Locality and nonlocality in the interaction-free measurement

We present a paradox involving a particle and a mirror. They exchange a nonlocal quantity, modular angular momentum Lz mod 2 , but there seems to be no local interaction between them that allows such an exchange. We demonstrate that the particle and mirror do interact locally via a weak local current 〈Lz mod 2 〉w. In this sense, we transform the “interaction-free measurement" of Elitzur and Vaidman, in which two local quantities (the positions of a photon and a bomb in the two arms of a Mach-Zehnder interferometer) interact nonlocally, into a thought experiment in which two nonlocal quantities (the weak modular angular momentum of the particle and of the mirror) interact locally. 1 The quantum Cheshire Cat So-called “weak values" [1] have taken their place alongside eigenvalues and expectation values as possible measured values in quantum mechanics. But while an ordinary ensemble suffices for measuring eigenvalues and expectation values, weak values require a “preand post-selected" (PPS) ensemble. Though unconventional, a PPS ensemble with initial state |ψin〉 and final state |ψ f in〉 is (in principle) easy to prepare: we measure an operator that has |ψin〉 as an eigenstate, and then an operator with |ψ f in〉 as an eigenstate, on as many systems as we like; and then we keep only those systems with those respective eigenstates. In between, we measure whatever we like, but with a measurement interaction weak enough to be consistent with the PPS ensemble. If the interaction is weak enough, the result of measuring an operator A is the weak value 〈A〉w of A: 〈A〉w = 〈ψ f in|A|ψin〉 〈ψ f in|ψin〉 . (1) In this way, weak values enable us to answer questions about quantum systems that we otherwise cannot even ask. An example of a weak value is the “quantum Cheshire cat" [2, 3], named after the Cheshire Cat in Alice in Wonderland [4] who could disappear while leaving its grin behind. In the weak-value version, a photon takes one path through a Mach-Zehnder interferometer while its net polarization vanishes on that path but not on the other. In this experiment, the photon and its polarization separate at a welldefined moment as the Cat passes through the first beam-splitter of the interferometer. There is also [5] an experiment in which the separation is continuous: the Cat is confined to one side of a potential ae-mail: rohrlich@bgu.ac.il


The quantum Cheshire Cat
So-called "weak values" [1] have taken their place alongside eigenvalues and expectation values as possible measured values in quantum mechanics. But while an ordinary ensemble suffices for measuring eigenvalues and expectation values, weak values require a "pre-and post-selected" (PPS) ensemble. Though unconventional, a PPS ensemble with initial state |ψ in and final state |ψ f in is (in principle) easy to prepare: we measure an operator that has |ψ in as an eigenstate, and then an operator with |ψ f in as an eigenstate, on as many systems as we like; and then we keep only those systems with those respective eigenstates. In between, we measure whatever we like, but with a measurement interaction weak enough to be consistent with the PPS ensemble. If the interaction is weak enough, the result of measuring an operator A is the weak value A w of A: In this way, weak values enable us to answer questions about quantum systems that we otherwise cannot even ask. An example of a weak value is the "quantum Cheshire cat" [2,3], named after the Cheshire Cat in Alice in Wonderland [4] who could disappear while leaving its grin behind. In the weak-value version, a photon takes one path through a Mach-Zehnder interferometer while its net polarization vanishes on that path but not on the other. In this experiment, the photon and its polarization separate at a welldefined moment as the Cat passes through the first beam-splitter of the interferometer. There is also [5] an experiment in which the separation is continuous: the Cat is confined to one side of a potential a e-mail: rohrlich@bgu.ac.il  barrier, while its grin tunnels through to the other side. Here, we apply this continuous Cheshire Cat separation to the "interaction-free measurement" (IFM) of Elitzur and Vaidman [6], where it challenges us to reconsider whether the effect-a paradigm of quantum nonlocality-is nonlocal after all.

The Cheshire Cat and interaction-free measurements
Recall that Elitzur and Vaidman imagined a bomb so sensitive that any interaction makes it explode.
To detect such a bomb in a region S, we build a Mach-Zehnder interferometer with one of its arms crossing the region S. Fig. 1(a) shows the interferometer with no bomb in the region S. A halfsilvered mirror splits the incident photon beam into two equal parts. The parts recombine at another half-silvered mirror. By adjusting the length of each arm, we can make the parts of the beam interfere constructively in one direction and destructively in the other. Then all the photons leave the interferometer in the same direction. If, however, a detector records which path the photon actually takes through the interferometer, interference disappears (according to the complementarity principle) and the photon may leave the interferometer in either direction. If there is a bomb in the region S and it does not explode, it records the fact that the photon did not pass through S. The photon went through the other arm of the interferometer. Then interference disappears and the photon may come out either way. (See Fig. 1(b).) So if we see a single photon leave the interferometer in the direction of destructive interference, we have detected the bomb without exploding it. What is so striking about this effect is that the bomb is revealed by a photon that, we can be quite sure, never came near it. But, analogously, we can consider a one-dimensional cavity of length 2L with a finite potential barrier positioned symmetrically at its center. To the left of the barrier is, initially, a single neutron. (See Fig. 2(a).) If the right end of the cavity were not blocked, the neutron would ultimately tunnel through the barrier and leave the left side completely. However, the neutron reflects completely from the right end of the cavity. Let us assume that the neutron initially approaches the potential barrier in a gaussian wave packet with momentum p 0 which is large compared to its uncertainty: p 0 ∆p. The neutron hits the barrier, and we can approximate its immediate evolution there via the matrix where is, for now, arbitrary. 1 Note that U( ) is unitary, as it must be, and that [U( )] 2 = U(2 ).
It is straightforward to prove, by induction, that [U( )] n = U(n ). This equation has the following application. Let us assume that at time t = 0, the neutron is at the left end of the cavity. It takes a time Lm/p 0 (where m is the neutron mass) to reach the finite potential barrier. When the neutron reaches the barrier, it passes through with amplitude i sin and reflects with amplitude cos . Both the transmitted and reflected wave packets reflect completely off the right and left ends, respectively, of the cavity, and meet again at the potential barrier after an additional time 2Lm/p 0 . They continue to do so every 2Lm/p 0 , and at time t n = 2nLm/p 0 , the amplitude for the neutron to be at the left end of the cavity is given by the first diagonal element of U(n ), which is cos(n ). Hence by choosing n = π/2, we can guarantee that the neutron will be on the right side of the potential barrier, at the right end of the cavity; it happens at time T ≡ t N such that N = π/2 , namely T = πLm/ p 0 . At time t = 2T the neutron will again be fully at the left end of the cavity, at time t = 3T at the right end again, and so on. And now-continuing with the analogy-let there be a bomb at the right end of the cavity, as in Fig. 2(b). Again, the bomb is so sensitive that any interaction makes it explode. The unitary matrix 1 If the diagonal elements of U( ) are real, then the off-diagonal elements must be imaginary, as the following argument shows: Let the barrier be a δ-function potential located at z = 0. Without loss of generality we can let the incident neutron wave be e ikx , the reflected wave be re −ikx and the transmitted wave be te ikx . Continuity of the wave function at z = 0 requires 1 + r = t. Conservation of current requires 1 = |r| 2 + |t| 2 . Then |1 + r| 2 = |t| 2 , hence r + r * = 0 and r is pure imaginary.  U( ) still describes the immediate evolution when the neutron first hits the potential barrier, and we specialize to the case where is very small, such that cos ≈ 1 − 2 /2 and sin ≈ . Now, for each interval of time 2Lm/p 0 (during which the neutron, on the left side, hits the barrier once), the amplitude that the neutron tunnels through the barrier and hits the bomb is , up to a phase, and the probability that the bomb does not explode is therefore proportional to 1 − 2 . For a given finite time T , we can take N as large as we like; and since N = π/2 , the total probability for the bomb not to explode is (1 − 2 ) N = (1 − π 2 /4N 2 ) N ≈ e −π 2 /4N , which approaches 1 in the limit N → ∞. So the bomb never explodes! But-from the fact that at time T we can find the neutron on the left of the potential barrier-we can be sure that the bomb is indeed on the right. Here is the IFM in a new guise close to the quantum Zeno effect [7].

A local IFM?
To test further the notion that the IFM can be local, let us now consider a variation of this experiment.
Let there be two one-dimensional cavities, identical to the one above, parallel to each other and to the z axis, symmetrically above and below the z axis, both having length 2L and a barrier in the middle. (See Fig. 3.) However, these two cavities differ from the one cavity in the following respect: although the left ends (at z = −L) of the cavities are closed, and the neutron reflects perfectly from them, the right ends (at z = L) are open. We now "populate" the two cavities with a single neutron (with horizontal coordinate z) and a single mirror (with horizontal coordinate Z) that is much heavier that the neutron and perfectly reflects it. We define neutron states ψ suppose we prepare the neutron and mirror in an initial state Ψ in (z, Z, 0), where Now Ψ in (z, Z, 0) is a product of the neutron and mirror wave functions, and is invariant under interchange of U and D, both for the wave functions ψ (ν) 2 of the neutron and mirror separately, and for their product. Formally, we can write e iL (ν) z π/ Ψ in (z, Z, 0) = Ψ in (z, Z, 0) = e iL (µ) Z π/ Ψ in (z, Z, 0) where L (ν) z and L (µ) Z are the angular momentum operators for the neutron and mirror, respectively. We denote the total angular momentum as L z = L (ν) z + L (µ) Z and define a modular [8] angular momentum L z mod . Since e iL (ν) z π/ e iL (µ) Z π/ applied to Ψ in (z, Z, 0) equals Ψ in (z, Z, 0), we obtain that L z mod 2 = 0.
How does the state Ψ in (z, Z, 0) evolve in time?
In particular, what is the state of the neutron and mirror at time T ≡ t N = πLm/ p 0 ? The answer to this question is contained in our previous calculations. Expanding the product in Ψ in (z, Z, 0), we get a sum of four terms: Each term tells a story, and the first and last term tell similar stories that differ qualitatively from the stories of the intermediate two terms. For in the first and last terms, there is a mirror at the right end of the cavity to reflect the neutron; and in such cases, as we know, the chance of finding the neutron in the left half of the cavity, whether the upper or the lower, vanishes at time T . For the inner two terms, the mirror is located where it has no influence on the neutron-which therefore can eventually escape from whichever cavity it initially inhabited. But the neutron escapes on a time scale much longer than T , namely N 2 2 , which is of order NT . At time t = T , the neutron is virtually assured to be at the left end of the cavity. Thus a natural post-selection is a projection of Ψ in (z, Z, T ) onto the left side of either cavity, i.e. the projection which leaves the neutron entangled with the mirror. But note, the post-selection did not create the entanglement: the neutron and the mirror states were entangled already before the post-selection, as a result of their evolution in time.
Note that our post-selection, which projected the entangled state onto the left side of either cavity, could not have affected either L (ν) z mod 2 and L (µ) Z mod 2 , since it commutes with them. Explicitly, let us define two projection operators, Π UL and Π DL , that project the neutron wave function onto the left side (from the left wall to the barrier) of the upper and lower cavities, respectively. Then we have e iL (ν) z π/ Π UL = Π DL and e iL (ν) z π/ Π DL = Π UL , hence e iL (ν) z π/ and L ν z mod 2 commute with Π UL + Π DL . At the same time, e iL (µ) Z π/ and L µ Z mod 2 commute with Π UL + Π DL because they act on orthogonal subspaces. (L µ Z does not act on the neutron at all.) Therefore projection of the overall state onto the left side of the cavities cannot affect its symmetry under the operators e iL (ν) z π/ and e iL (µ) Z π/ in any way, and non-conservation of L ν z mod 2 and L µ Z mod 2 could not be traced to post-selection of the left side of the cavities.
Thus, what is striking about this result is that the entangled state is not an eigenstate of the modular angular momentum of either the neutron or the mirror, i.e. it is not an eigenstate of either e iL (ν) z π/ or e iL (µ) Z π/ ; but it is still an eigenstate of the total modular angular momentum L z mod 2 . We thus conclude that L (ν) z mod 2 and L (µ) Z mod 2 were not separately conserved during the interaction of the neutron and the mirror, yet their sum was conserved. This conclusion would not be a paradox if it were not for the fact that, according to the post-selection, the neutron and mirror could never have met; for while the mirror was stationed at the right end of the cavity, the neutron was confined to the left end! If the neutron and the mirror did interact-as apparently they did, since they exchanged modular angular momentum-they apparently did so nonlocally-which challenges our proposed local interpretation of interaction-free measurements.
Here is a contrasting experiment. Equations (2-3) define the state Ψ in (z, Z, 0), i.e. the combined state of the neutron and mirror at time t = 0. This state is an eigenvector of e iL (ν) z π/ and of e iL (µ) z π/ , both with eigenvalue 1. What is Ψ in (z, Z, 2T ), i.e. their combined state at time t = 2T ? At that time, we know, the neutron has either never leaked to the right (for the two middle terms of Eq. (4)) or has leaked to the right and leaked back to the left. So does Ψ in (z, Z, 2T ) equal Ψ in (z, Z, 0), up to an overall phase? No, it does not, because the first and fourth terms have acquired a phase factor of −1. That is, t = 2T corresponds to n = 2N = π/ , hence cos(n ) = cos π = −1. Combining the four terms of Ψ in (z, Z, 2T ) into a product, we obtain  same barrier, we know from the previous sections that a series of wave packets will "leak" through the barrier and escape from the right of the same cavity; but since the amplitude of each "leakage" is of order , it proves to be negligible over the time scale T we consider. (See Fig. 4.) However, it is worth noting that the time evolution relevant to our calculation is not only the evolution forwards in time after t = T , but also the evolution backwards in time for t ≤ T . This evolution must be the time reverse of the evolution we considered in previous sections; in particular, it must include wave packets entering the cavity from the right (wherever there is no mirror in the way) and approaching the barrier from the right. For concreteness, it is useful to break up the evolution into pairs of time intervals of duration Lm/p 0 . In the forward time evolution, a neutron on the right of the barrier approaches the barrier in the first of these two intervals, and leaves the barrier, moving to the right, in the second interval. By contrast, a neutron on the left end of the cavity produces order-wave packets only in the second of the two Lm/p 0 intervals; in the first interval, it is still on the way to the barrier on its right. Now the forwards evolution of the post-selected state in the second interval translates into a backwards evolution of the post-selected state in the first interval, with order-wave packets moving to the left. Thus in the first interval of each pair of intervals, we have a coincidence between an orderwave packet moving to the left in one cavity and a generally larger wave packet moving to the left in the other cavity. These wave packets have the same z coordinate. Since they are in different cavities, the weak value of the projection on each wave packet's position vanishes, but the weak value of e iπL (ν) z / does not vanish! A "weak" current of modular angular momentum flows from the right end of the cavities towards the barrier! Here is a qualitative explanation of how the mirror and the neutron can exchange modular angular momentum, even when the neutron never reaches the mirror. Can we make this qualitative explanation quantitative? To show that we can, let us calculate the change in modular angular momentum to the left of the barrier during each paired interval. The pre-selected state evolves forward in time such that the weak value of e iπL (ν) z / , calculated over the wave packet to the left of the barrier (in the closed cavity), is cos(n ) at time t n = 2nLm/p 0 . Thus during every interval of time t n+1 − t n = 2Lm/p 0 , the region to the left of the barrier gains cos([n + 1] ) − cos(n ) = cos(n ) cos( ) − sin(n ) sin( ) − cos(n ), which equals − sin(n ) up to terms