How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100 without using weak measurements
aa r X i v : . [ qu a n t - ph ] J u l How the result of a measurement of a component of the spin of a spin-1/2 particle canturn out to be 100 without using weak measurements
S. Ashhab
1, 2 and Franco Nori
1, 2 Advanced Science Institute, The Institute of Physical and Chemical Research (RIKEN), Wako-shi, Saitama 351-0198, Japan Physics Department, Michigan Center for Theoretical Physics,The University of Michigan, Ann Arbor, Michigan 48109-1040, USA (Dated: October 20, 2018)We discuss two questions related to the concept of weak values as seen from the standard quantum-mechanics point of view. In the first part of the paper, we describe a scenario where unphysicalresults similar to those encountered in the study of weak values are obtained using a simple experi-mental setup that does not involve weak measurements. In the second part of the paper, we discussthe correct physical description, according to quantum mechanics, of what is being measured in aweak-value-type experiment.
I. INTRODUCTION
The first part of the title of this paper (all but thelast four words) is taken from the title of a paper writtenby Aharonov, Albert and Vaidman (AAV) over twentyyears ago [1]. In that paper AAV introduced the con-cept of weak values. This concept immediately causedcontroversy [2], but over the years it has proved to bea useful paradigm for considering questions related toquantum measurement and the foundations of quantummechanics. For example, the observation of paradoxicalvalues in a weak-value-type measurement has been linkedto the violation of the Leggett-Garg inequality, which canbe used to test realism [3, 4, 5].In the setup considered by AAV, a beam of spin-1/2 particles propagates through a non-uniform magneticfield in a Stern-Gerlach-type experiment, where the tra-jectory of a given particle is affected by the spin state ofthe particle. The modification from the original Stern-Gerlach experiment is that, in the path of its propaga-tion, the beam encounters two regions in space with mag-netic fields. The magnetic field gradient in the first regionis designed such that it creates a tendency for particleswhose x -component of the spin (which we denote by S x )is positive to develop a finite component of the momen-tum in the positive x direction and for particles whose S x is negative to develop a finite component of the mo-mentum in the negative x direction. After exiting thisregion in space, the beam enters a second region wherea z -component in the momentum develops based on the z -component of the spin ( S z ). Either one of these stageswould constitute a measurement of the spin along somedirection: by setting up a screen that the beam hits suf-ficiently far from the field-gradient region, the positionwhere a given particle hits the screen serves as an indi-cator of the particle’s spin state. When combined, theycreate a situation where two non-commuting variablesare being measured in succession. If (1) the first mea-surement stage is designed to be a weak measurement,(2) the particles in the beam are created in a certain ini-tial state [e.g. close to being completely polarized alongthe positive z -axis] and (3) only those particles for which the second measurement produces a certain outcome [inthis example, a negative z -component of the spin], thenthe average value of the spin’s x -component indicatorcan suggest values of this component of spin being muchlarger than 1/2, a situation that seems paradoxical.A number of studies have already pointed out thatsince in the AAV setup two non-commuting variables arebeing measured in succession, quantum mechanics for-bids treating them as independent measurements whoseoutcomes do not affect one another [2]. In this paper westart by presenting an example that demonstrates therole of interpretation in obtaining unphysical results ina weak-measurement-related setup. The setup is chosento be very simple in order to remove any complicationsin the analysis related to the successive measurement ofnon-commuting variables. In the second part of the pa-per, we present the proper analysis (from the point ofview of quantum mechanics) of the measurement resultsobtained in an AAV setup. II. QUESTION 1: UNPHYSICAL RESULTS OFTHE AAV TYPE IN AN ALTERNATIVE SETUP
Let us consider the following situation: An ex-perimenter purchases a device for measuring the z -component of a spin-1/2 particle. The device producesone of two readings, 0 or 1. The experimenter goes to thelab and calibrates the device. The calibration is done bypreparing 10 particles in the spin up state, measuringthem one by one, and then doing the same for the spindown state. Let us say that the result of the calibrationprocedure is that for the spin up state the device showsthe reading “1” in 50.25% of the experimental runs andthe reading “0” in 49.75% of the runs. For the spin downstate, the probabilities are reversed. Clearly, the readingof the measurement device is only weakly correlated withthe spin state of the measured particle. The experimentertakes this fact into account and reaches the following con-clusion: If I have a large number of identically preparedspin-1/2 particles and measure them using this device,I will obtain a probability for the reading “1”. Usingthe results of the calibration procedure, the expectationvalue of the spin z -component for the prepared state willbe given by the formula: h S z i = (Prob − . × . (1)If the probability of obtaining the outcome “1” is 0.5025,the above formula gives 1/2. If the probability of obtain-ing the outcome “1” is 0.4975, the above formula gives-1/2. It looks like the device is ready to be used. Theexperimenter now performs an experiment that involves,as its final step, a measurement of S z . Surprisingly, themeasurement device shows the reading “1” every timethe experiment is repeated, leading the experimenter toconclude that the value of the spin is in fact 100. Thusone has a paradox.The resolution of the paradox in the above story liesin the fact that the device was not a weak-measurementdevice as the experimenter assumed, but a strong-measurement device whose reading is perfectly corre-lated with the spin state of the measured particle. Theonly problem is that at some point before the measure-ment device was calibrated, its spin-sensing part was ro-tated from being parallel to the z -axis to an axis thatmakes an angle 89.7135 with the z -axis (note here thatcos (89 . / ≈ . III. QUESTION 2: CORRECT EXPLANATIONOF RESULTS IN AN AAV SETUP
We now turn to the question of the correct interpre-tation of the AAV experiment according to quantummechanics. Instead of the original, Stern-Gerlach-typeexperiment analyzed by AAV, we formulate the prob-lem slightly differently. We consider a spin-1/2 particlethat is subjected to two separate measurements. As afirst step, a weak measurement is performed in the basis {| + i , |−i} , where |±i = ( |↑i ± |↓i ) / √ |↑i and |↓i are the eigenstates of ˆ S z . This measurement canproduce any one of a large number of possible outcomes,with probability distributions as shown in Fig. 1. Thismeasurement constitutes a weak measurement of ˆ S x . Asdiscussed in [6], each possible outcome is associated with FIG. 1: (color online) Schematic diagram of the probabilitydistributions of the possible measurement outcomes of a weakmeasurement (labelled by the index k ) for the two states ofthe measurement basis, | + i and |−i . a measurement matrix ˆ U x,k , where the index k repre-sents the outcome that is observed in a given run of theexperiment. If the outcome k occurs with probability P x,k for the system’s maximally mixed state, i.e. whenaveraged over all possible initial states, and it providesmeasurement fidelity F x,k (in favor of the state | + i ), themeasurement matrix ˆ U x,k is given byˆ U x,k = p P x,k np F x,k | + i h + | + p − F x,k |−i h−| o = p P x,k (cid:18) p F x,k + p − F x,k p F x,k − p − F x,k p F x,k − p − F x,k p F x,k + p − F x,k (cid:19) . (2)We shall use the convention where a measurement thatfavors the state |−i has a negative value of F x,k and ˆ U x,k is given by the same expression as above. It is worthmentioning here that the overall, or average, fidelity ofthis measurement can be obtained by averaging over all possible initial states and all possible outcomes: h F x i = X k P x,k | F x,k | . (3)After the weak x -basis measurement, a strong measure-ment in the {|↑i , |↓i} basis is performed. This strongmeasurement step can be described by two outcomes withcorresponding measurement matricesˆ U z, = |↑i h↑| = (cid:18) (cid:19) ˆ U z, = |↓i h↓| = (cid:18) (cid:19) . (4)As mentioned above, paradoxes arise if one treats the x -basis and z -basis measurements as two separate mea-surements that provide complementary information. In-stead, one should treat each pair of outcomes as a single combined-measurement outcome. The maximum amount of information in a given run of the experiment can beextracted as follows [6]: given that the outcome pair { k, l } was observed, one can construct the combined-measurement matrixˆ U Total ,k,l = ˆ U z,l ˆ U x,k . (5)From the matrices ˆ U Total ,k,l one can construct a so-calledpositive operator-valued measure (POVM) defined by thematrices ˆ M k,l : ˆ M k,l = ˆ U † Total ,k,l ˆ U Total ,k,l , (6)where the superscript † represents the transpose conju-gate of a matrix. In particular,ˆ M k, = P x,k (cid:0)p F x,k + p − F x,k (cid:1) F x,k F x,k (cid:0)p F x,k − p − F x,k (cid:1) ! = P x,k (cid:0) | ψ k, i h ψ k, | − (cid:12)(cid:12) ψ k, (cid:11) (cid:10) ψ k, (cid:12)(cid:12)(cid:1) = 2 P x,k | ψ k, i h ψ k, | (7)where | ψ k, i = cos θ k |↑i + sin θ k |↓i (cid:12)(cid:12) ψ k, (cid:11) = sin θ k |↑i − cos θ k |↓i sin θ k = F x,k . (8)Similarly one can find thatˆ M k, = 2 P x,k | ψ k, i h ψ k, || ψ k, i = sin θ k |↑i + cos θ k |↓i , (9)with θ k given by the same expression as above.As discussed in Ref. [6], one can obtain the measure-ment basis and fidelity that correspond to the outcomedefined by { k, l } by diagonalizing the matrix ˆ M k,l . Sinceˆ M k,l is a hermitian matrix, its two eigenvalues ( m k,l, and m k,l, , with m k,l, ≥ m k,l, ) will be real and its twoeigenstates ( | ψ k,l i and (cid:12)(cid:12) ψ k,l (cid:11) ) will be orthogonal quan-tum states that define a basis (the measurement basis).Note that because the second measurement in the prob-lem considered here is a strong measurement, we alwayshave m k,l, = 0.The different outcomes produce different measurementbases, thus this measurement cannot be thought of in theusual sense of measuring S n with n being some fixeddirection. Therefore, the measurement basis is deter-mined stochastically for each (combined) measurement(note that after the strong z -basis measurement, the sys-tem always ends up in one of the states {|↑i , |↓i} , even though the combined-measurement basis can be differentfrom the basis {|↑i , |↓i} ). By analyzing all the measure-ment data, one can perform partial quantum state to-mography and determine the x and z -components in theinitial state of the system (assuming of course that allcopies are prepared in the same state, which can be pureor mixed). Note that in this setup no information about S y can be obtained from the measurement outcome.We now ask whether information can be extracted fromthe x -basis and z -basis measurements separately, i.e. bydisregarding the outcome of one of the two measurementsteps. The answer is yes, provided care is taken in inter-preting the results. Extracting an x -basis measurementfrom a given measurement outcome is straightforward.All one has to do is disregard the outcome of the z -basismeasurement, since this measurement is performed afterthe x -basis measurement and cannot affect the outcomeof the x -basis measurement. Therefore, by disregardingthe outcome of the z -basis measurement, one obtains an x -basis measurement with overall fidelity h F x i . The sit-uation is somewhat trickier if one wants to extract a z -basis measurement from the measurement outcome. Onecan disregard the outcome of the x -basis measurement,but one must take into account the fact that this mea-surement generally changes the state of the system be-fore the z -basis measurement is performed. The effect ofthe x -basis measurement is to reduce the fidelity of the z -basis measurement. One can calculate this reduced fi-delity as follows: Let us assume that the system startsin the initial state |↑i . After the x -basis measurement isperformed and the outcome k (with fidelity F x,k ) is ob-served, the state of the system is transformed into a newpure state | ψ int i with |h ψ int | ˆ σ x | ψ int i| = F x,k . Since |h ψ int | ˆ σ x | ψ int i| + |h ψ int | ˆ σ y | ψ int i| + |h ψ int | ˆ σ z | ψ int i| = 14(10)for any pure state and here we have |h ψ int | ˆ σ y | ψ int i| =0, we find that after the x -basis measurement4 |h ψ int | ˆ σ z | ψ int i| is reduced from 1 to q − F x,k . If F x,k is independent of k , one obtains the relation (in this con-text, see e.g. Ref. [7]) h F x i + h F z i = 1 . (11)We now take one final look at the AAV gedankenex-periment. We choose a specific form for the x -basis mea-surement, which is essentially the same one used by AAV P x,k = 1 p πk exp (cid:26) − k k (cid:27) F x,k = r π h F x i k rms k, (12)with k running over all integers from −∞ to + ∞ and k rms assumed to be a large number. Note that the aboveexpression violates the constraint that F x,k <
1. How-ever, provided that h F x i ≪
1, the above expression canbe treated as a good approximation of the realistic sit-uation for all practical purposes. A simple calculationshows that in this case h F z i = ∞ X k = −∞ q − F x,k P x,k ≈ − π h F x i , (13)such that h F x i + h F z i ≈ − π − h F x i . (14)If the measured system is prepared in one of the states |±i , the average value of k that is obtained in an ensemble of measurements (all with the same initial state) is h k i |±i = ± h F x i k rms . (15)The small difference between h k i | + i and h k i |−i is thereason why the x -basis measurement qualifies as a weakmeasurement of S x . We now consider the full measure-ment procedure. If one prepares the measured system ina state that is very close to |↑i , most z basis measure-ments will produce the outcome l = 1. Only a smallfraction of the experimental runs will produce the out-come l = 2. If the initial state deviates slightly from |↑i ,i.e. | ψ i i = cos α |↑i + sin α |↓i , (16)then outcomes with negative values of k and l = 2 willbe suppressed the most (assuming α is positive), becausethese outcomes correspond to states that are orthogonalor almost orthogonal to the initial state (making their oc-currence probabilities particularly small). One thereforefinds that among the measurements that produced l = 2,the average value of k can be much larger than h k i | + i forproperly chosen parameters. This situation leads to theAAV paradox. IV. CONCLUSION
In conclusion, we have presented explanations accord-ing to quantum mechanics of two questions that are rele-vant to discussions of weak values. First we presented anexample that emphasizes the role of interpretation in ob-taining unphysical results in an AAV setup. We have alsopresented the correct interpretation (according to quan-tum mechanics) of the measurement results obtained inan AAV setup. We believe that our discussion is usefulfor understanding the origin of the possible observationof unphysical values in a weak-value experimental setup.This work was supported in part by the National Secu-rity Agency (NSA), the Laboratory for Physical Sciences(LPS), the Army Research Office (ARO) and the Na-tional Science Foundation (NSF) grant No. EIA-0130383. [1] Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev.Lett. , 1351 (1988).[2] A. J. Leggett, Phys. Rev. Lett. , 2325 (1989); A. Peres, ibid , 2326 (1989); Y. Aharonov and L. Vaidman, ibid , 2327 (1989).[3] A. J. Leggett and A. Garg, Phys. Rev. Lett. , 857(1985); A. J. Leggett, J. Phys. Condens. Matter , R415(2002).[4] N. S. Williams and A. N. Jordan, Phys. Rev. Lett. ,026804 (2008). [5] For other studies on the subject, see e.g. A. Romito, Y.Gefen, and Y. M. Blanter, Phys. Rev. Lett. , 056801(2008); V. Shpitalnik, Y. Gefen, and A. Romito, Phys.Rev. Lett. , 226802 (2008).[6] S. Ashhab, J. Q. You, and F. Nori, Phys. Rev. A ,032317 (2009); arXiv:0903.2319.[7] Y. Kurotani, T. Sagawa, and M. Ueda, Phys. Rev. A76