Amplification limit of weak measurements: a variational approach
aa r X i v : . [ qu a n t - ph ] J u l Amplification limit of weak measurements: a variational approach
Shengshi Pang , ∗ Todd A. Brun , † Shengjun Wu , , ‡ and Zeng-Bing Chen § Department of Electrical Engineering, University of Southern California, Los Angeles, California 90089, USA Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics,University of Science and Technology of China, Hefei, Anhui 230026, China and Kuang Yaming Honors School, Nanjing University, Nanjing 210093, China
Post-selected weak measurement has been widely used in experiments to observe weak effectsin various physical systems. However, it is still unclear how large the amplification ability of aweak measurement can be and what determines the limit of this ability, which is fundamental tounderstanding and applying weak measurements. The limitation of the conventional weak valueformalism for this problem is the divergence of weak values when the pre- and post-selections arenearly orthogonal. In this paper, we study this problem by a variational approach for a generalHamiltonian H int = gA ⊗ Ω δ ( t − t ) , g ≪ . We derive a general asymptotic solution, and show thatthe amplification limit is essentially independent of g , and determined only by the initial state of thedetector and the number of distinct eigenvalues of A or Ω . An example of spin- particles with a pairof Stern-Gerlach devices is given to illustrate the results. The limiting case of continuous variablesystems is also investigated to demonstrate the influence of system dimension on the amplificationlimit. PACS numbers: 03.65.Ta, 03.65.Ca, 03.67.Ac, 42.50.-p
I. INTRODUCTION
The von Neumann projective measurement model iswell known theory for standard quantum measurements,in which the readings of a measurement are the eigen-values of an observable, and the system is projected intothe corresponding eigenstate of the observable. To real-ize such an ideal measurement, the spread width of thedetector’s wave functions must be sufficiently narrow, orthe interaction between the system and the detector mustbe sufficiently strong, so that the detector’s final states—translated by different eigenvalues of the observable—canbe distinguished with high probability.In contrast to von Neumann measurements, weak mea-surements (coined by Aharonov, Albert, and Vaidman in1988 [1]) exploit the opposite conditions: the initial de-tector state has a very wide spread, or the measurementstrength is ultra-weak. Such a weak measurement makesthe detector’s final states, translated by different eigen-values of the system observable, significantly overlap witheach other. And a further step of this protocol, postselec-tion, superposes these states. Interference between themcan dramatically change the original state of the detector(not only by a translation). A remarkable effect inducedby this interference is that the output from the postse-lected detector can be much larger than the eigenvaluespectrum of the system observable, due to the coherencein the superposed state of the detector canceling the ma-jor part of the original detector wave function [2].This striking difference from standard von Neumann ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] measurements makes weak measurement particularlyuseful to amplify small physical quantities. Experimentshave successfully realized the amplification of many dif-ferent physical effects by postselected weak measure-ments, including the spin Hall effect of light [3, 4], opticalbeam deflection [5, 6], optical frequency shift [7], opticalphase shift [8, 9], temperature shift [10], longitudinal ve-locities [11], etc. More experimental protocols have beenproposed [12–20]. Moreover, weak measurements havebeen realized on various physical systems besides opti-cal systems, including SQUIDs [21, 22] and NMR [23],among others.Despite the considerable existing research on the the-ory of weak measurements, and their increasing appli-cation in experiments, a fundamental problem is stillunclear: what is the ultimate limit of amplification ina postselected weak measurement? Usually, weak mea-surements are studied in the weak value formalism (see[24] for a general framework, and [25, 26] for reviews).However, in the weak value formalism, the amplificationof a weak measurement can be infinitely large if the in-ner product between the pre- and postselections of thesystem is sufficiently small. From a practical view, thisis obviously impossible. The root of this contradiction isthat the weak value formalism is valid only when the am-plification is small, since it is a first-order approximationtheory that works only when the response of the detec-tor is linear in the signal. When the amplification is toostrong, the response of the detector to the signal becomesnonlinear, so the weak value formalism breaks down. Itcannot give a valid result for the limit of amplification ina postselected weak measurement.The significance of this problem is manifold: (1) it de-termines clearly to what extent postselected weak mea-surements can amplify small signals in practical experi-ments, thus revealing the limits of the quantum advan-tage in this task; (2) it can show what determines theamplification limit of a weak measurement, thus provid-ing guidance for designing experiments; (3) it goes be-yond the limitation of the weak value formalism, so itsresult will be an important supplement to the currentknowledge of weak measurement. Given the wide appli-cation of weak measurements in many different branchesof physics, the solution of this problem will be broadlyuseful to the physics community.Despite the importance of this problem, few resultshave been known to date, and a complete solution is stillmissing. Numerical upper bounds were observed for somecases in [24, 27]. Certain special cases were studied withspecific assumptions on the observable A (e.g., A = I in [28, 29]), or on the detector states (e.g., a qubit sys-tem with a Gaussian detector in [30]). Orthogonal andasymptotically orthogonal pre- and postselections wereconsidered in [31]. An optimal detector for a given experi-mental setup was provided in [32]. In a more recent paper[33], a refined weak value method was attempted; but itis still not sufficient to give a rigorous solution without in-cluding higher order weak values—particularly when thedimension of the system or the detector is high—sincehigher order weak values can dominate over the lowestorder weak value. (This has been verified by weak mea-surements with OAM pointer states [34].)In this paper, we fill this gap by exploiting a variationalapproach to study the problem in a rigorous way. Wederive a general analytical solution for a weak couplingHamiltonian H int = gA ⊗ Ω δ ( t − t ) , g ≪ , and revealthe surprising property that the solution is independentof the coupling strength g when g ≪ , depending only onthe initial state of the detector and the dimension of thesystem or the detector, whichever is less (if A and Ω arenondegenerate). This is in marked contrast to the weakvalue formalism. For degenerate A or Ω , the degeneracywill decrease the amplification limit, which depends onthe number of distinct eigenvalues. The results are illus-trated in detail by an example of spin- particles passingthrough a pair of Stern-Gerlach devices. We also con-sider continuous variable systems as a limiting example,to show how the dimension of the system can significantlyinfluence the amplification limit, which also is missed bythe weak value formalism. II. PRELIMINARY: THE WEAK VALUEFORMALISM
We start by revisiting the weak value formalism. Sup-pose the initial state of the system is | Ψ i i , the posts-elected state is | Ψ f i , the initial state of the detector is | Υ i , and the interaction Hamiltonian between the systemand the detector is H int = gA ⊗ Ω δ ( t − t ) . (1)Let ~ = 1 . The final state of the detector afterthe interaction followed by the postselection is | Υ f i = h Ψ f | exp( − i gA ⊗ Ω) | Ψ i i| Υ i , where | Υ f i is unnormalized.If we measure an observable M on the final detector state,the detector will have a shift of h Υ f | M | Υ f ih Υ f | Υ f i − h Υ | M | Υ i inthe expected value of M .When g is sufficiently small, the final state of the de-tector is approximately exp( − i gA w Ω) | Υ i , where A w = h Ψ f | A | Ψ i ih Ψ f | Ψ i i , (2)is called the weak value . It can be derived that the aver-age output of the detector is g Im A w ( h Υ |{ Ω , M }| Υ i − h Υ | Ω | Υ ih Υ | M | Υ i )+ i g Re A w h Υ | [Ω , M ] | Υ i . (3)The second term of Eq. (3) is real because the average ofa commutator must be imaginary.One can see from (3) that the average output of a weakmeasurement approximately amplifies g by the real orimaginary part of the weak value. This is the basis of allweak measurement amplification protocols. It is worthmentioning that Ref. [35] demonstrated the roles of thereal and imaginary parts of the weak value in the positionor momentum shift of the detector in a postselected weakmeasurement. Eq. (3) is a generalization of those resultsto an arbitrary observable M on the detector, and when [Ω , M ] = 0 or i , (3) reduces to the results of [35].An obvious shortcoming of the weak value formalismis that when h Ψ f | Ψ i i → , A w → ∞ , implying that theoutput of a weak measurement could be infinite, whichis impossible in practice. This issue is rooted in the firstorder approximation in deriving the weak value formal-ism, so it is not a proper tool to study the amplificationlimit of a weak measurement. III. AMPLIFICATION LIMIT: A VARIATIONALAPPROACH
In the following, we shall show a variational approachto this problem that can avoid the divergence of the weakvalue formalism and give a valid result for the amplifica-tion limit.Let’s define a shift operator ∆ M = M −h Υ | M | Υ i . Theaverage shift of the detector is h ∆ M i = h Υ f | ∆ M | Υ f ih Υ f | Υ f i . (4)When we choose different pre- and postselections of thesystem, the detector will give different outputs at the endof the measurement. Intuitively, the shift of the detectorshould be bounded. Our goal is to find the maximum h ∆ M i over all possible pre- and postselections. Thismaximum is the amplification limit .When h ∆ M i attains an extremal value h ∆ M i e , its vari-ation with respect to | Υ f i is zero: δ h ∆ M i e = 1 h Υ f | Υ f i (( δ h Υ f | )(∆ M | Υ f i − | Υ f ih ∆ M i e )+ ( h Υ f | ∆ M − h ∆ M i e h Υ f | )( δ | Υ f i )) = 0 , (5)according to the variational principle. Since | Υ f i is de-termined by | Ψ i i and | Ψ f i , the variation of | Υ f i can beexpressed in terms of the variations of | Ψ i i and | Ψ f i : δ | Υ f i = ( δ h Ψ f | ) exp( − i gA ⊗ Ω) | Ψ i i| Υ i + h Ψ f | exp( − i gA ⊗ Ω)( δ | Ψ i i ) | Υ i . (6)Thus, the variation of h ∆ M i e becomes δ h ∆ M i e = h Υ |h Ψ i | exp(i gA ⊗ Ω)(∆ M | Υ f i − | Υ f ih ∆ M i e )( δ | Ψ f i )+ ( δ h Ψ i | ) h Υ | exp(i gA ⊗ Ω) | Ψ f i (∆ M | Υ f i − | Υ f ih ∆ M i e )+ c . c . = 0 . (7)Note that the variations δ | Ψ i i and δ | Ψ f i are arbitraryand independent, it follows from (7) that h Υ | exp(i gA ⊗ Ω) | Ψ f i (∆ M − h ∆ M i e ) | Υ f i = 0 , h Υ |h Ψ i | exp(i gA ⊗ Ω)(∆ M − h ∆ M i e ) | Υ f i = 0 . (8)It is crucial to observe that the left sides of the twoequations in (8) are vectors in the system Hilbert space,so their amplitudes in the eigenbasis of A must all bezero. Suppose the observable A has r A eigenvalues a , · · · , a d s with eigenstates | a i , · · · , | a d s i . If | Ψ i i = P k α k | a k i , | Ψ f i = P k β k | a k i , the two equations of (8)give d s X k =1 α k | a k ih Υ | exp(i ga k Ω)(∆ M − h ∆ M i e ) | Υ f i = 0 , d s X k =1 β ∗ k h a k |h Υ | exp(i ga k Ω)(∆ M − h ∆ M i e ) | Υ f i = 0 . (9)Since either | Ψ i i or | Ψ f i can be arbitrary, we can chooseall α k = 0 or all β k = 0 . Then the above equation shows h Υ | exp(i ga i Ω)(∆ M − h ∆ M i e ) | Υ f i = 0 (10)for all i = 1 , · · · , r A .Let Ξ g be the matrix (cid:2) e − i ga Ω | Υ i| · · · | e − i ga d s Ω | Υ i (cid:3) .We can write | Υ f i as a linear combination of exp( − i ga k Ω) | Υ i , k = 1 , · · · , d s , i.e., | Υ f i = Ξ g µ , µ =( µ , · · · , µ d s ) T , and µ k = β ∗ k α k . Then (Ξ † g ∆ M Ξ g − Ξ † g Ξ g h ∆ M i e ) µ = 0 , (11)Eq. (11) is a homogeneous linear equation with respectto µ , so the necessary and sufficient condition for theexistence of nonzero µ is det(Ξ † g ∆ M Ξ g − Ξ † g Ξ g h ∆ M i e ) = 0 , (12) which is the major equation that h ∆ M i e must satisfy.Eq. (12) implies that an extremum of h ∆ M i must be aneigenvalue of (Ξ † g Ξ g ) − Ξ † g ∆ M Ξ g (Ξ † g Ξ g ) − . Therefore,the largest h ∆ M i e is its largest eigenvalue: |h ∆ M i| max = | λ ((Ξ † g Ξ g ) − Ξ † g ∆ M Ξ g (Ξ † g Ξ g ) − ) | max . (13)Note that above we have assumed Ξ g to be full rank, sothat (Ξ † g Ξ g ) is invertible. If the eigenvalues of A aredegenerate, or d s > d D , the rank of Ξ g will be less than d s ; in that case, one needs to pick out a maximal linearlyindependent subset from exp( − i ga i Ω) | Υ i , i = 1 , · · · , d s to construct the matrix Ξ g .A formal asymptotic solution for the amplificationlimit can be obtained from (13) by Gelfand’s theorem[36] which connects the spectral radius of a matrix to its(arbitrary) norm. If we choose the norm k·k to be thetrace norm, then |h ∆ M i| max = lim n →∞ (tr(Ξ † g ∆ M Ξ g (Ξ † g Ξ g ) − ) n ) n . (14)Usually, one can choose a finite k to derive an approxi-mate solution to |h ∆ M i| max from (14), and large k willgive higher precision to the approximation.In a weak measurement, the coupling strength g is usu-ally very small. This can lead to a simplified and g -independent form of (12), which is helpful in finding theessential factors that determine the amplification limit.The key is to prove that when g ≪ , the sup-port of Ξ g , i.e., the subspace spanned by all trans-lated detector states exp( − i ga k Ω) | Υ i , k = 1 , · · · , d s , is g -independent, and can be spanned approximately by | Υ i , Ω | Υ i , · · · , Ω r A − | Υ i (unorthonormalized), where r A is the number of different eigenvalues of A . The proofis given in Appendix A. If we define e Ξ to be the matrix (cid:2) | Υ i| Ω | Υ i | · · · | Ω r A − | Υ i (cid:3) , the equation for h ∆ M i e (12)can be simplified to det( e Ξ † ∆ M e Ξ − e Ξ † e Ξ h ∆ M i e ) = 0 . (15)If e Ξ is full rank, Eq. (15) is equivalent to the eigenvalueequation for ( e Ξ † e Ξ) − e Ξ † ∆ M e Ξ( e Ξ † e Ξ) − , so |h ∆ M i| max = | λ (( e Ξ † e Ξ) − e Ξ † ∆ M e Ξ( e Ξ † e Ξ) − ) | max . (16)If the rank of e Ξ is less than r A , one just needsto pick out a maximal linearly independent set from | Υ i , Ω | Υ i , · · · , Ω r A − | Υ i to reconstruct e Ξ . So (16) willstill hold.Eq. (16) is a general solution for the amplification limit |h ∆ M i| max . An explicit asymptotic solution can also bederived from (16) by Gelfand’s theorem [36], which con-nects the spectral radius of a matrix to its (arbitrary)norm. If we choose the norm k·k in [36] to be the tracenorm, then |h ∆ M i| max = lim n →∞ (tr( e Ξ † ∆ M e Ξ( e Ξ † e Ξ) − ) n ) n . (17)We see from (16) and (17) that the amplification limit |h ∆ M i| max is independent of g , and determined only bythe subspace spanned by | Υ i , Ω | Υ i , · · · , Ω r A − | Υ i . De-note this subspace as H D . Usually, when r A increases, H D will become larger, so |h ∆ M i| max will increase aswell. But if r A > r Ω , where r Ω is the number of distincteigenvalues of Ω , then this subspace will be r Ω dimen-sional at most, and the increase of r A will no longer affect |h ∆ M i| max . In addition, the size of the support of | Υ i on the eigenbasis of Ω also influences this subspace. Forexample, if | Υ i is an eigenstate of Ω , then H D is one di-mensional, whatever r A and r Ω are. A detailed analysisis given in Appendix B. IV. EXAMPLESA. Spin- particles To illustrate our result, we consider an example of spin- particles with a pair of Stern-Gerlach devices [1].When a beam of spin- particles with the same spindirection moving in the x direction pass through a Stern-Gerlach device which has a nonuniform external magneticfield in the z direction, it will be coupled to the magneticfield due to the interaction H I = − µ ∂B z ∂z zσ z δ ( x − x ) , (18)where µ is the magnetic moment of a single particle, and δ ( x − x ) means that the support of the magnetic field isvery narrow so that the duration of the interaction is ex-tremely short. Immediately after the first Stern-Gerlachdevice, let the beam of particles pass through a secondStern-Gerlach device, where the magnetic field is in the y direction. Then the particles will split into two beamswith spins pointing to the ± y directions respectively. Ifwe keep track of one of the two beams, say the beamwith spins + y , then we actually postselect the particlesin the state of spin + y . If the gradient of the magneticfield in the first Stern-Gerlach device is sufficiently small,i.e. | ∂B z ∂z | ≪ , and the initial direction of the spins isproperly chosen to be close to the − y direction, then the + y beam will have a large displacement in the z directiondue to the amplification effect of the postselected weakmeasurement.Now, we can apply the results derived above to find thelargest possible displacement of the postselected beamover all directions of the magnetic field in the secondStern-Gerlach device. In this example, g = − µ ∂B z ∂z , Ω = z, M = z, p z . Suppose the initial spatial wave functionof the particles in the z direction is Υ( z ) and is symmetricabout its center for simplicity. Since d s = 2 for spin- particles, the final spatial wave function of the particlescan be spanned by Υ( z ) and z Υ( z ) (unnormalized). Ifthe position or the momentum of the particle is measuredin the z direction after the second Stern-Gerlach device, Detector state Detector wave function |h ∆ z i| max |h ∆ p z i| max Gaussian ((2 π ) K ) − exp( − z K ) K K − Lorentzian ( πK ) − z/K ) K K − Exponential K − exp( − | z | K ) √ K √ K − Table I. Results for three typical examples with d s = 2 . Themaximal position shift |h ∆ q i| max can be very large for all threeexamples if the spread width of the states (proportional to K )is sufficiently large, or the maximal momentum shift |h ∆ p i| max can be large if K is sufficiently small. This also verifies thecomplementarity relation (20). then from Eq. (16) one can obtain (see Appendix C) |h ∆ z i| max = p h ∆ z i Υ , |h ∆ p z i| max = 12 s h{ z, ∆ p z }i h ∆ z i Υ , (19)where h Υ |·| Υ i is denoted h·i Υ for brevity. From (19), onecan see that neither |h ∆ z i| max nor |h ∆ p z i| max depend onthe magnetic moment µ or the external field B z , and theyare determined only by the initial wave function of theparticle, which confirms the previous result.We applied (19) to three typical spatial wave functionsfor the particles: the Gaussian state, the Lorentzian stateand the exponential state; the results are summarized inTable I. In addition, the results for the exponential stateare plotted in Fig. I for different weak values. It can beseen that upper bounds always exist whatever g is, andthe upper bounds are nearly the same when g ≪ , whichverifies the previous result.An important complementarity relation between |h ∆ z i| max and |h ∆ p z i| max can be obtained from (19): |h ∆ z i| max |h ∆ p z i| max = 12 q h{ z, ∆ p z }i ≥ . (20)If the wave function Υ( z ) is real, then h{ z, ∆ p z }i Υ =0 , so the complementarity relation becomes an equality: |h ∆ z i| max |h ∆ p z i| max = .According to the Robertson-Schrödinger uncertaintyinequality [37], the second equation of (19) leads to |h ∆ p z i| max ≤ p h ∆ z i Υ , so it can be combined with thesecond equation of (19) as |h ∆ M i| max ≤ p h ∆ M i Υ , M = z, p z . (21)The upper bound of the amplification limit (21) has anintuitive physical picture: the final spatial wave functionof a single particle is a superposition of several wave func-tion that are translated very little from the initial wavefunction. When the postselection on the particle’s spinis properly chosen, the superposition can cancel the ma-jor part of the initial wave function, resulting in a largedeviation of the particle’s spatial wave function from itsoriginal position. But such a displacement cannot be toolarge, because the wave function is still bounded. So the ImA w < ∆ z > Figure 1. (Color online) The figure shows the average shiftof spin- particles h ∆ z i as Im A w increases. The particles arein an exponential state exp( −| q | ) initially, and the Hamilto-nian is (18). Different curves are plotted for different − µ ∂B z ∂z ,ranging from − to × − (steeper curves with larger − µ ∂B z ∂z ). Explicit turning points can be found in this figure,which indicates the upper bound of the amplification effect.It can be seen that the largest h ∆ z i are almost the same fordifferent − µ ∂B z ∂z , which confirms the independence from g of |h ∆ M i| max . particle can be only shifted to the edge of the spread ofits wave function at most. This is what Eq. (19) implies.The above results can help to choose appropriate set-tings in designing real weak measurement experiments.The initial detector states can be chosen by (19) or (21)to realize a desired amplification limit. For example, ifone wants to enlarge the maximal position shift of thedetector, one should choose an initial state with widerspread for the detector; to enlarge the maximal momen-tum shift of the detector, the initial spread of the detec-tor should be narrower. The width of the initial detectorwave function decides the amplification limit of a weakmeasurement with that detector. B. Continuous variable systems
The role of the dimension of the system is often ne-glected in the study of weak measurement amplification.To show how the dimension of the system can influencethe amplification limit, we now consider continuous vari-able systems, i.e., d s = ∞ , as a limiting case of highdimensional systems. For simplicity, we assume A and Ω are nondegenerate, i.e. r A = d s and r Ω = d D .When d s = ∞ , the subspace H D of all possible finaldetector states can be spanned by | Υ i , Ω | Υ i , · · · , Ω ∞ | Υ i .Obviously, this subspace is equal to the whole Hilbertspace of the detector only if | Υ i has full support on theeigenbasis of Ω . This implies that the final detector canbe (but is not limited to) any eigenstate of M . There-fore, |h ∆ M i| max = | λ ( M ) | max in this case. In particular, if the detector is also a continuous variable system and M = q, p , then |h ∆ M i| max = ∞ , but is not bounded bythe spread of the detector state p h ∆ M i Υ as in (21), adramatic difference from the results for d s = 2 .Of course, one cannot always postselect a continuousvariable system to be in an arbitrary state, so |h ∆ M i| max will still be finite, or only approach ∞ asymptotically, inpractice. Yet this limiting example demonstrates how sig-nificantly the dimension of the system can influence theamplification limit of a weak measurement, particularlywhen the system dimension is high, which clearly showsthe advantage of the variational method, since such a re-sult cannot be derived from the first-order weak valueformalism [33]. And it reveals the rich structure andcomplexity of this problem. V. GENERALIZATION TO MIXED DETECTORSTATES
In this section, we generalize the main results to mixeddetector states. Suppose the initial state of the detectoris mixed: ρ D = X k η k | Υ k ih Υ k | . (22)The generalization to this case is mostly straightforward,but one must take care to avoid a tricky pitfall. At firstglance, as ρ D represents the ensemble { η k , | Υ k i} , it seemsthat the maximum shift of the detector should be theaverage maximum shift of the detector over the ensemble: |h ∆ M i| max ? = X k η k |h ∆ M i| | Υ k i max , (23)where the superscript | Υ k i indicates the dependence of |h ∆ M i| max on | Υ k i . But the optimal choice of pre- andpostselections to reach the maximum shift depends onthe initial detector state, so for different | Υ k i ’s, the opti-mal choices of pre- and postselections are different. Onecannot make the optimal choice for all | Υ k i ’s simulta-neously. So (23) is not the maximal shift for a mixeddetector state, in general, but rather an upper bound onthe shift.In fact, by carrying out the previous variational pro-cedure for mixed detector states, it can be shown that |h ∆ M i| max is the largest absolute value over all solutionsto det X k η k ( e Ξ † k ∆ M e Ξ k − e Ξ † k e Ξ k h ∆ M i e ) = 0 , (24)where e Ξ k is the matrix with | Υ k i , Ω | Υ k i , · · · , Ω d s − | Υ k i as its columns, as defined above. So the maximum de-tector shift is |h ∆ M i| max = | λ (( X i η i e Ξ † i e Ξ i ) − X k η k e Ξ † k ∆ M × e Ξ k ( X j η j e Ξ † j e Ξ j ) − ) | max , (25)and an asymptotic solution is |h ∆ M i| max = lim n →∞ (tr( X k η k e Ξ † k ∆ M e Ξ k ( X j η j e Ξ † j e Ξ j ) − ) n ) n . (26) ACKNOWLEDGMENTS
Shengshi Pang thanks Justin Dressel and Antonio DiLorenzo for helpful discussions. Shengshi Pang and ToddA. Brun acknowledge support from the ARO MURI un-der Grant No. W911NF-11-1-0268 and the NSF grantCCF-0829870. Shengjun Wu thanks support from theNSFC under Grant No. 11275181. Zeng-Bing Chenthanks support from NNSF of China under Grant No.61125502, the CAS, the National High Technology Re-search and Development Program of China, and the Na-tional Fundamental Research Program under Grant No.2011CB921300.
APPENDIX A. THE SUBSPACE OF THE FINALDETECTOR STATES
In this appendix, we prove by induction that | Υ i , Ω | Υ i , · · · , Ω ( r A − | Υ i is an approximate(unorthonormalized) spanning set for the sub-space spanned by the translated detector states exp( − i ga k Ω) | Υ i , k = 1 , · · · , r A , when g | λ ( A ) | max ≪ .Suppose that A has r A distinct eigenvalues. Thesubspace of final detector states, which we denote as H D , can be approximately spanned by the vectors { exp( − i ga k Ω) | Υ i} , k = 1 , · · · , r A . Define the matrix e Ξ = (cid:2) | Υ i| Ω | Υ i | · · · | Ω r A − | Υ i (cid:3) . (27)It is easy to verify that the columns of e Ξ( e Ξ † e Ξ) − are anorthonormal basis of H D .Now suppose that A has r A +1 distinct eigenvalues, andwe have constructed the matrix e Ξ as above using the first r A eigenvalues. By the Gram-Schmidt orthogonalizationprocedures, the next state in the basis (if there is one)can be obtained by | e r A +1 i = exp( − i ga r A +1 Ω) | Υ i− e Ξ( e Ξ † e Ξ) − ( e Ξ † e Ξ) − e Ξ † exp( − i ga r A +1 Ω) | Υ i = exp( − i ga r A +1 Ω) | Υ i− e Ξ( e Ξ † e Ξ) − e Ξ † exp( − i ga r A +1 Ω) | Υ i . (28)Note that we assumed e Ξ to have a full rank so that e Ξ † e Ξ is invertible in (28). If e Ξ does not have a full rank, thenone should use the pseudoinverse of e Ξ † e Ξ (the inverse onits support) instead. Since g | λ ( A ) | max ≪ , exp( − i ga r A +1 Ω) | Υ i = r A X k =0 ( − i ga r A +1 ) k k ! Ω k | Υ i + o ( g r A )= e Ξ X + ( − i ga r A +1 ) r A r A ! Ω r A | Υ i + o ( g r A ) , (29)where X = (cid:18) , − i ga r A +1 , · · · , ( − i ga r A +1 ) r A − ( r A − (cid:19) T . (30)Therefore, (28) can be simplified to | e r A +1 i = e Ξ X + ( − i ga r A +1 ) r A r A ! Ω r A | Υ i + o ( g r A ) − e Ξ( e Ξ † e Ξ) − e Ξ † (cid:18)e Ξ X + ( − i ga r A +1 ) r A r A ! Ω r A | Υ i + o ( g r A ) (cid:19) = ( − i ga r A +1 ) r A r A ! (cid:16) Ω r A | Υ i − e Ξ( e Ξ † e Ξ) − e Ξ † Ω r A | Υ i (cid:17) + o ( g r A ) . (31)Since the columns of e Ξ are | Υ i , Ω | Υ i , · · · , Ω r A − | Υ i , itcan be seen from (31) that | e r A +1 i is a linear combinationof | Υ i , Ω | Υ i , · · · , Ω r A | Υ i . Thus, the subspace spannedby exp( − i ga k Ω) | Υ i , k = 1 , · · · , r A + 1 can be spannedby | Υ i , Ω | Υ i , · · · , Ω r A | Υ i . This completes the proof byinduction. APPENDIX B. ANALYSIS OF THE DIMENSIONOF H D In this appendix, we analyze the dimension of the sub-space H D spanned by all possible final detector states,so as to show what determines the amplification limit |h ∆ M i| max . The vectors | Υ i , Ω | Υ i , · · · , Ω r A − | Υ i can bechosen to be a spanning set for H D , as proved in Ap-pendix A, so the dimension of H D is equal to the rank ofthe matrix e Ξ = (cid:2) | Υ i| Ω | Υ i | · · · | Ω r A − | Υ i (cid:3) .Let Ω have r Ω distinct eigenvalues ω , · · · , ω r Ω , andlet the projectors of the corresponding eigensubspacesbe P ω k , k = 1 , · · · , r Ω . Then | Υ i = P r Ω k =1 P ω k | Υ i , then Ω i | Υ i = P r Ω k =1 ω ik P ω k | Υ i , and e Ξ = CD , where C = c c . . . c r Ω , D = ω · · · ω r A − ω · · · ω r A − ... ... . . . ... ω r Ω · · · ω r A − r Ω . (32)The computational basis is P ωk | Υ i √ h Υ | P ωk | Υ i , and c k = p h Υ | P ω k | Υ i .When | Υ i has a full support on the eigenbasis of Ω ,i.e. c k = 0 for all k = 1 , · · · , r Ω , dim( H D ) = rank( e Ξ) = rank( D ) . Since ω , · · · , ω r Ω are distinct from each other,it can be inferred that dim( H D ) = ( r A , r A ≤ r Ω ,r Ω , r A > r Ω . (33)When | Υ i does not have full support on the eigenbasisof Ω , say c n +1 = · · · = c r Ω = 0 , then similarly to theabove equation, it can be shown that dim( H D ) = ( r A , r A ≤ n,n, r A > n. (34)In summary, the initial detector state | Υ i , the numberof distinct eigenvalues r A and r Ω of A and Ω (respec-tively), and the support of | Υ i in the eigenbasis of Ω determine the amplification limit. APPENDIX C. GENERAL DISCUSSION FOR d s = 2 In the main text, we showed an example of spin- par-ticles passing through two Stern-Gerlach devices to il-lustrate the main results. In this appendix, we want togive a general discussion about the results for two dimen-sional systems, which is of great interest in the field ofquantum information and quantum computing. We as-sume A to be nondegenerate, since otherwise A would beproportional to the identity and the interaction would betrivial. A. General results for d s = 2 . When d s = 2 , the subspace spanned by all possiblefinal detector states is approximately spanned by | Υ i and Ω | Υ i . We assume that | Υ i is not an eigenstate of Ω sothat | Υ i and Ω | Υ i are linearly independent. We shallconsider three typical cases below. In this section, weshall denote h Υ | · | Υ i by h·i Υ for short.For d s = 2 , h ∆ M i e satisfies det (cid:18) −h ∆ M i e h ∆ M Ω i Υ − h Ω i Υ h ∆ M i e h Ω∆ M i Υ − h Ω i Υ h ∆ M i e h Ω∆ M Ω i Υ − h Ω i Υ h ∆ M i e (cid:19) = 0 , (35)where we have used h ∆ M i Υ = 0 . |h ∆ M i| max can bestraightforwardly derived from the above equation: |h ∆ M i| max = W Υ h ∆Ω i Υ , (36) where h ∆Ω i Υ = h Ω i Υ − h Ω i , and W Υ = |h Ω i Υ h{ Ω , ∆ M }i Υ − h Ω∆ M Ω i Υ | + (( h Ω i Υ h{ Ω , ∆ M }i Υ − h Ω∆ M Ω i Υ ) + h ∆Ω i Υ ( h{ Ω , ∆ M }i − h [Ω , ∆ M ] i )) . (37)Note that h ∆ M i Υ = 0 , so according to the Robertson-Schrödinger uncertainty inequality [37], an upper boundfor |h ∆ M i| max is |h ∆ M i| max ≤ |h Ω i Υ h{ Ω , ∆ M }i Υ − h Ω∆ M Ω i Υ |h ∆Ω i Υ + p h ∆ M i Υ , (38)where h ∆ M i Υ = h M i Υ − h M i . B. Symmetric detector states
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