≤ λ ≥ λg = 0 . (A26)Since we require g <
0, condition (A26) implies λ = 0 for which condition (A23) becomes − cosh 2 r − cosh 2 r r
10 0 1 cosh 2 r s k k s = − r − r . (A27)For r = 0, this has the solution s = k = 0, s = k = csch r which gives g = csch r > solution Putting it all together, the smallest solution satisfying the Karush-Kuhn-Tucker condi-tions, and hence the minimum of h and the Holevo bound is C H = 4 exp( − r ). [1] A. S. Holevo. Noncommutative analogues of the cram´er-Rao inequality in the quantum mea-surement theory. Lecture Notes in Mathematics, pages 194–222, 1976.[2] Alexander S Holevo. Probabilistic and statistical aspects of quantum theory, volume 1.Springer Science & Business Media, 2011.
3] MG Genoni, MGA Paris, G Adesso, H Nha, PL Knight, and MS Kim. Optimal estimation ofjoint parameters in phase space. Physical Review A, 87(1):012107, 2013.[4] Yang Gao and Hwang Lee. Bounds on quantum multiple-parameter estimation with gaussianstate. The European Physical Journal D, 68(11):1–7, 2014.[5] CW Helstrom. Minimum mean-squared error of estimates in quantum statistics. Physicsletters A, 25(2):101–102, 1967.[6] Carl W Helstrom. Quantum detection and estimation theory. Journal of Statistical Physics,1(2):231–252, 1969.[7] H Yuen and Melvin Lax. Multiple-parameter quantum estimation and measurement of non-selfadjoint observables. IEEE Transactions on Information Theory, 19(6):740–750, 1973.[8] Matteo GA Paris. Quantum estimation for quantum technology. International Journal ofQuantum Information, 7(supp01):125–137, 2009.[9] D. Petz and C. Ghinea. Introduction to Quantum Fisher Information, chapter 15, pages261–281. World Scientific, Jan 2011.[10] Masahito Hayashi and Keiji Matsumoto Asymptotic performance of optimal state estimationin quantum two level system. arXiv:quant-ph/0411073, 2006.[11] Koichi Yamagata, Akio Fujiwara, and Richard D. Gill. Quantum local asymptotic normalitybased on a new quantum likelihood ratio. The Annals of Statistics, 41(4):2197–2217, Aug2013.[12] K Matsumoto. A new approach to the Cramer-Rao-type bound of the pure-state model.Journal of Physics A: Mathematical and General, 35(13):3111–3123, Mar 2002.[13] Akio Fujiwara. Linear random measurements of two non-commuting observables. Math. Eng.Tech. Rep, 94(10), 1994.[14] Akio Fujiwara and Hiroshi Nagaoka. An estimation theoretical characterization of coherentstates. Journal of Mathematical Physics, 40(9):4227–4239, 1999.[15] Akio Fujiwara. Multi-parameter pure state estimation based on the right logarithmic deriva-tive. Math. Eng. Tech. Rep, 94(9):94–10, 1994.3] MG Genoni, MGA Paris, G Adesso, H Nha, PL Knight, and MS Kim. Optimal estimation ofjoint parameters in phase space. Physical Review A, 87(1):012107, 2013.[4] Yang Gao and Hwang Lee. Bounds on quantum multiple-parameter estimation with gaussianstate. The European Physical Journal D, 68(11):1–7, 2014.[5] CW Helstrom. Minimum mean-squared error of estimates in quantum statistics. Physicsletters A, 25(2):101–102, 1967.[6] Carl W Helstrom. Quantum detection and estimation theory. Journal of Statistical Physics,1(2):231–252, 1969.[7] H Yuen and Melvin Lax. Multiple-parameter quantum estimation and measurement of non-selfadjoint observables. IEEE Transactions on Information Theory, 19(6):740–750, 1973.[8] Matteo GA Paris. Quantum estimation for quantum technology. International Journal ofQuantum Information, 7(supp01):125–137, 2009.[9] D. Petz and C. Ghinea. Introduction to Quantum Fisher Information, chapter 15, pages261–281. World Scientific, Jan 2011.[10] Masahito Hayashi and Keiji Matsumoto Asymptotic performance of optimal state estimationin quantum two level system. arXiv:quant-ph/0411073, 2006.[11] Koichi Yamagata, Akio Fujiwara, and Richard D. Gill. Quantum local asymptotic normalitybased on a new quantum likelihood ratio. The Annals of Statistics, 41(4):2197–2217, Aug2013.[12] K Matsumoto. A new approach to the Cramer-Rao-type bound of the pure-state model.Journal of Physics A: Mathematical and General, 35(13):3111–3123, Mar 2002.[13] Akio Fujiwara. Linear random measurements of two non-commuting observables. Math. Eng.Tech. Rep, 94(10), 1994.[14] Akio Fujiwara and Hiroshi Nagaoka. An estimation theoretical characterization of coherentstates. Journal of Mathematical Physics, 40(9):4227–4239, 1999.[15] Akio Fujiwara. Multi-parameter pure state estimation based on the right logarithmic deriva-tive. Math. Eng. Tech. Rep, 94(9):94–10, 1994.