On the Resolution of Quantum Paradoxes by Weak Measurements

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The measurement f therefore defines a decomposition By analogy, the same should be true for Schrödinger’s of the initial density matrixρ î into a mixture of sub- cat. Theoretically, there exists a microscopic interfer- ensemble density matrices Rîf associated with the indi- ence pattern that arises from the superposition of death vidual outcomes f, and survival. If we were to perform weak measurement tomography on the cat before confirming its death or sur- ρî = p(f i) Rîf , (3) vival, we would find that (a) the dead cats were already X | f dead when the tomography was performed and (b) both dead and living cats showed the microscopic signature of where p(f i) = Tr ρîΠˆ f and a “+” superposition before the final measurement con- | { } firmed their death or survival. ˆ 1 ˆ ˆ Rif = ρîΠf + Πf ρî . (4) 2Tr ρîΠˆ f { } IV. ENTANGLEMENT AND NON-LOCALITY This decomposition is valid even if the density matrixρ î describes a pure state that is a coherent superposition of Entanglement is widely considered to be the most different outcomes f. In other words, eq. (3) shows that counter-intuitive feature of quantum mechanics. In many there is no contradiction between a statistical interpreta- introductions of entanglement, it is assumed that the tion of the alternatives f1 and f2 and quantum coherence non-local change of the wavefunction caused by a lo- between the eigenstates associated with those outcomes. cal measurement is an effect that has no classical ex- Specifically, the problem is resolved by attributing half planation. However, such claims are rather misleading, of the coherence to f1 and the other half to f2. since the same non-local change is caused by a classical (Bayesian) update of probabilities if the initial probability distribution (or prior) includes correlations between ¨ III. SCHRODINGER’S CAT AND DOUBLE the two systems. In the following, I will show that this SLIT INTERFERENCE is indeed the correct analogy. Consider a maximally entangled state E of two d- The significance of the result derived above emerges level systems, A and B. The notion of non-locality| i arises when it is applied to the examples that are used to il- because any measurement represented by a projection on lustrate the strangeness of quantum mechanics. Firstly, a pure state f in A results in a corresponding pure state eq.(3) suggests that the death or survival of Schrödinger‘s f ∗ in B, | i cat is not decided by the measurement - the measurement | i merely uncovers which of the alternatives has already 1 ∗ A f E AB = f B. (6) happened. h | i √d | i It may be surprising that a simple argument about However, the density matrix of the maximally entangled quantum measurements should lead to new insights into state can already be written as a mixture of d alternatives the nature of quantum coherence. However, it seems that f given by much of the perplexity caused by Schrödinger’s cat was due to the analogy of quantum states and classical waves, d RÊf = ( E E )AB ( f f )A where interference patterns appear to require the pres- 2 | ih | | ih | ence of two sources. It is therefore useful to consider the +( f f )A( E E )AB statistical explanation of coherence in terms of the dou- | ih | | ih | ble slit interference of a single particle. Here, it is com- √d ∗ ∗ = ( E f; f + f; f E ) . (7) monly argued that the interference pattern could not ex- 2 | ih | | ih | ist if the particle either went through one slit or through Here, the coherences between f; f ∗ and orthogonal the other slit. However, weak measurement tomography components of E represent the| non-locali correlations shows that the states defined by a two slit superposition | i ∗ 1 + 2 and a subsequent which-path measurement are between properties of A and B other than f and f . | i | i Weak measurement tomography therefore shows that (a) ∗ 1 the property f was already present in B before the mea- Rˆ+1 = 1 1 + ( 1 2 + 2 1 ) | ih | 2 | ih | | ih | surement in A, and (b) the two systems are strongly correlated in the properties of the systems other than those 1 ∗ Rˆ+2 = 2 2 + ( 1 2 + 2 1 ) . (5) determined by f and f until the correlations are lost in | ih | 2 | ih | | ih | the back-action related randomization of the local prop- Thus, weak measurement shows that the assumption that erties in system A. which-slit information prevents interference is unfounded One significant consequence of this analysis is that the - the interference pattern associated with the coherences physical back-action of the measurement is completely between path 1 and path 2 shows up equally in the weak local. The changes of the quantum state in B are com- measurements performed on photons found in path 1 and pletely explained by the selection of an alternative al- on photons found in path 2. ready present in the initial density matrix. To confirm 3 that system B is already in the pure state f ∗ even be- system. The operators Π(ˆ X, Y ) describing the joint prob- | i fore the measurement in A is performed, we can trace out abilities can be obtained from the projectors (1 Xˆ)/2 ˆ ± system A to find the local statistics described by REf , and (1 Yˆ )/2. They read ± ∗ ∗ TrA RÊf = d f E E f = f f . (8) 1 { } h | ih | i | ih | Π(+1;+1)ˆ = (1 + Xˆ + Yˆ ) Initially, system B is in an incoherent mixture of the 4 ∗ 1 states f , and the measurement in A merely identifies Π(+1;ˆ 1) = (1 + Xˆ Yˆ ) the alternative| i that is actually realized in a given individ- − 4 − 1 ual case. Thus, the non-locality of entanglement is not Π(ˆ 1;+1) = (1 Xˆ + Yˆ ) really different from the non-locality of classical corre- − 4 − 1 lated statistics. The difference between classical physics Π(ˆ 1; 1) = (1 Xˆ Yˆ ). (11) and quantum physics lies elsewhere. − − 4 − − The eigenvalues of each of these operators are (1 √2)/4, ± V. NEGATIVE PROBABILITY PARADOXES corresponding to probabilities of 60 % and -10 %. The eigenstates are those of the spin operator Sˆ+ along the ˆ ˆ ˆ ˆ All quantitative paradoxes in quantum mechanics are diagonal between X and Y for Π(+1; +1) and Π( 1; 1), −ˆ − based on the simultaneous assignment of values to mea- and those of the orthogonal spin component S− for surement results that cannot be obtained at the same Π(+1;ˆ 1) and Π(ˆ 1; +1). If the initial state is maxi- − − time. Weak measurements show that the joint proba- mally entangled, the values of S+ = 1 and S− = 1 in ± ± bilities associated with such simultaneous assignments of system B can be determined by measurements in system eigenvalues can be negative. In weak measurement to- A. The joint probabilities for X, Y , S+ and S− then mography, the possibility of negative joint probabilities result in the following total probabilities, shows up in the form of negative eigenvalues of the den- 60% for (X + Y )S+ = +2 sity matrix Rîf . and (X Y )S− =0 If Φˆ g is the operator for the probability of obtaining an − 60% for (X + Y )S+ =0 outcome g in a given measurement, then the probability and (X Y )S− = +2 of g obtained in weak measurements is given by − -10% for (X + Y )S+ = 2 − ˆ ˆ and (X Y )S− =0 p(g i,f) = Tr Rif Φg , (9) − | { } -10% for (X + Y )S+ =0 as in conventional measurement theory. This formal- and (X Y )S− = 2. ism accurately describes the negative probabilities ob- − − served in experimental resolutions of quantum paradoxes Clearly, the value of (X + Y )S+ + (X Y )S− is either +2 or 2. However, local negative probabilities− result in by weak measurements such as the one reported in [7]. − Alternatively, it is possible to define the joint probability an average of 2√2. of f and g for the initial stateρ î, as shown in [8]. The This result can be confirmed experimentally by weak result is a joint probability given by measurements of the joint probabilities. Significantly, the negative probabilities originate not from some mysteri- 1 ous action at a distance, but from the local correlations p(f,g i)= Tr ρî Πˆ f Φˆ g + Φˆ gΠˆ f . (10) | { 2 } between the spin components of the same system. The Since this definition of joint probabilities is the only one paradox of Bell’s inequality violation is therefore resolved consistent with weak measurement tomography, it is in- in favor of locality and against realism. dependent of the specific experimental realization which is used to confirm it. In particular, it does not matter whether g or f is obtained in the weak measurement. VI. THE PROBLEM WITH REALITY Moreover, the joint probability of g and f is consistent with all strong measurements of g and f. It is there- Weak measurement tomography shows that a measure- fore possible to explain quantum paradoxes constructed ment result f can be considered real even before the mea- from the measurement results of strong measurements in surement is performed. Specifically, the quantum statis- terms of the negative joint probabilities observed in weak tics of a pure stateρ î = ψ ψ can be decomposed | ih | measurements. into a mixture of non-positive statistical operators Rîf Since it was shown in the previous section that en- in anticipation of the measurement f. However, the de- tanglement does not involve any physical non-locality, it composition is only justified if the measurement of f is may be of particular interest to explain Bell’s inequal- actually performed. The negative joint probabilities of ity violation in terms of negative joint probabilities. In measurements that cannot be performed jointly indicate the formulation of Bell’s inequalities, values of 1 are as- that reality cannot be simultaneously attributed to the signed to orthogonal components Xˆ and Yˆ of a± two level outcomes f and g of both measurements. 4

Weak measurements therefore indicate that quantum depend on whether the kind of realism we have in mind paradoxes arise from the assumption of a measurement has any impact on our appreciation of the plot. independent reality. In everyday life, we naturally assume that such a reality exists. Obviously, it would be crazy to assume that a room ceases to exist when we close VII. CONCLUSIONS the door from the outside. However, it may be foolish to extrapolate this experience without reflecting on its foun- dations. In fact, almost every philosopher who thought Empirically, quantum mechanics is a statistical the- deeply about the origin of our sense of reality noticed ory that predicts only the probabilities of events. In that it is somehow rooted in experience and perception. the case of weak measurements, it is particularly im- The list includes famous names such as Descartes (“I portant to keep this in mind since the results of weak think therefore I am”), Berkeley (“to exist is to be per- measurements are averages over a large number of indi- ceived”), Kant (no experience of the “thing in itself”), vidual measurements. Nevertheless weak measurements and Schopenhauer (no object without subject). In prac- can provide a complete description of a quantum state tice, objective reality is established through the consis- defined by both initial conditions i and final conditions ˆ tency of observations. As Kant explained in perhaps ex- f. The non-positive density matrix Rif of this state can cessive detail, we identify realities by extending our expe- be determined experimentally by weak measurement to- rience using chains of causality relations. Normally, this mography [8]. results in a tightly knit web with very little wiggle space Weak measurement tomography shows that the col- - especially in the physical sciences. Nevertheless, the lapse of the wavefunction caused by the final measure- primary condition for the reality of an object is always ment f is very similar to the collapse of a classical prob- the possibility of experiencing its effects. ability distribution when additional information becomes The uncertainty principle of quantum mechanics de- available. In fact, the initial stateρ î can always be writ- ˆ fines an absolute limit for the effects of an individual ten as a mixture of the non-positive density matrices Rif quantum object on the web of empirical reality. However, associated with the different possible outcomes f. Co- quantum mechanics can provide a complete description of herences between different f are divided equally between the measurement statistics obtained from a certain class the alternative outcomes. of objects. Intuitively, we try to identify the individ- In the case of entangled states, the non-locality of the ual representative with the whole class, and are therefore collapse of the wavefunction can be explained entirely in confused by the appearance of irreconcilable differences terms of the statistical correlations between the systems. between representatives observed in different ways. How- The non-locality is therefore statistical and corresponds ever, there is really no reason why a particle with the to the non-locality of an update of probability in classi- observable property f should also have a hidden prop- cally correlated systems. Quantum paradoxes arise only ˆ erty g, just because this property has been observed on from the negative eigenvalues of Rif that define negative a different member of the same class of objects. joint probabilities for the outcome f and the outcome g Ultimately, our imagination is pushed to its limits of an alternative measurement. when it comes to quantum mechanics. Nevertheless, we While it is possible to attribute reality to the measure- might achieve some insights by identifying the correct ment result f even before the measurement happens, it analogies and rejecting false and misleading ones. As is not possible to attribute reality to a measurement re- far as reality is concerned, it may be useful to discard sult that is not actually obtained. The negativity of joint the notion of fundamental material objects in favor of a probabilities shows that any assignment of hidden vari- model that emphasizes the role of perspective. Quantum ables would be inconsistent with the empirical results objects seem to show us only their front, whereas the obtained in weak measurements. Weak measurements backside is hidden by uncertainty. When we look at dif- therefore indicate that non-empirical realism (that is, the ferent objects of the same class, we can construct some- dogmatic insistence on a measurement independent re- thing that looks like a complete image, but there seem to ality) is inconsistent with the statistical predictions of be inconsistencies when we try to identify the backside quantum mechanics. Thus, weak measurements resolve of an individual object. It is almost as if the individ- quantum paradoxes by identifying non-empirical realism ual quantum objects were stage props showing different as the crucial fallacy in the formulation of the paradox. sides of the same building in different scenes of a play. In our minds, we construct the complete building, and for the purpose of understanding the action on the stage, Acknowledgements this is quite useful. But a look backstage would show that the people who built the set did not really bother to construct the whole building, but achieved their pur- Part of this work has been supported by the Grant-in- pose much more economically with just a bit of wood and Aid program of the Japanese Society for the Promotion canvas. Now, should we be disappointed by this absence of Science, JSPS of “realism”? If the world is a stage, this might actually 5

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