On the resolution of quantum paradoxes by weak measurements Holger F. Hofmann∗ Graduate School of Advanced Sciences of Matter, Hiroshima University, Kagamiyama 1-3-1, Higashi Hiroshima 739-8530, Japan and JST,CREST, Sanbancho 5, Chiyoda-ku, Tokyo 102-0075, Japan In this presentation, I argue that weak measurements empirically support the notion of quantum superpositions as statistical alternatives. In short, weak measurements show that Schr¨odinger’s cat is already dead or alive before the measurement. The collapse of the wavefunction in a strong measurement should therefore be separated into the statistical selection of one of the available alternatives and a physical interaction that causes decoherence. The application to entanglement reveals that measurements in A have no physical effect in B, resolving the paradox of Bell‘s inequality violation in favor of locality and against (non-empirical) realism. Keywords: wavefunction collapse, entanglement, non-locality I. INTRODUCTION raphy [8]. Specifically, weak measurement tomography shows that the only empirically correct answer to Bell’s As Schr¨odinger famously illustrated in his paradox of question is that quantum superpositions are actually sta- a cat in a superposition of life and death [1], it is not at tistical alternatives. The quantum mechanical “and” is a all obvious how one can reconcile the intuitive notion of misinterpretation, it is always an “or”. Once this misun- an external reality with the concept of quantum coher- derstanding is cleared up, it is possible to resolve quan- ence between distinguishable alternatives. Although de- tum paradoxes by identifying the fallacies that cause the coherence may seem to solve the problem, Bell correctly contradictions with reality. pointed out that the theory would have to clarify at what point the quantum “and” turns into a classical “or” [2]. II. WEAK MEASUREMENT TOMOGRAPHY Bohr would probably have objected to this criticism by pointing out that only the results of experiments are real, and that it is the purpose of the theory to describe this The key to understanding weak measurements is quan- reality, not to define it. In fact, our only hope for a scien- tum statistics. An individual weak measurement results tific resolution of the quantum measurement problem is in a completely random outcome, with no discernible re- the design of experiments that provide empirical answers lation to the properties of the quantum state. However, to our questions. the average of a large number of weak measurements re- In 1988, Aharonov, Albert, and Vaidman proposed veals small differences in the probability distribution that such an empirical approach in the form of weak mea- can be identified with the quantum statistics of the in- surements [3]. Essentially, weak measurements can ac- put state. In terms of measurement theory, the operator cess the quantum statistics of a system between its ini- defining the probability of obtaining an outcome m in tial preparation i and a final measurement f with neg- any given measurement can be written as ligible measurement interaction. Recently, this method Eˆ = w (1+ˆ ǫSˆ ), (1) has been applied to experimental realizations of quantum m m m paradoxes, suggesting that an empirical resolution of the where ǫSˆm 1. Thus, the small differences be- arXiv:0911.0071v1 [quant-ph] 31 Oct 2009 paradoxes can be achieved by interpreting the weak val- tween the|| experimentally|| ≤ observed probabilities p(m) = ues observed in the measurements as negative probabili- Tr ρˆiEˆm and wm can be used to determine the expec- ties [4, 5, 6, 7]. This seems to be an exciting possibility, { } ˆ but it assumes that the negative probabilities observed tation values of Sm. The complete density matrixρ ˆi can be reconstructed from the expectation values of at least in weak measurements are well understood and not at 2 ˆ all paradoxical. Otherwise, we have merely replaced one d linearly independent operators Sm. However, weak paradox with another. measurements have negligible back-action, so the quan- tum state is still available for further measurements. It In the following, I will argue that the real resolution is therefore possible to determine the quantum statistics of quantum paradoxes by weak measurement should be of a system between the initial preparation i given byρ ˆ based on the complete quantum statistics of the states i and a final measurement outcome f given by a measure- between preparation i and measurement f that can be ment operator Πˆ . As explained in more detail in [8], the experimentally determined by weak measurement tomog- f joint probabilities of the measurement outcomes m and f can be given as 1 ∗ p(m,f i) = Tr Eˆm ρˆiΠˆ f + Πˆ f ρˆi . (2) Electronic address: [email protected] | { 2 } 2 The measurement f therefore defines a decomposition By analogy, the same should be true for Schr¨odinger’s of the initial density matrixρ ˆi into a mixture of sub- cat. Theoretically, there exists a microscopic interfer- ensemble density matrices Rˆif 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- ρˆi = p(f i) Rˆif , (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 ρˆiΠˆ f and a “+” superposition before the final measurement con- | { } firmed their death or survival. ˆ 1 ˆ ˆ Rif = ρˆiΠf + Πf ρˆi . (4) 2Tr ρˆiΠˆ f { } IV. ENTANGLEMENT AND NON-LOCALITY This decomposition is valid even if the density matrixρ ˆi 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 probabil- ity 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 Schr¨odinger‘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 Schr¨odinger’s cat was due to the analogy of quantum states and classical waves, d RˆEf = ( 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 cor- related 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.