Quantum Foundations

The two-state vector formalism of Aharonov and collaborators introduces a backwards-evolving state in order to restore time symmetry to quantum measurement theory. The question then arises, does any time-symmetric account of quantum theory necessarily involve retrocausality (influences that travel backwards in time)? In [1], Huw Price argued that, under certain assumptions about the underlying ontology, an interpretation of quantum theory that is both realist and time-symmetric must be retrocausal. Price’s argument is based on an analysis of a photon travelling between two polarizing beam-splitters. One of his assumptions is that the usual forward-evolving polarization vector of the photon is a beable, i.e. part of the ontology. He argues, on the basis of this and his other assumptions, that a backward-evolving polarization vector must also be a beable. The assumption that the forward evolving polarization vector is a beable is an assumption of the reality of the quantum state. But one of the reasons for exploring retrocausal interpretations of quantum theory is that they offer the potential for evading the unpleasant conclusions of no-go theorems, such as Bell’s theorem and, in particular, recent proofs of the reality of the quantum state [2]. In this talk, I will show how Price’s argument can, in fact, be generalized so that it does not assume the reality of the quantum state. I also reformulate the common assumptions of Price’s and our arguments to make them more generally applicable and to pin down the notion of time-symmetry involved more precisely. The notion of time-symmetry used in the argument is stronger than the notion of time-symmetry usually used in physics, but is still a true symmetry of quantum theory that ought to be taken seriously. This talk is based on joint work with Matt Pusey. [1] H. Price. Does time-symmetry imply retrocausality? How the quantum world says “maybe”. Stud. Hist. Phil. Mod. Phys., 43(2):75–83, 2012. arXiv:1002.0906 [2] For a review see M. Leifer. Is the quantum state real? An extended review of psi-ontology theorems. Quanta, 3:67-155, 2014. arXiv:1409.1570

According to the many worlds interpretation (MWI), quantum mechanics in its simplest form (no collapse or hidden variables) is complete. A primary objection to the MWI is that it fails to account for the Born rule. The most prominent response to this objection comes from the decision-theoretic program, which aims to derive a rationality postulate according to which a believer in the MWI ought to act as if the Born rule is true. I argue that the existence of alternative coherent rationality postulates undermines this response. A different response, based on self-locating uncertainty, avoids this objection and may explain the Born rule in the MWI. I conclude by considering whether this framework is capable of explaining the weak trace of particles in certain difficult cases.

The weak value, as an expectation value, requires an ensemble to be found. Nevertheless, we argue that the physical meaning of the weak value is much more close to the physical meaning of an eigenvalue than to the physical meaning of an expectation value. Theoretical analysis and experimental results performed in the MPQ laboratory of Harald Weinfurter are presented. Quantum systems described by numerically equal eigenvalue, weak value and expectation value cause identical average shift of an external system interacting with them during an infinitesimal time. However, there are differences between the final states of the external system. In the case of an eigenvalue, the shift is the only change in the wavefunction of the external system. In case of the expectation value, there is an additional change in the quantum state of the same order, while in the case of the weak value the additional distortion is negligible. The understanding of weak value as a property of a single system refutes recent claims that there exist classical statistical analogue to the weak value.

The products of weak values of quantum observables have interesting implications in deriving quantum uncertainty and complementarity relations for both weak and strong measurement statistics. We show that a product representation formula allows the standard Heisenberg uncertainty relation to be derived from a classical uncertainty relation for complex random variables. This formula also leads to a strong uncertainty relation for unitary operators which displays a new preparation uncertainty relation for quantum systems. Furthermore, the two system observables that are weakly and strongly measured in a weak measurement context are shown to obey a complementarity relation under the interchange of these observables, in the form of an upper bound on the product of the corresponding weak values. Moreover, we derive general tradeoff relations, between weak purity, quantum purity and quantum incompatibility using the weak value formalism. Our results may open up new ways of thinking about uncertainty and complementarity relations using products of weak values.

I discuss the outcome statistics of sequential weak measurement of general observables. In sequential weak measurement of canonical variables, without post-selection, correlations yield the corresponding correlations of the Wigner function. Outcome correlations in spin-1/2 sequential weak measurements without post-selection coincide with those in strong measurements, they are constrained kinematically so that they yield as much information as single measurements. In sequential weak measurements with post-selection, a new anomaly occurs, different from the weak value anomaly in single weak measurements. I consider trivial post-selection, i.e.: re-selection |f>=|i>, which should intuitively not differ from no post-selection since weak measurements are considered non-invasive. Indeed, re-selection does not matter, compared with no-selection, for single weak measurement. It does so, however, for sequential ones. I illustrate it in spin-1/2 weak measurement.

The state vector describing the physical situation of the magnetic A-B effect should depend upon all three quantizeable entities in the problem, the electron orbiting the solenoid, the moving charged particles in the solenoid and the vector potential. One may imagine three approximate solutions to the exact dynamics, where two of the three entities do not interact at all, and the third, quantized, entity interacts with a classical approximation. Thus, fifty-five years ago, A-B showed that, if the interaction is between the quantized electron current and the classical approximation to the solenoid’s vector potential, the state vector acquires a measurable phase shift. Four years ago Vaidman showed that, if the interaction is between the quantized solenoid current and the classical approximation to the electron’s vector potential, the state vector acquires the A-B phase shift. I shall first show why these two results have to be the same. Then, I shall show that, if the interaction is between the quantized vector potential and the classical approximation to the electron and solenoid currents, the state vector acquires the A-B phase shift. Lastly, I shall show how to reconcile these three mathematically and conceptually different calculations.

In this brief talk we will show how weak values appear in a wide range of physical contexts beyond the usual context of weak measurements. Among others, we will discuss how weak values appear in: the physics of classical parameters in a quantum evolution; the statistics of strong measurements; formulas for probability amplitudes in quantum mechanics; and finally, in the classical correspondence of quantum mechanics.