aa r X i v : . [ qu a n t - ph ] F e b State-dependent rotations of spins by weak measurements
D. J. Miller ∗ Centre for Time, University of Sydney,Sydney NSW 2006, Australia and School of Physics,University of New South Wales, Sydney NSW 2052, Australia (Dated: November 17, 2018)
Abstract
It is shown that a weak measurement of a quantum system produces a new state of the quantumsystem which depends on the prior state, as well as the (uncontrollable) measured position ofthe pointer variable of the weak measurement apparatus. The result imposes a constraint onhidden-variable theories which assign a different state to a quantum system than standard quantummechanics. The constraint means that a crypto-nonlocal hidden-variable theory can be ruled outin a more direct way than previously.
PACS numbers: 03.65.Ta,03.65.Ca,03.65.Ud ∗ [email protected] . INTRODUCTION In a standard von Neumann measurement, or normal measurement (NM), the state ofthe quantum system is transformed to one of the eigenstates of the observable A that isbeing measured. In a weak measurement (WM) [1, 2] of A , the state of the quantum systemis transformed to a linear combination of the eigenstates of A . For a measurement of spin,which we will be concerned with here, the transformation of state as a result of either typeof measurement can be visualized as a rotation of the direction of spin. In Sect. II, it isshown that the new spin-direction as a result of a WM depends on both the spin direction ofthe quantum system before measurement (the “prior” state) and the pointer position whichrecords the result of the WM.Since the new spin-direction depends on the spin direction of the prior state, it is inter-esting to apply the result to WMs of maximally entangled spin-states. This is because instandard quantum mechanics the prior state is the reduced state of each of the entangledsubsystems which does not specify a direction in space for the subsystem’s spin. On theother hand, some hidden-variable (HV) theories of quantum mechanics do assign spin statesto the subsystems of an entangled pair so the spin-direction dependent WM rotation pro-vides an additional means of testing such theories. Leggett [3] has shown that a nonlocalHV theory of a certain type (a “crypto-nonlocal” (CNL) theory) is ruled out by a differentset of inequalities from the Bell inequalities [4]. The new inequalities have been tested ex-perimentally [5, 6]. In Sec. IIIB, the rotations due to WMs are used to give a more directdemonstration that the CNL HV theory of Ref. 3 is not viable. II. WEAK MEASUREMENTS
The concept of a WM was introduced by Aharonov and co-workers (for a discussion, seeSection 3 of Ref. [2]). A WM involves a “weak” (in the sense described below) interactionbetween a measuring instrument, in this case the weak measurement apparatus (WMA),and the quantum system. The WM outcome is recorded by a NM of the “pointer” position Q of the WMA.The observable of the quantum system which is to be measured is the Pauli spin operatorˆ σ z , twice the component of the spin of the quantum system along the z -axis (which is a2irection set by the WMA) in units of ¯ h . The interaction Hamiltonian for the WM isˆ H ( t ) = ag ( t ) ˆ σ z ˆ P . (1)Here ˆ P is the operator which is conjugate to ˆ Q and the constant a ( − a ) is the averagedistance the WM pointer moves when a quantum system with spin up (down) along the z -axis interacts with the WMA. We assume that, except for the interaction, the quantumsystem and the WMA evolve freely, meaning that ˆ σ z and ˆ P are constants of the motionbetween the preparation at t of the quantum system and the WMA and the time when theNM of the WMA pointer-position is made at t W M (the interaction between the quantumsystem and the WMA may cease prior to t W M ). The real, scalar function of time g ( t ) isnormalized, R t WM t g ( t ) dt = 1. The only relevant factor of the time-evolution operator isˆ U ( t W M , t ) = exp (cid:20) − i ¯ h Z t WM t ag ( t ) ˆ σ z ˆ P dt (cid:21) = exp h − i a ¯ h ˆ σ z ˆ P i . (2)The WMA is prepared in the state | ψ ; t i at time t with h ψ ; t | ˆ P | ψ ; t i = 0 for t < t Consider a single spin prepared in the state | p + i = cos θ p e − iφ p / | + i + sin θ p e iφ p / |−i , (4)i.e. with spin up in a direction ˆ p specified by the polar coordinates θ p , φ p , making an angle θ p with the z -axis (specified by the WMA) and φ p with the x -axis (in this case chosen3rbitrarily). After the interaction and an observation identifying the WMA pointer-positionto be Q = Q , the normalized state of the quantum system and WMA is1 √ N | Q ih Q | e − i a ¯ h ˆ σ z ˆ P | p + i| ψ ; t i = 1 √ N (cid:20) h Q | ˆ S ( a ) | ψ ; t i cos θ p e − iφ p / | + i + h Q | ˆ S ( − a ) | ψ ; t i sin θ p e iφ p / |−i (cid:21) | Q i = 1 √ N (cid:20) ψ ( Q − a, t W M ) cos θ p e − iφ p / | + i + ψ ( Q + a, t W M ) sin θ p e iφ p / |−i (cid:21) | Q i (5)where ψ ( Q, t ) = h Q | ψ ; t i is the wave function of the WMA pointer and the normalizationconstant N = | ψ ( Q − a, t W M ) | cos θ p + | ψ ( Q + a, t W M ) | sin θ p .Thus, as a result of the WM, the spin of the quantum system is up in a direction ˆ q specified by the polar angles θ q and φ q wheretan θ q | f ( Q ) | tan θ p φ q = φ p + arg f ( Q ) (7)where f ( Q ) = ψ ( Q + a, t W M ) ψ ( Q − a, t W M ) . (8)Throughout the following we will assume that φ q = φ p and independent of time, which is agood approximation if, for example, the initial distribution of Q -values is Gaussian and issufficiently broad so that further broadening with time between t and t W M can be neglected.Under those conditions f ( Q ) is real and positive.As a result of the WM, the spin of the quantum system is rotated through an angle∆ θ ( θ p , Q ) away from the z -axis where∆ θ ( θ p , Q ) = 2 arctan (cid:20) f ( Q ) tan θ p (cid:21) − θ p (9)which depends on both the initial angle θ p between the spin-direction and the direction ofthe positive z -axis set by the WMA and the (uncontrollable) result Q of the WM. Notethat if f ( Q ) < 1, the spin is rotated towards the positive z -axis and becomes aligned alongthat direction as f ( Q ) → f ( Q ) > 1, the spin is rotated away from the positive z -axis and becomes aligned opposite to that direction as f ( Q ) → ∞ . A NM correspondsto either f ( Q ) = 0 or f ( Q ) = ∞ . The angle of rotation ∆ θ ( θ p , Q ) is shown in Fig. 1 as afunction of Q for a spin originally in the xy -plane ( θ p = π/ 2) and the WMA prepared in aGaussian state with width ∆ ψ = a . 4 Q /a !" (90 , Q ) oooooooooo ! =90 o "! (90 o , Q ) FIG. 1: A spin is prepared in a direction making angle θ p with respect to the positive z -axis which isa direction set by a weak measurement apparatus. The pointer of the weak measurement apparatusoriginally has a Gaussian wave function of width a . After the weak measurement interaction thepointer is observed to have the position Q . As a result, the direction of the spin is rotated throughan angle ∆ θ ( θ p , Q ) away from the z -axis. The dependance of the angle of rotation with Q isshown for the case when the original direction θ p = 90 ◦ . III. WEAK MEASUREMENTS AND HV THEORIES FOR ENTANGLEDSTATESA. CNL HV model We will consider the CNL theory of Ref. 3 for spins rather than the photons of theoriginal formulation (the equivalence of the two cases was pointed out in Ref. 3). In thetheory, instead of emitting pairs of spins in an entangled state Ψ, the source emits pairswith spins in directions ˆ u and ˆ v respectively with probability density F ( ˆ u , ˆ v ) in the four-dimensional space U V of the spin directions. The measurement outcomes on each pair ofspins, including the nonlocal correlations, are controlled by a HV λ with probability density g ˆ u ˆ v ( λ ) in the space Λ of the HVs. The probability densities F ( ˆ u , ˆ v ) and g ˆ u ˆ v ( λ ) must satisfy1 ≥ g ˆ u ˆ v ( λ ) ≥ ≥ F ( ˆ u , ˆ v ) ≥ Z Λ dλg ˆ u ˆ v ( λ ) = 1 and Z U d ˆ u Z V d ˆ v F ( ˆ u , ˆ v ) = 1 . (11)The spins will be identified as the left-hand side (lhs) and right-hand side (rhs) spins. Aftertime evolutions ˆ U l ( t ) on the lhs and ˆ U r ( t ) on the rhs (with ˆ U l ( t ) ˆ U r ( t ) = ˆ U ( t ) below), NMsare performed on the lhs spin in the ˆ a -direction and on the rhs spin in the ˆ b -direction.The eigenstates of the measurements will be denoted respectively by | a α i and | b β i , α, β = ± 1, with corresponding eigenvalues α and β in units of ¯ h/ 2. We consider the probabilityProb hv [ a α , b β | ˆ U ( t )] of outcomes α and β on the lhs and rhs respectively (the dependence onthe preparation state Ψ will be repressed). According to the CNL HV theory, the probabilityProb hv [ a α , b β | ˆ u , ˆ v , ˆ U ( t )] of the outcomes α and β given that the source emits spins in thedirections ˆ u , ˆ v is given byProb hv [ a α , b β | ˆ u , ˆ v , ˆ U ( t )] = Z Λ Prob hv [ a α , b β | ˆ u , ˆ v , λ ( t ) , ˆ U ( t )] g ˆ u ˆ v ( λ ( t ) , t ) dλ ( t ) . (12)It is a requirement of the theory [3] that for the marginal probabilities on the lhs and rhs(but not the joint probability), the ˆ u , ˆ v spins behave “normally” when averaged over theHV λ . That is, for the lhs, the marginal probability must give the quantum mechanicalresult Prob qm [ a α | ˆ u , ˆ U ( t )] for outcome α of a measurement in the ˆ a direction of a spin in theˆ u direction (which corresponds to the state | u + i in the present notation)Prob hv [ a α | ˆ u , ˆ v , ˆ U ( t )] = Prob qm [ a α | ˆ u , ˆ U ( t )] = |h a α | ˆ U l ( t ) | u + i| . (13)This will be called the secondary condition in the following. Of course, in order to agreewith experiment, when averaged over ˆ u , ˆ v , both the joint and marginal probabilities mustgive the quantum mechanical result. For the marginal probability on the lhs, this means Z UV Prob hv [ a α | ˆ u , ˆ v , ˆ U ( t )] F ( ˆ u , ˆ v ) d ˆ u d ˆ v = Prob qm [ a α | ˆ U ( t )] = |h a α | ˆ U l ( t ) | Ψ i| . (14)This will be called the primary condition. Combining the primary and secondary conditionsin Eqs. (13) and (14) means thatProb hv [ a α | ˆ U ( t )] = Z UV |h a α | ˆ U l ( t ) | u + i| F ( ˆ u , ˆ v ) d ˆ u d ˆ v = |h a α | ˆ U l ( t ) | Ψ i| . (15)If ˆ U l ( t ) is unitary, the transformation of | u + i in its two-dimensional spin space can bereplaced by a rotation of the direction of measurement ˆ a in three-dimensional space [8].6 M ˆ b ˆ a = ˆ z Q l NM S NM R x ( ! ) FIG. 2: A singlet state is produced by source S. The left-hand side subsystem is subjected first toa rotation through an angle α about the x -axis and then a weak measurement along the z -axis andfinally a normal measurement also along the z -axis. For the right-hand side subsystem, a normalmeasurement is performed in the ˆ b -direction. The result of the normal measurement of the pointerposition of the weak measurement apparatus is Q l . Since the primary and secondary requirements must already be satisfied for any direction ˆ a ,unitary time-evolution imposes no further condition on the HV theory. In the next section,we show that including the non-unitary time-evolution operator involving a WM does imposea further condition on the HV theory. B. WM and the HV model In order to investigate the CNL HV theory using WMs, it is sufficient to consider theexperimental set-up shown in Fig. 2. Singlet states are produced by the source S and the lhssubsystem is subjected first to a rotation R x ( α ) through an angle α about an axis specifyingthe x -axis and then by a WM whose orientation specifies the z -axis. The time-evolutionˆ U l ( t ) operator for the WM is given by Eq. (2). It is also sufficient for the present purposesto consider the case when the final NM on the lhs is along the z -axis (the direction specifiedby the WMA). The rhs subsystem is measured in the direction ˆ b .7he quantum systems and the WMA are prepared at t = 0 in the state | Ψ i = 1 √ | z + i l | z − i r | − | z − i l || z + i r ) | ψ l i (16)where | ψ l i is the initial state of the WMA. We will be concerned with probabilities condi-tional on the outcome Q l of the WM (the WMs with different outcomes can be consideredto be discarded).If, according to the CNL HV model, the source emits a spin oriented in the direction ˆ u specified by the polar angles θ u , φ u , after the rotation the spin is oriented in the directionspecified by the polar angles θ u ( α ) , φ u ( α ) where θ u ( α ) = arccos(cos θ u cos α + sin θ u sin φ u sin α ) (17)If the outcome of the WM pointer-position is Q l , then from Eq. (6), the direction of the spinwith respect to the z -axis becomes θ l = 2 arctan (cid:20) f ( Q l ) tan θ u ( α )2 (cid:21) . (18)Therefore, for the specified time-evolution, the probability involved in the secondary require-ment in Eq. (13) isProb qm [ a α | ˆ u , ˆ U ( t )] = |h z + | ˆ U l ( t ) | u + i| = cos (cid:18) arctan (cid:20) f ( Q l ) tan θ u ( α )2 (cid:21)(cid:19) . (19)A straightforward calculation using Eq. (16) leads to the primary requirement for this case,i.e. the marginal probability (conditional on Q l ) for the outcomes spin-up in the ˆ z -directionon the lhs, Prob qm [ a α | ˆ U ( t )] = |h z + | ˆ U l ( t ) | Ψ i| = 11 + ( f ( Q l )) . (20)From Eqs. (15), (19) and (20), the primary and secondary requirements of the HV theoryfor this particular case mean that Z UV cos (cid:20) (cid:18) f ( Q l ) tan θ u ( α )2 (cid:19)(cid:21) F ( ˆ u , ˆ v ) d ˆ u d ˆ v = 11 + f ( Q l ) . (21)From Eq. (17), θ u ( α ) depends on α which can be chosen arbitrarily, which means thatEq. (21) cannot be satisfied in general. Therefore the time-evolution involving the WMrules out the CNL HV. The previous argument [3, 5, 6] that the CNL theory was notviable was that the CNL could not satisfy certain inequalities involving results for severalcombinations of ˆ a and ˆ b . The present demonstration is simpler and more direct because itinvolves results for one value of ˆ a and is independent of ˆ b .8 V. CONCLUSION The main result is that a WM causes a change in a quantum system to a new statewhich depends on the prior state of the quantum system and the (uncontrollable) measuredposition of the pointer variable of the WMA. This provides a way of manipulating a statewhich is significantly different from a NM because, after a NM, the new state of the quantumsystem is one of the eigenstates of the observable that is measured is therefore independentof the prior state of the quantum system. In a NM the only role of the prior state is toinfluence the probabilities with which the eigenstates of the observable are taken up. Therotation of the spin subjected to a WM interaction has been discussed before [9] but in termsof the operator conjugate to the pointer-variable operator ˆ Q and not, as here, in terms of theWM outcome Q . It is significant that the WM, together with a NM of the pointer variableof the WMA, causes a non-unitary evolution of the quantum system.It follows that consideration of a WM of a quantum system can impose a significantconstraint on possible HV formulations of quantum mechanics. This is because a HV theorymay assign states to a quantum system which are different from the quantum mechanicalstate. A unitary transformation changes all states in a way which is independent of theprior state and so, if the HV theory mimics standard quantum mechanics before the unitarytransformation, it will continue to do so afterwards. In contrast, the WM will change theHV states in a way which depends on those assigned states and will also change the quantummechanical state by an amount which depends on it. The extra constraint is that the HVtheory must continue to mimic standard quantum mechanics both with and without thosedisparate changes due to the WM. In Sec. IIIB it was shown that those constraints rule outthe setting-dependent CNL HV model proposed by Leggett [3] in a more direct way thanpreviously. [1] Y. Aharonov and L. Vaidman, Phys. Rev. A , 11 (1990).[2] Y. Aharonov and L. Vaidman, in Time in Quantum Mechanics , edited by J. G. Muga, R. SalaMayato, and I. L. Egusquiza (Springer, Berlin, 2002), p. 369; e-print quant-ph 0105101[3] A. J. Leggett, Found. Phys. , 1469 (2003).[4] J. S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge University Press, ambridge, 1987), pp. 14-21.[5] S. Gr¨oblacher, T. Paterek, R. Kaltenbaek, ˘C. Brukner, M. ˙Zukowski, M. Aspelmeyer, and A.Zeilinger, Nature , 871 (2007).[6] C. Branciard, N. Brunner, N. Gisin, C. Kurtsiefer, A. Lamas-Linares, A. Ling, and V. Scarani,Nature Physics , 681 (2008).[7] C. Cohen-Tannoudji, B. Diu, and F. Lalo¨e, Quantum Mechanics , translated by S. R. Hemley,N. Ostrowsky, and D. Ostrowsky (Wiley, New York, 1977), pp. 187-189.[8] M. Redhead, Incompleteness, Nonlocality, and Realism (Clarendon Press, Oxford, 1987), pp.103-105.[9] Y. Aharonov and A. Botero, Phys. Rev. A , 052111 (2005)., 052111 (2005).