PHYSICAL REVIEW A 87, 014105 (2013) Peculiar features of entangled states with postselection Yakir Aharonov,1,2 Shmuel Nussinov,1,2 Sandu Popescu,3 and Lev Vaidman1,2,* 1School of Physics and Astronomy, Tel Aviv University, Tel Aviv, Israel 2Institute for Quantum Studies, Chapman University, 1 University Drive, Orange, California, USA 3H. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, United Kingdom (Received 31 December 2012; published 28 January 2013) We consider quantum systems in entangled states postselected in nonentangled states. Such systems exhibit unusual behavior, in particular, when weak measurements are performed at intermediate times. DOI: 10.1103/PhysRevA.87.014105 PACS number(s): 03.65.Ta Measurements performed on pre- and postselected quantum and systems often exhibit peculiar results. One particular example N is that of a single particle, which is found with certainty (n) (k) (l) =− σx σy σy 1, (5) in any one of a large number of boxes if only this box n=k,l is opened [1,2]. In spite of experimental implementations of these ideas [3–5], there is still some controversy about respectively. this example [6–20]. Even more surprising is the fact that Next consider three particular cases of Eq. (5) where the outcomes of weak measurements [21], namely, standard von pairs of particles for which σy was measured were chosen from Neumann measurements with weakened interaction, yield a triplet of particles, s, t, and r. We then have weak values which might be far away from the range of N (n) = possible eigenvalues. This feature led to practical applications σx 1, for high-precision measurements [22,23]. Here we consider n=1 the peculiar features of quantum systems when the preselected N (n) (s) (t) =− state is entangled. σx σy σy 1, First, let us consider the case where the preselected n=s,t state is the following Greeberger-Horne-Zeilinger (GHZ)–like (6) N entangled state of N spin- 1 particles: 2 σ (n)σ (s)σ (r) =−1, x y y N N n=s,r | = √1 |↑ + |↓ 1 z n z n . (1) N 2 = = (n) (t) (r) =− n 1 n 1 σx σy σy 1. = Just like the original three-particle GHZ state, it can be used n t,r to show that no local hidden variable theory is consistent with Had there been local hidden variables, then all variables should the predictions of quantum theory. have values, i.e., prior to the measurement every particle should To this end, consider the following sets of measurements. “know” the value of all spin components and, in particular, the The first set consists of σx measurements performed over all x and the y components. These values should fulfill all four particles. The second set consists of σx measurements for all Eqs. (6). Just like in the original GHZ case, this is impossible particles except for two particles for which σy are measured. since the product of the left-hand sides is the product of squares Since (or fourth powers) of the spin variables and the product of the right-hand sides is −1. N (n)| =| In order to see surprising results we consider the particular σx 1 1 (2) postselected state |2 in which all spins were found with n=1 σx = 1, i.e., and N | = |↑ N 2 x n. (7) (n) (k) (l) | =−| n=1 σx σy σy 1 1, (3) n=k,l A unique feature of this particular pre- and postselection is that at intermediate times no pair of particles can be found with the the outcomes of the two sets of measurements should fulfill same σy ; otherwise, the requirement (5) cannot be fulfilled. N To see this in a more dramatic way we map states of N σ (n) = 1(4)1 x spin- 2 particles into the states of N distinguishable particles n=1 which can reside in two spatially separated boxes A and B: |↑ ≡| |↓ ≡| y A , y B , (8) where |A (|B) is the state of the particle in box A (B). Thus, *[email protected] we have N (which might be as large as we want) particles in 1050-2947/2013/87(1)/014105(4)014105-1 ©2013 American Physical Society BRIEF REPORTS PHYSICAL REVIEW A 87, 014105 (2013) two boxes, but if we try to find any specific pair of particles of O given by the following expression: in one box, we are bound to fail! This is a very paradoxical | | 2 O 1 situation for N 3. The probability of the postselection which Ow ≡ . (14) 1 2|1 leads to this situation is 2N ,butforN which is not too large it is feasible to implement experimentally. The weak value might be far away from the eigenvalues of O The “paradox” in this example follows from (i) the obvious and the standard quantum formalism can explain the outcome classical observation that when three or more particles are by surprising universal interference phenomenon of the pointer located in two boxes, then at least one pair has to be in one of the measuring device. box, and (ii) the quantum property, which follows from the The following two theorems connect weak and “strong” joint pre- and postselection (but not from pre- or postselection (standard von Neumann) measurements [1]: (i) if, for a pre- and separately), that no pair of particles can be found in a single postselected system, the outcome oi of the strong measurement box. of O is known with certainty, then the weak value is the Hardy has already proposed [24] a related experiment on a same, Ow = oi ; (ii) if, for a pre- and postselected system, pre- and postselected system. It can be viewed as an experiment the weak value of a dichotomic variable O is equal to one of with two particles in two boxes [25]. The particles were the eigenvalues oi , then a strong measurement of O will yield prepared in the entangled state, this value oi with certainty. In the above generalized examples with N particles, weak 1 |1=√ (|A1|B2 +|B1|A2 +|B1|B2). (9) measurements suggest that, in some sense, all particles are in 3 box A in spite of the fact that, at most, one particle can be found The preparation started with the nonentangled state in A. Indeed, the additive property of weak values implies | = 1 | +| | +| N N 2 ( A 1 B 1)(A 2 B 2), (10) (n) = (n) = g PA PA w N. (15) and entanglement was achieved by projecting out the state n=1 w n=1 | | A 1 A 2. Then it was postselected in the state Note, however, that the weak measurement of the product of | = 1 |A −|B |A −|B . the projection operators vanishes, 2 2 ( 1 1)( 2 2) (11) N N The paradoxical feature here is that at intermediate times P(n) = 0 = P(n) , (16) each particle is found in box A if it is searched for there, but A A w n=1 w n=1 the two particles cannot be found together in A. The fact that each particle is (in this particular sense) in A follows from the because the strong value of the product vanishes with certainty. joint pre- and postselection, while the fact that both are not Let us return to the example with N pre- and postselected there follows directly from the preselection. particles in two boxes, in which every particular pair of A generalization of this for the case of N particles is the particles cannot be found in the same box. It is instructive preselection of the state to consider this example for identical spinless bosons. The ⎡ ⎤ statement that strong measurement cannot find any particular N N N pair of particles in any single box is then meaningless because 1 ⎣ ⎦ |1=√ (N −1) |Bn + |An |Bj there is no “particular pair” among N identical particles. 2 − + N N 1 n=1 n=1 j=n Nevertheless, we still can make a statement about weak (12) measurements. Let us assume that when two particles are in the same box, there is a potential V between them. and the postselection of the state Weak measurement of the interaction energy between N particles in the two boxes in our pre- and postselected state 1 N should then yield 0! This can be seen in a straightforward |2=√ (|An −|Bn). (13) N way by calculating the weak value of the potential energy 2 n=1 (n) (m) n=m V PA PB for a system preselected in state (1) and Every one of the N particles is found with unit probability in postselected in state (7). According to the mapping (8) the box A if it is searched for there. However, not only can they particles are described (up to normalization) by the following not all be found in A, but any number of particles larger than two-state vector: 1 cannot be found there. N N N All these peculiarities of pre- and postselected systems (A|n +B|n) (|An − i|Bn) + (|Bn − i|An) . can be seen, albeit in a much more complicated way, in n=1 n=1 n=1 the standard formalism of quantum mechanics with a single (17) quantum state evolving from the past to the future. The most surprising features of pre- and postselected quantum Some intuition and an alternative proof of this result are systems are manifested in the outcomes of weak measurements as follows. The potential energy is the same for identical [21]. Any standard measuring procedure with weak enough and nonidentical particles. For nonidentical particles, the coupling performed on a quantum system preselected in state measurement of the potential energy of any specific pair yields |1 and postselected in state |2 yields, instead of one of 0 for this particular pre- and postselection.