Preferred Measurements: Optimality and Stability in Quantum Parameter Estimation
PPreferred Measurements: Optimality and Stability in Quantum Parameter Estimation
Gabriel A. Durkin ∗ Quantum Laboratory , NASA Ames Research Center, Moffett Field, California 94035, USA (Dated: October 30, 2018)We explore precision in a measurement process incorporating pure probe states, unitary dynamics and com-plete measurements via a simple formalism. The concept of ‘information complement’ is introduced. It under-mines measurement precision and its minimization reveals the system properties at an optimal point. Maximallyprecise measurements can exhibit independence from the true value of the estimated parameter, but demandingthis severely restricts the type of viable probe and dynamics, including the requirement that the Hamiltonianbe block-diagonal in a basis of preferred measurements. The curvature of the information complement near aglobally optimal point provides a new quantification of measurement stability.
PACS numbers: 03.65.Ta,06.20.Dk,,42.50.St,42.50.Dv
Scientists strive for an understanding of Nature by a physi-cal interaction introducing correlations between observer andobserved, and the process is called measurement. Fundamen-tal limitations on the precision of any measurement exist dueto the geometric distinguishability of quantum states [1], yetfor a quantum observable there is still a classical probabilitydistribution over measurement outcomes. It may depend onsome real-valued system parameter θ such as an interactiontime or interferometric phase that has no associated Hermitianobservable and is not measurable directly. Yet, the estimationof θ may be the true goal of the measurement. Inferring θ within a confidence interval (precision) ∆ θ from frequenciesof measurement outcomes is a standard challenge in classicalinformation theory with an established methodology [2]. Ina quantum context, the conventional approach stretches backdecades [3], employing techniques from Riemannian geome-try to find precision limits via ‘quantum Fisher information’(QFI) [4, 5, 6, 7] , a function of the input state and dynamicsalone. It defines a precision limit for measurements but shedslittle light on what measurement to use – those proposed asoptimal are typically functions of the unknown θ [8]. (It isof limited utility that the parameter estimation should requireprior knowledge of its true value. Adaptive techniques em-ploying feedback have been proposed to circumvent this prob-lem [9].)In this letter we harness a quantum formalism for purestates emerging directly and naturally from a single resultof classical information theory. The method incorporates allthree instrument components – input, dynamics (Hamiltonian)and measurement choice – on an equal footing without re-course to QFI or the mathematical apparatus supporting it.Important metrological results will be confirmed in a straight-forward manner along the way towards developing new con-ditions for optimality and stability in quantum measurements.A preferred measurement ˆ M has three important proper-ties: (1) high precision, or in the many-body case, ‘supra-classical’ precision. (By this it is meant that the precision isbetter for collective dynamics than is possible for componentparts evolving in a separable state [6].) (2) If possible, ˆ M isindependent of the estimated parameters [5, 10], and (3) ˆ M is highly stable, i.e. the precision exhibits robustness against small perturbations in the state or measurement alignment.Below, these goals are given explicit mathematical expression;optimality criteria imposed on probe and dynamics.Consider a maximal test [11] having outcomes labelled k and an associated probability distribution P ( θ ) = { p k ( θ ) } that depends on a continuous real parameter θ . The parame-ter estimation task involves an inference of θ from { p k ( θ ) } .Classical Fisher information [12] is defined as J ( θ ) = (cid:88) k p k ( θ ) (cid:2) ∂ θ ln p k ( θ ) (cid:3) , (1)a measure of the information contained in the distribution P ( θ ) about the parameter θ [2]. Unlike the quantum coun-terpart [1] it defines a unique distance metric on probabilityspace [13]. An explicit lower bound for the standard error ofan unbiased estimate ˜ θ on the true value θ is given by the re-ciprocal of the Fisher information, ( δ ˜ θ ) ≥ / J ( θ ) , calledthe Cram´er-Rao bound [12]. For optimal precision one musttherefore maximize J ( θ ) .Take a complete measurement observable ˆ M with out-comes { m k } associated with distribution { p k } . Apply ˆ M toa quantum system that previously evolved from a known ini-tial state | ψ (cid:105) under the dynamics of some time-independentHamiltonian ˆ H for time θ . The Schr¨odinger equation gov-erns the dynamics i∂ θ | ψ θ (cid:105) = ˆ H | ψ θ (cid:105) , (where (cid:126) = 1 ) andthe time evolution is explicitly | ψ θ (cid:105) = exp {− i ˆ Hθ }| ψ (cid:105) . Writing the spectral decomposition of the measurement as ˆ M = (cid:80) k m k | k (cid:105)(cid:104) k | then complex amplitudes (cid:104) k | ψ θ (cid:105) = r k exp { iφ k } give probabilities p k = (cid:104) k | ψ θ (cid:105)(cid:104) ψ θ | k (cid:105) = r k , where { p k , r k } ∈ [0 , and φ k ∈ [0 , π ) are all real-valuedfunctions of θ . Replacing p k with r k in Eq.(1) gives: J ( θ ) = 4 (cid:88) k ˙ r k , (2)which we can now use to find an operator expression for theclassical Fisher information. Differentiating r k gives r k r k = (cid:104) k | ψ θ (cid:105)(cid:104) ˙ ψ θ | k (cid:105) + (cid:104) k | ˙ ψ θ (cid:105)(cid:104) ψ θ | k (cid:105) . (3)Now, (cid:104) k | ˙ ψ θ (cid:105) = ∂ θ ( r k e iφ k ) = e iφ k ( ˙ r + ir ˙ φ k ) = |(cid:104) k | ˙ ψ θ (cid:105)| e i ( φ k + τ k ) where we define a velocity vector with ra- a r X i v : . [ qu a n t - ph ] S e p FIG. 1: An optimal measurement scheme for a Hamiltonian ˆ H span-ning any number of dimensions may be restricted to the qubit sub-space of its extremal eigenvectors | λ ↑ (cid:105) and | λ ↓ (cid:105) . (Other measure-ment elements ∈ {| k (cid:105)} span an orthogonal subspace.) On the Blochsphere: the diagonal basis of ˆ H defines the z -axis, and the opti-mal measurement projectors | k ± (cid:105) the x -axis. An optimal probe state | ψ opt (cid:105) lies anywhere in the equatorial plane. dial and transverse components { ˙ r k , r k ˙ φ k } , and inclination: τ k = tan − ( r k ˙ φ k / ˙ r k ) = arg (cid:104) k | ˙ ψ θ (cid:105) − arg (cid:104) k | ψ θ (cid:105) . (4)Eq.(3) yields ˙ r k = cos τ k |(cid:104) k | ˙ ψ θ (cid:105)| . From theSchr¨odinger equation we also have i (cid:104) k | ˙ ψ θ (cid:105) = (cid:104) k | ˆ H | ψ θ (cid:105) .Substituting this expression and squaring gives ˙ r k =cos τ k (cid:104) ψ θ | ˆ H | k (cid:105)(cid:104) k | ˆ H | ψ θ (cid:105) . Summing over all outcomes ‘ k ’gives the Fisher information using Eq.(2): J ( θ ) =4 (cid:80) k cos τ k (cid:104) ψ θ | ˆ H | k (cid:105)(cid:104) k | ˆ H | ψ θ (cid:105) . Thus we can define a non-linear, positive and hermitian operator ˆ F θ , diagonal in themeasurement basis: ˆ F θ = 4 (cid:88) k cos τ k | k (cid:105)(cid:104) k | = 4 (cid:88) k c ψ,k | k (cid:105)(cid:104) k | , (5)such that the Fisher information is then J ( θ ) = (cid:104) ψ θ | ˆ H ˆ F θ ˆ H | ψ θ (cid:105) = (cid:104) ψ | ˆ H ˆΦ θ ˆ H | ψ (cid:105) , (6)where ˆΦ θ = e i ˆ Hθ ˆ F θ e − i ˆ Hθ = 4 (cid:80) k c ψ,k | k (cid:48) (cid:105)(cid:104) k (cid:48) | is the uni-tarily transformed operator. Due to the non-linear nature of ˆ F θ , a basis transformation | k (cid:105) (cid:55)→ | k (cid:48) (cid:105) gives a different result: ˆ F θ (cid:55)→ ˆ F (cid:48) θ = 4 (cid:80) k c ψ,k (cid:48) | k (cid:48) (cid:105)(cid:104) k (cid:48) | , not equivalent to ˆΦ θ , since c ψ,k (cid:48) (cid:54) = c ψ,k generally. This explicit definition of ˆ F θ will beuseful in the qubit optimization to come. Fixed Probe Optimization : Now let us establish an upperbound for J ( θ ) in terms of a fixed input and dynamics, butvarying the measurement. The completeness of the measure-ment basis provides a resolution of the identity (cid:80) k | k (cid:105)(cid:104) k | = , and therefore, with cos τ k = 1 − sin τ k , Eq.(6) becomes: J ( θ )4 = (cid:104) ˆ H (cid:105) − (cid:88) k sin τ k (cid:104) ψ θ | ˆ H | k (cid:105)(cid:104) k | ˆ H | ψ θ (cid:105) = (cid:104) ˆ H (cid:105) − (cid:88) k ( r k ˙ φ k ) = (cid:104) ˆ H (cid:105) − K ( θ ) , (7) where we define the ‘ information complement ’ : K ( θ ) = (cid:88) k ( r k ˙ φ k ) = (cid:88) k p k ˙ φ k = (cid:104) ˙ ϕ (cid:105) c (8)a non-negative functional of the probe, measurement andHamiltonian that can only reduce J . Here ˙ ϕ is a classicalrandom variable taking values from the set { ˙ φ k } and the c subscript denotes the expectation value is classical. We nowlook for a basis {| k (cid:105)} and associated set { r k , ˙ φ k } that mini-mizes K for a fixed input | ψ (cid:105) . First, a new description of theHamiltonian expectation value is needed: (cid:104) ˆ H (cid:105) = i (cid:104) ψ | ˙ ψ (cid:105) = i (cid:88) k (cid:104) ψ | k (cid:105)(cid:104) k | ˙ ψ (cid:105) = i (cid:88) k ( r k ˙ r k + ir k ˙ φ k )= −(cid:104) ˙ ϕ (cid:105) c + i ∂ θ (cid:18) (cid:88) k r k (cid:19) = −(cid:104) ˙ ϕ (cid:105) c (9)Therefore, K − (cid:104) ˆ H (cid:105) = ∆ c ˙ ϕ ≥ . Comparing Eq.(7) it fol-lows directly that J ≤ ˆ H for a fixed input | ψ (cid:105) . (Thiswas derived by a different method in [5].) The bound is satu-rated by a particular qubit input, as we will now show. Optimizing for a Qubit:
Probes that are eigenstate of ˆ H give ˙ r k (cid:55)→ and J = 0 from Eq.(2) because any eigen-state | λ (cid:105) of ˆ H only gains a phase during its evolution. Thus r k ( θ ) = |(cid:104) k | e i ˆ Hθ | λ (cid:105)| = |(cid:104) k | e iλθ | λ (cid:105)| = |(cid:104) k | λ (cid:105)| = r k (0) ,and ˙ r k = 0 . For optimality over all | ψ (cid:105) and {| k (cid:105)} the inputstate must thus be a superposition of at least two Hamiltonianeigenvectors, | ψ (cid:105) (cid:55)→ cos γ | λ (cid:105) + e iχ sin γ | λ (cid:105) . Choose sucha qubit probe and a measurement basis {| k (cid:105) , | k (cid:105)} spanningthe same C as | λ , (cid:105) : | k (cid:105) = cos α | λ (cid:105) + sin α | λ (cid:105) , | k (cid:105) = − sin α | λ (cid:105) + cos α | λ (cid:105) . This is a two-dimensional subspaceof the full Hilbert space supporting ˆ H . Here it has been cho-sen that {| k , (cid:105) defines the x axis on the Bloch sphere ofFIG.1, hence Im (cid:104) k | λ (cid:105) =0. By confining | k , (cid:105) to the span of | λ , (cid:105) then (using the fact it is diagonal in the measurementbasis) one restricts interest to the component of ˆΦ θ within thisqubit space too: ˆΦ θ = e i ˆ Hθ { c ψ,k | k (cid:105)(cid:104) k | + c ψ,k | k (cid:105)(cid:104) k | + . . . } e − i ˆ Hθ = c ψ,k (cid:18) c e − i ( λ − λ ) θ sce + i ( λ − λ ) θ sc s (cid:19) + c ψ,k (cid:18) s − e − i ( λ − λ ) θ ) sc − e + i ( λ − λ ) θ ) sc c (cid:19) + . . . where c ( s ) is cos α ( sin α ). We ignore elements of ˆΦ θ that project onto the remaining Hilbert space, orthogonal to {| λ (cid:105) , | λ (cid:105)} . Defining β = χ − ( λ − λ ) θ , then angles { α, β, γ } give an expectation value (cid:104) ψ | ˆ H ˆΦ θ ˆ H | ψ (cid:105) : J ( α, β, γ ) = − λ − λ ) s [2 α ] s [2 γ ] s [ β ]( c [2( α − γ )] + c [2( α + γ )] + 2 c [ β ] s [2 α ] s [2 γ ] − c [2( α − γ )] + c [2( α + γ )] + 2 c [ β ] s [2 α ] s [2 γ ] + 2) (10)writing sin as ‘ s ’ and cos as ‘ c ’. J is optimized by angles { α, γ } (cid:55)→ π/ , independent of the value of β and giving asaturable bound: (cid:104) ψ | ˆ H ˆΦ θ ˆ H | ψ (cid:105) ≤ ( λ − λ ) . This is theupper bound on the Fisher information for any superpositionof two eigenstates of the Hamiltonian. It is saturated by aprobe state | ψ opt (cid:105) = ( | λ (cid:105) + e iχ | λ (cid:105) ) / √ , where χ ∈ [0 , π ) ,see FIG.1. The result α (cid:55)→ π/ dictates an optimal measure-ment scheme with components: | k ± (cid:105) = ( | λ (cid:105) ± | λ (cid:105) ) / √ , (11)also in FIG.1. Other basis elements ∈ {| k (cid:105)} span an orthogo-nal subspace. It is significant that the optimal measurement isindependent of β , and hence the estimated parameter θ . Generalization to Higher Dimensions:
The above resultshows that for a given ˆ H , the maximal Fisher information isbounded from below by ( λ ↑ − λ ↓ ) = || ˆ H || where λ ↑ ( λ ↓ )is the max (min) eigenvalue of ˆ H , and || ˆ H || = ( λ ↑ − λ ↓ ) isthe operator seminorm of the Hamiltonian [7]. The variancehas the seminorm as an upper bound: || ˆ H || ≥ ˆ H , cre-ating a bridge between the qubit result with that for a fixed | ψ (cid:105) in a higher dimensional space. We saw for a fixed | ψ (cid:105) that J ≤ ˆ H . Therefore the qubit maximum variancestate must be the universally optimal state over the full Hilbertspace; it saturates the variance bound. Concisely:max | ψ (cid:105) , {| k (cid:105)} J = || ˆ H || . (12)(This bound has been discussed previously in terms of quan-tum Fisher information [7].) A corollary of Eq.(12) is that nogreater number of superposed energy eigenstates can be usedas an input to improve on the Fisher information provided bythe (qubit) maximum variance state ( | λ ↑ (cid:105) + e iχ | λ ↓ (cid:105) ) / √ . Theoptimal measurement set can be chosen as the one with twoelements straddling the qubit subspace of extremal eigenval-ues in Eq.(11). Importantly, it retains independence from thetrue value of θ . Optimal Measurement Criterion : There may certainly existmore than one optimal measurement set for a given input, andwe can define an optimal measurement as one that satisfies K = (cid:104) ˆ H (cid:105) , i.e. for measurements saturating J ≤ ˆ H : ∆ c ˙ ϕ = K − (cid:104) ˆ H (cid:105) = (cid:88) k r k (cid:0) ˙ φ k − (cid:104) ˙ ϕ (cid:105) c (cid:1) (cid:55)→ (13)Using Eq.(9), for optimality ˙ φ k = (cid:104) ˙ ϕ (cid:105) c = −(cid:104) ˆ H (cid:105) , ∀ k . This isequivalent to a condition presented in [4], Im (cid:104) ψ θ | k (cid:105)(cid:104) k | ψ ⊥ (cid:105) =0 , ∀ k , where | ψ ⊥ (cid:105) = | ˙ ψ (cid:105) − (cid:104) ψ θ | ˙ ψ (cid:105)| ψ θ (cid:105) is the component or-thogonal to | ψ θ (cid:105) in the qubit space spanned by {| ψ θ (cid:105) , | ˙ ψ (cid:105)} .Measurements that are superpositions of {| ψ θ (cid:105) , | ψ ⊥ (cid:105)} , e.g. those of [8], are generally only instantaneously optimal – asthe state evolves, precision decreases. Demanding parameterindependent optimality means ˙ φ k = −(cid:104) ˆ H (cid:105) at all times, i.e. φ k = −(cid:104) ˆ H (cid:105) θ + ζ k , ∀ k, θ (14)Enforcing this condition will limit the viable probes, measure-ments and dynamics. Starting with e iφ k ( ˙ r + ir ˙ φ k ) = (cid:104) k | ˙ ψ (cid:105) = − i (cid:104) k | ˆ H | ψ (cid:105) from just after Eq.(2) and substituting Eq.(14) re-stricts the re-zeroed Hamiltonian, ˜ H = ˆ H − (cid:104) ˆ H (cid:105) ˆ I : ˜ H ( I ) | ψ θ (cid:105) = i | ˙ ψ (cid:105) , ˜ H ( R ) | ψ θ (cid:105) = 0 . (15)Here we have chosen an optimal | k (cid:105) basis, in which ˜ H hasa real (imaginary) part denoted by R ( I ) . Only the imagi-nary part determines dynamics [15], and | ψ θ (cid:105) is confined tothe null space of ˜ H ( R ) . That ˜ H ( R ) exp {− i ˜ H ( I ) δθ }| ψ (cid:105) = 0 implies [ ˜ H ( R ) , ˜ H ( I ) ] = 0 . Therefore, in the optimal basis theHamiltonian is block diagonal: ˜ H = ˜ H ( R ) ⊕ ˜ H ( I ) . Note thenull space of ˜ H ( R ) needs to be at least two-dimensional for | ψ (cid:105) to evolve at all – we have seen that maximum precisionis possible in just such a qubit space. (Then ˜ H ( I ) must be pro-portional to the Pauli σ y .) Complete measurements like ˆ M are not covariant; see [5, 10, 14] for a discussion of over-completecovariant measurements. Stability at Optimal Point:
Fulfilling the conditions abovemay indeed give maximum precision but what if small devi-ations in the measurement orientation lead to a dramatic re-duction in the Fisher information? If the measurement basisis rotated slightly by | k (cid:105) (cid:55)→ exp {− i ˆ hδω }| k (cid:105) then the over-lap becomes (cid:104) k | ψ θ (cid:105) (cid:55)→ (cid:104) k | e i ˆ hδω e − i ˆ Hδθ | ψ θ (cid:105) , combining dy-namics for measurement drift and state evolution. The in-formation complement is now a function of two variables, K ( θ, ω ) . At a turning point we have ∂ θ K = ∂ ω K = 0 ,but this doesn’t indicate much. However, at the global min-imum ( gm ) for parameter-independent evolution Eq.(14) ap-plies and therefore ¨ φ k = − ∂ θ (cid:104) ˆ H (cid:105) = 0 , ∀ k . Here derivativesare ∂ θ K = ∂ θ K = 0 , meaning that to enforce the optimal-ity condition for all θ values confirms a zero curvature of J in the direction of evolution. Denoting derivatives of θ by adot and of ω by a dash, the other second order derivatives forparameter-independent evolution are: ∂ K ∂ω∂θ (cid:12)(cid:12)(cid:12)(cid:12) gm = − (cid:104) ˆ H (cid:105) (cid:88) k ˙ p k ˙ φ (cid:48) k ,∂ K ∂ω (cid:12)(cid:12)(cid:12)(cid:12) gm = 2 (cid:104) ( ˙ ϕ (cid:48) ) (cid:105) c − (cid:104) ˆ H (cid:105) (cid:20) (cid:104) ˙ ϕ (cid:48)(cid:48) (cid:105) c +2 (cid:88) k p (cid:48) k ˙ φ (cid:48) k (cid:21) (16)Probes returning (cid:104) ˆ H (cid:105) = 0 produce a minimum in K for de-viations δω from the optimal measurements – then ∂ ω K = 0 ˆ J z ˆ J y J j ˆ J x π ( a ) ( b ) ( c ) π/ j ω y +0 . − . δω ⊥ ω y δ ω ⊥ FIG. 2: For a spin Hamiltonian ˆ H (cid:55)→ ˆ J y the maximum variancestate | ψ (cid:105) = ( | j, + j (cid:105) y + exp { iχ }| j, − j (cid:105) y ) / √ (sometimes called aNOON state [16, 17]) yields greatest precision: J = || ˆ J y || = 4 j .Eigen-equations are ˆ J | j, m (cid:105) i = j ( j + 1) | j, m (cid:105) i and ˆ J i | j, m (cid:105) i = m | j, m (cid:105) i for i ∈ { x, y, z } . The optimal measurement set is notunique; one is given by Eq.(11) (with λ , (cid:55)→ ± j ) but others include ˆ J x . Spherical surfaces ( a ) and ( b ) are coloured by precision: Fisherinformation for rotational drift | k (cid:105) (cid:55)→ exp { i ˆ J z ω z } exp { i ˆ J y ω y }| k (cid:105) of optimal measurements away from ˆ J x in ( a ) or from the set ofEq.(11) in ( b ) . Blue regions correspond to precision below the clas-sical bound (associated with J max for n = 2 j individual spin / particles, J ≤ j ). Red regions indicate supra-classical precision j < J ≤ j where quantum correlations contribute to the pre-cision [18]. The clamshell structure of ( a ) is characteristic of themeasurement landscape at all j with the red zones or ”hotspots”numbering j around the equator, each decreasing in size as j in-creases. In ( a ) precision is highest and curvature is zero along theequator, showing that measurements cos ξ ˆ J x + sin ξ ˆ J z are optimaland parameter independent for dynamics ˆ J y . However, curvature isnon-zero along lines of longitude and dependent on drift angles ω y,z so | ψ (cid:105) may offer greatest precision but low stability. Precision con-tours in ( c ) mark the classical precision boundary for NOON statesof spin j = 3 / (dashed) and / (unbroken). The angle δω ⊥ in-dicates how much transverse drift can be tolerated while maintainingsupra-classical precision. and ∂ ω K ≥ . (Precision is optimal in both evolution anddrift variables.) Ideally (cid:104) ˆ H (cid:105) = ∂ ωθ φ k = 0 , ∀ k , then optimalmeasurements are perfectly stable, i.e. K has a zero Hessian.Well, (cid:104) ˆ H (cid:105) = 0 is easily obtained by a recalibration of the zeroenergy point, ˆ H (cid:55)→ ˜ H without changing the physics of thesystem. However, the mixed second derivative of φ k cannotbe zero if drift dynamics rotate the measurement basis closerto the eigenbasis of ˆ H . (Parameter information becomes zerowhen measurements are ˆ H eigenstates.) The ideal scenario ofEqs.(15) produces Tr [ ˆ M ˜ H ( I ) ] = 0 and maintaining this underdrift e − iω ˆ h ˆ M e iω ˆ h implies an orthogonality condition:Tr (cid:16) ˜ H ( I ) [ ˆ M , ˆ h ] (cid:17) = 0 (17) Summary and Outlook : The formalism we have developedincorporates all aspects of quantum parameter estimation ex-plicitly; probe | ψ (cid:105) , dynamics ˆ H , and measurement, {| k (cid:105)} ,clarifying how precision is determined by the interplay of allthree. The information complement K was introduced, a mea-surement dependent functional that undermines the Fisher in-formation. At the global optimum for parameter-independent measurement, phases φ k = arg (cid:104) k | ψ θ (cid:105) vary linearly with theparameter θ in proportion to their average energy, restrictingdynamics to imaginary Hamiltonians in the optimal measure-ment basis.Greatest possible precision within the Hilbert spacespanned by the Hamiltonian exists in the qubit subspace ofthe maximal variance input state and ultimate precision overall probes and measurements is completely defined by the ex-tremal energy eigenvalues. No additional dynamical structureis relevant, nor is the dimension of the Hilbert space. The mostprecise measurement is also parameter free.Stability of optimal measurements was quantified in termsof the curvature of K in the vicinity of its global minimum.For dynamics ˆ h causing unitary drift of measurement ori-entation the curvature indicates whether the apparatus is ofpragmatic utility, quantifying its immunity to alignment er-rors at the optimal setting. States with zero average energy (cid:104) ˆ H (cid:105) are associated with optimal parameter-independent mea-surements that become suboptimal given measurement drift,at a rate determined by the magnitude of mixed second orderderivatives of phases φ k . See FIG.2 for an illustration of theprecision terrain for spin states proposed previously for pa-rameter estimation.In future, it may prove fruitful to develop the approach pre-sented here to incorporate evolution governed by completelypositive maps and generalised measurements. This work wascarried out under a contract with Mission Critical Technolo-gies at NASA Ames Research Center. The author thanksHugo Cable, Gen Kimura and Vadim Smelyanskiy for usefuldiscussions ∗ Electronic address: [email protected][1] A. Rivas and A. Luis, Phys. Rev. A , 063813 (2008); S. Luoand Q. Zhang, Phys. Rev. A. , 032106 (2004); D. Petz and C.Sudar, J. Math. Phys , 2662 (1996).[2] T. M. Cover and J. A. Thomas, Ch.12, Elements of InformationTheory (Wiley, 1991).[3] C. W. Helstrom, Ch.VIII,
Quantum Detection and EstimationTheory (Academic, NY, 1976).[4] S. L. Braunstein and C.M. Caves, Phys. Rev. Lett , 3439(1994).[5] S. L. Braunstein, C.M. Caves, and G. J. Milburn, Annals ofPhysics , 135 (1996).[6] V. Giovannetti et al. , Phys. Rev. Lett , 010401 (2006).[7] S. Boixo et al. , Phys. Rev. Lett , 090401 (2007).[8] One optimal measurement is formed by the eigenvector set forthe density matrix’s symmetric logarithmic derivative, takenwith respect to the estimated parameter. See [4, 5].[9] M. Kacprowicz et al. , arXiv: 0906.3511 [quant-ph] (2009).[10] B. C. Sanders and G. J. Milburn, Phys. Rev. Lett , 2944(1995).[11] A. Peres, Ch.2, Quantum Theory (Wiley, 1993).[12] H. Cram´er,
Mathmatical Methods of Statistics (Princeton Uni-versity Press, 1946); R. A. Fisher, Proc. Camb. Phil. Soc. ,700 (1925).[13] N. N. ˇCencov, Statistical Decision Rules and Optimal Infer- ence. (Providence, R.I.: Amer. Math. Soc., 1982).[14] D. W. Berry and H. M. Wiseman, Phys. Rev. Lett , 5098(2000).[15] Generally, only some traceless Hermitian matrices can be uni-tarily transformed into a purely imaginary form, so this is in-deed a restriction on ˜ H if ˜ H ( I ) is to exist at all.[16] G. A. Durkin and J. P. Dowling, Phys. Rev. Lett , 070801 (2007).[17] J. J. Bollinger et al. , Phys. Rev. A , R4649 (1996); H. Lee etal. , J. Mod. Opt. , 2325 (2002).[18] C. M. Caves, Phys. Rev. D , 1693 (1981); Z.Y. Ou, Phys.Rev. A , 2598 (1997); V. Giovannetti et al. , Science306