Quantum Foundations

In quantum information, we frequently consider (for instance, whenever we talk about entanglement) a composite system consisting of two separated subsystems. A standard axiom of quantum mechanics states that a composite system can be modeled as the tensor product of the two subsystems. However, there is another less restrictive way to model a composite system, which is used in quantum field theory: we can require only that the algebras of observables for each subsystem commute within some larger subalgebra. For finite-dimensional systems, these two axioms are equivalent, but this is not necessarily true for infinite-dimensional systems. Tsirelson's question (which comes in several variants) asks whether the correlations arising from commuting-operator models can always be represented by tensor-product models. I will give examples of linear system non-local games which cannot be played perfectly with tensor-product strategies, but can be played perfectly with commuting-operator strategies, resolving (one version of) Tsirelson's question in the negative. From these examples, we can also derive other consequences for the theory of non-local games, such as the undecidability of determining whether a non-local game has a perfect quantum strategy. 

Bell's theorem shows that our intuitive understanding of causation must be overturned in light of quantum correlations. Nevertheless, quantum mechanics does not permit signalling and hence a notion of cause remains. Understanding this notion is not only important at a fundamental level, but also for technological applications such as key distribution and randomness expansion. It has recently been shown that a useful way to determine which classical causal structures give rise to a given set of correlations is to use entropy vectors. We consider the question of whether such vectors can lead to useful certificates of non-classicality. We find that for a family of causal structures that include the usual bipartite Bell structure they do not, in spite of the existence of non-classical correlations. Furthermore, we find that for many causal structures non-Shannon entropic inequalities give additional constraints on the sets of possible entropy vectors in the classical case. They hence lead to tighter approximations of the set of realisable entropy vectors, which enables a sharper distinction of different causal structures. Whether these improved characterisations are also valid for the quantum case remains an open problem whose resolution would have implications for the discrimination of classical and quantum causes.

In this talk, I will outline the current state of the art in the study of the reality of the quantum state.  The main theme will be that, although you cannot derive the reality of the quantum state in an ontological model without additional assumptions, you can place constraints on the amount of overlap between probability measures that begin to make psi-epistemic theories look implausible.  These overlap bounds come from noncontextuality inequalities, and there are two types in the literature: those based on Cabello-Severini-Winter type inequalities and those based on Yu-Oh type inequalities.  The latter type of overlap bound was not originally derived from noncontextuality, but thinking of them this way yields a new proof of the Yu-Oh inequality and gives rise to family of related inequalities.  I will also explain why I think that most papers on ovelap bounds (including my own) have adopted sub-optimal measures, introduce better ones, and explain how this affects the choice of the best experimental protocol for demonstrating overlap bounds.  

The ideas of no-signalling, nonlocality, Bell inequalities, and quantum correlations can all be understood as implications of a presumed causal structure. In particular, the causal structure of the Bell scenario implies the Bell inequalities whenever the shared resource is presumed to act like a classical hidden random variable. If the shared resource in the scenario is a quantum system, however, then the quantum causal structure can give rise to a larger set of correlations, including probability distributions which violate Bell inequalities up to Tsirelson's bound. It is hard to generically distinguish between the classical and quantum correlations, though, because the standard method for computing Bell inequalities cannot be generalized to general causal scenarios. We therefore introduce a method (the "Inflation DAG" technique) to heuristically constrain the set of correlations compatible with a given classical causal structure, and we demonstrate how it may be used to derive explicit inequalities in terms of probabilities. We also discuss deriving physics-independent constraints, i.e. non-trivial inequalities which are nevertheless satisfied even by all quantum correlations, thereby quantifying some absolute limits as to what the universe allows.

[Unpublished results of E.W., Rob Spekkens, Tobias Fritz]

In this talk I will go over the recent paper by Daniela Frauchiger and Renato Renner, "Single-world interpretations of quantum theory cannot be self-consistent" (arXiv:1604.07422).

The paper introduces an extended Wigner's friend thought experiment, which makes use of Hardy's paradox to show that agents will necessarily reach contradictory conclusions - unless they take into account that they themselves may be in a superposition, and that their subjective experience of observing an outcome is not the whole story.

Frauchiger and Renner then put this experiment in context within a general framework to analyse physical theories. This leads to a theorem saying that a theory cannot be simultaneously (1) compliant with quantum theory, including at the macroscopic level, (2) single-world, and (3) self-consistent across different agents.

In this talk I will (1) describe the experiment and its immediate consequences, (2) quickly review how different interpretations react to it, (3) explain the framework and theorem in more detail.


 

Certain superposition states of the 1-D infinite square well have transient zeros at locations other than the nodes of the eigenstates that comprise them. It is shown that if an infinite potential barrier is suddenly raised at some or all of these zeros, the well can be split into multiple adjacent infinite square wells without affecting the wavefunction. While the average energy of the state was unchanged, this splitting effects a change of the energy eigenbasis of the state to a basis that does not commute with the original, and a subsequent measurement of the energy now reveals a completely different spectrum, which we call the interference energy spectrum of the state. Numerical simulations were used to verify that a barrier can be rapidly raised at a zero of the wavefunction without significantly affecting it. The concept of state inherent energy spectra is further explored as it relates to quantum state time evolution, probability theory, and weak value measurements. Of particular interest, this procedure can result in measurable energies that are greater than the energy of the highest mode in the original superposition, raising questions about the conservation of energy akin to those that have been raised in the study of superoscillations.

Most physicists take it for granted that the experimental violation of Bell's inequality provides evidence that it is not possible to completely describe the state of a physical system in terms of purely local information when this system is entangled with some other system. We disagree. Provided we redefine appropriately what is the information-theoretic state of a quantum system, it becomes possible to recover the whole from the description of its parts. This is in sharp contrast with the standard formalism of quantum mechanics in which the density matrix provides all there is to say about the state of a system. According to our formalism, there is no need to invoke supernatural nonlocality in order to explain everything standard quantum mechanics tells us that we can observe. We show, however, that this is inconsistent with the usual belief held among Everettians that the universal wavefunction can be taken as the complete representation of reality. Inspired by Plato and Kant, we introduce and contrast the notions of noumenal and phenomenal states of physical systems: the former corresponds to the complete but unknowable state of the system and the latter to what can be perceived about it with the help of arbitrary technology. We exhibit an explicit epimorphism from the former to the latter, which explains the relationship between all that there is and all that can be apprehended. Joint work with Paul Raymond-Robichaud

For a spin 1/2 (a qubit), Hamiltonian evolution is equivalent to an elliptic rotation of the (Bloch) spin vector in 3D space. In contrast, measurement alters the state norm, so may not be described as such a rotation. Nevertheless, extending the 3D spin vector to a 4D "spacetime" representation allows weak measurements to be interpreted as hyperbolic (boost) rotations. The combined Hamiltonian and measurement dynamics in continuous weak measurement trajectories are then equivalent to (stochastic) Lorentz transformations. Notably, in superconducting circuit QED implementations, the choice between which type of stochastic rotation occurs may be made long after the qubit and measurement field interact.

Considering a conformally coupled massless scalar field with arbitrary sign of kinetic energy and show that the conformal fluctuations in flatspace will increase without bounds if the mass of the scalar field is greater than a critical value.

We provide a classification of entangled states that uses new discrete entanglement invariants. The invariants are defined by algebraic properties of linear maps associated with the states. We prove a theorem on a correspondence between the invariants and sets of equivalent classes of entangled states. The new method works for an arbitrary finite number of finite-dimensional state subspaces. As an application of the method, we considered a large selection of cases of three subspaces of various dimensions. We also obtain an entanglement classification of four qubits, where we find 27 fundamental sets of classes.

 

Among QM's (in)famous oddities, perhaps the most intriguing is the capability of an event that did not occur, only could have, to exert a causal effect. How can a non-event leave a trace as concrete as a detector's click? I discuss this question and a novel insight into it offered by Cohen and Elitzur's "Quantum Oblivion" (20014).