Simultaneous Measurement of Non-Commuting Observables

a r X i v :q u a n t -p h /0702013v 1 1 F eb 20071SIMULTANEOUS MEASUREMENT OF NON-COMMUTINGOBSER V ABLESG.AQUINO ∗and B.MEHMANIInstitute for Theoretical Physics,University of Amsterdam,Valckenierstraat 65,1018XE,Amsterdam,The Netherlands∗E-mail:gaquino@science.uva.nlIt is shown that the full unknown state of a spin-12system,S,can statistically bedetermined by means of a repeated Stern-Gerlach experiment.In this process,the x -and y -component of the polarization vector are destroyed as a consequence of the non-commutation of the spin operator in the transversal directions.Other ex-perimental settings seem therefore necessary to measure the unknown polarization vector of S.Its three components are represented by incompatible observables,the Pauli operators,and their direct determination requires three macroscopic appara-tuses,differing by a change of orientation of the magnets and detectors.Likewise,the state of any two-level system,represented by a 2×2density matrix ˆρcan be fully determined only through measurement of three linearly independent observ-ables which do not commute and cannot be simultaneously measured.Nevertheless,we will prove that the whole unknown density matrix of such a system2S,in particular the full polarization vector of a spin-12system as an assistant in a known pure state to determinethe initial state of S.Then,in section4,we show that it is also possible to use an electromagneticfield in a coherent state to determine the whole elements of the unknown density matrix of S.2.Statement of the ProblemThe idea of mapping the stateˆρof an unknown spin-13 Repeated measurements ofˆΩwhich means repeated simultaneous measurements of ˆωandˆo,determines the joint probabilities to observeωi for S and o a for APα≡P ia=Tr[ˆRτ(ˆπi⊗ˆp a)],(5) whereˆRτis defined in(Eq.1).In fact the numbers Pαare the diagonal elements ofˆU†(ˆρ⊗ˆR)ˆU in the factorized basis which diagonalizesˆωandˆo.The whole elements of the density matrixˆρcan be determined by the mapping ˆρ→Pα.Like the idea offinding a universal observable,4ifˆH couples S and A properly,this mapping will be expected to be invertible for n≥m.We shall see that even simple interactions can achieve this condition.For given observablesˆωof S andˆo of A and for a known initial stateˆR,the precision of this procedure relies on the ratio between the experimental uncertainty of Pαand the resulting uncertainty onˆρ,which can be characterized by the determinant,∆,of the transformation (5).For∆=0it is impossible to determineˆρfrom Pα.This means,the system is unstable with respect to small errors made during experimental determination of Pαor equivalently( ˆσz , ˆs z , ˆs zˆσz ).Therefore,the HamiltonianˆH and time interval τshould be chosen so as to maximize|∆|over all possible unitary transformations.3.Spin-12system in a known pure state.The density matrix of a spin-12(ˆ1+ˆσz),ˆp a=12(ˆ1+ˆs z)into(Eq.5).Pα=uα+ vα· ρ,(7) whereuα=12[ˆU( ˆσ⊗ˆR)ˆU†]α,α,(8)withα={ia}={++,+−,−+,−−}and matrix elements have been represented in the standard representation of the Pauli matrices, ˆσand ˆs.The probabilities Pαshould be positive and normalized for any the density matrix,ˆρ,such that ρ2≤1. These conditions imply thatuα≥|vα|, αuα=1, α vα=0.(9)4The determinant of transformationˆρ→Pαcan be eitherv++·( v+−× v+−),(10) or any other permutations of three of the vectors v++, v+−, v−+and v−−.Therefore, the determinant of the transformation is four times the volume of the parallelepipedmade by three of these vectors.For example,∆=4 v++·( v+−× v+−).(11) If the unitary evolution operatorˆU is such that vectors vαare not coplanar,thetransformation(Eq.7)is invertible and one can determine ρfrom the set of Pα.Alternatively,ˆρis deduced from ˆσz , ˆs z and ˆs zˆσz at timeτ.The notationˆs zˆσz is used for simplicity instead ofˆs z⊗ˆσz which lives in the common Hilbert space of S and A.ˆσz,ˆs z andˆs zˆσz can be simultaneously measured and are in one to onecorrespondence with the set of probabilities Pα.Wefirst look for the upper bound of the determinant of transformation(Eq.7),|∆|implied by the conditions(Eq.9).First we note that|∆|increases with| vα|for eachα.We therefore maximize∆2under the constraintsα| vα|=1, α vα=0.(12) This yields a symmetric solution for all these vectorsuα=| vα|=1| vα|| vβ|=−112√4√4√4√4√3 ˆσz ,ρ2=√3 ˆs zˆσz ,(16) which gives directly the whole elements of the density matrix,ˆρ,in terms of the expectation values and the correlation of the commuting observablesˆσz andˆs z in thefinal state.Next step is tofind out the interaction Hamiltonian and the interaction timeτ5 which give such a description of the tested system,S.This correspondence can be achieved under the action of the HamiltonianˆH=12ˆσx(ˆs x cosφ+ˆs z sinφ)+1√√2(ˆs y sinφ+ˆs z)(18)ofˆH can be obtained by a rotation of ˆs and also achieves an optimal mapping ˆρ→Pα,providedˆs z→ˆs x sinφ+ˆs z cosφboth in the measured projectionsˆp a= 1/2(1±ˆs z)and in the initial stateˆR=ˆp+.Thefirst term in(Eq.18)describes,in the spin language,an Ising coupling,while the second term represents a transverse magneticfield acting on the assistant A.4.Assistant System as a Coherent State of LightIn this section we discuss the possibility of using light as an assistant to determine the elements of the density matrix of a spin-12system)and a single mode of radiation.This model is exactly solvable but stillrather non-trivial and itfinds direct experimental realization in quantum optics. The Hamiltonian reads:6ˆH=ˆHA +ˆH S+ˆH SA= ωˆa†ˆa+16One can then calculate exactly the relevant observables at time t using the Heisenberg equations of motion:˙ˆa=−iωˆa−iγˆσ−,˙ˆσ−=−iωˆσ−+iγˆσzˆa,˙ˆσz=2iγ(ˆa†ˆσ−−ˆσ+ˆa).(21) We will briefly outline the result.The exact solution of the above set of equations reads:ˆa(t)=e i(γˆV−ω)t cosγˆKt−iˆV sinγˆKtˆKˆσ−(0) ,ˆσ−(t)=e i(γˆV−ω)t cosγˆKt+iˆV sinγˆKtˆKˆa(0) ,(22)whereˆK= ˆV2+1.Note thatˆK andˆV commute so that their mutual ordering is irrelevant.We study the case in which the electromagneticfield is in a coherent state,a con-dition that coincides with the common experimental situation of a resonant laser mode interacting with a spin-12(1+ ˆσi ˆσi)⊗∞ k=0∞ m=0e−|α|2k!√2system at time t=0is possible.We report here the detailsof the calculation only for this relevant choice of observables.We have to calculate the expectation values of these three observables at time t.In order to perform this calculation,we will make use of the eigenfunctions of theˆV operator,which are: |φ±n =|n−1 |+ ±|n |−2(n≥1),|φ±0 =|φ0 =|0 |− (n=0),(24) and the operatorsˆV andˆK,when applied to these functions,evaluate to:ˆV|φ±n =±√n+1|φ±n .(25)7Let us introduce the following notation:ˆf (ˆV )=e i (γˆV −ω)t ⇒f ±n = φ±n |f (ˆV )|φ±n =ei (±γ√ˆK⇒S n = φ±n |S (ˆK )|φ±n =i sin γ√√n +1t ±√n !¯f +n ¯g +n −¯f −n ¯g −n n −¯S nn −|α|22λ1−¯S n¯f +n −¯f −n n2−√2λ∗2−¯S n¯f +n −¯f −n n !|g +n |2(n −|α|2)−S n √2+|S n |2λ1+S ng +n +g −nn ! nn¯f +n ¯g +n −¯f −n ¯g +n2g +n −g −n2(n −|α|2(n +1)λ1))+S n¯f +n ¯g +n +¯f −n ¯g −n2|α|2λ1)+|S n |2(¯f +n ¯g +n −¯f −n ¯g −nn+¯f +n +¯f −n √2¯f +n −¯f −nng +n −g −n2−|α|2n g −n+g +n 2) λ1+ n ¯f +n ¯g +n +¯f −n ¯g −n 2·|g +n |2√2¯g +n −¯g −n 2¯f +n −¯f −n n·g −n +g +n 2+¯f +n ¯g +n +¯f −n ¯g −n 2√α|S n |2g −n +g +n2+S n (|g +n |2¯f +n −¯f −n√2g +n +g −n n) λ28where we have also defined:λ1=1+ ˆσz(0)2system at time t=0.In order to assess if this systemof equations has solutions,we have to evaluate the determinant of the matrixˆM made up by the coefficients of ˆσx(0) , ˆσy(0) , ˆσz(0) appearing in the system of equations determined by the evaluation of(29).In order to achieve this goal,it is convenient to defineˆM ij= ∞n=0e−|α|2|α|2nα(√n+1t sinγ√nt sinγ√√nn+1sinγ√ntA13(n)=e iωt cosγ√ntA21(n)=(1+2n)(1+n−|α|2)−(1+n+|α|2)cos2γ√2(n+1)A22(n)=iαcosγ√ntn+1,A23(n)=A∗22(n)A31(n)=nA11(n),A32(n)=nA13(n),A33(n)=nA12(n)It is easy to check that these coefficients are related to the coefficients M ij(n)in the following way:M i1(n)=ℜ[A i1(n)],M i2(n)=ℜ[A i2(n)+A∗i3(n)],M i3(n)=ℑ[A i2(n)+A∗i3(n)](31) If we now calculate the determinant∆(t)of the matrixˆM,we obtain:∆(t)=∞ l=0∞ m=0∞ n=0e−3|α|2|α|2(l+n+m)9 amplitude,so that a time with a large enough determinant can be chosen to solve for the initial state of the spin-12system and avoidscases of ill-conditioned matrix inversion.Fig.1.of the model:line to |α|2=4,the dashed line to|α|2=9,withαreal and positive.5.ConclusionsWe have illustrated a procedure which allows to reconstruct the state of a spin-12system coupled to a coherentlaserfield.We have shown that in this case,after a proper choice of the commuting observables,it is always possible to reconstruct the initial state of the spin-12system. 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