Quantum Foundations

Every unitary map with a factorisation of domain and codomain into subsystems has a well-defined causal structure given by the set of influence relations between its input and output subsystems. A causal decomposition of a unitary map U is, roughly, one that makes all there is to know about U in terms of causal structure evident and understandable in compositional terms. We'll argue that this is more than just about drawing more intuitive pictures for the causal structure of U -- it is about really understanding it at all. Now, it has been known for a while that decompositions in terms of ordinary circuit diagrams do not suffice to this end and that at least so called 'extended circuit diagrams', or 'routed circuit diagrams' are necessary, revealing a close connection between causal structure and algebraic structures that involve a particular interplay of direct sum and tensor product. Though whether or not these sorts of routed circuit diagrams suffice has been an open question since. I will give an introduction and overview of this business of causal decompositions of unitary maps, and share what is an on-going thriller.

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There are numerous properties of quantum states that one might be interested in characterizing, including statistical moments of observables such as expectations or variances, or more generally purities, entropies, probabilities, eigenvalues, symmetries, marginals, etc. Given a fixed collection of properties, the realizability problem aims to determine which value-assignments to those properties are jointly exhibited by at least one quantum state. In addition to the decision problem of realizability, one might also be interested in quantifying what proportion of quantum states possesses those property values. 

Any property of a quantum state can always, at least in principle, be estimated empirically by suitably measuring an ensemble of many independently and identically prepared copies of that quantum state. The particular sequence of positive operator valued measures which estimates a given property is known as a property estimation scheme. The purpose of this talk is to discuss a strategy for tackling realizability problems by studying the large deviation behaviour of property estimation schemes.

The key idea of this approach is the following:
A given collection of properties is realized by a quantum state if and only if a random quantum state occasionally produces that collection of properties as estimates.
Under suitable conditions, this observation leads to a complete hierarchy of necessary conditions for realizability.

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The theory of polynomial optimisation considers a polynomial objective function subject to countable many polynomial constraints. In a seminal contribution Navascués, Pironio and Acín (NPA) generalised a previous result from Lassere, allowing for its application in quantum information theory by considering its non-commutative variant. Non-commutative variables are represented as bounded operators on potentially infinite dimensional Hilbert spaces. These infinite-dimensional non-commutative polynomials optimisation (NPO) problems are recast as a complete hierarchy of semidefinite programming (SDP) relaxations by a suitable partitioning of the underlying spaces.

The reformulation into convex optimisation problems allows for numerical analysis. We focus on an operator theoretical approach to the NPA hierarchy and show its equiv-
alence to the original NPA hierarchy. To do so, we introduce the necessary mathematical preliminaries from operator algebra theory and semidefinite programming. We conclude by showing how certain relations on operators translate to SDP relaxations yielding drastically reduced problem sizes.

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Abstract: TBD

Zoom Link: https://pitp.zoom.us/j/95212522185?pwd=eWx4R3o3cmZISmtPY0xwMmdxc2EzZz09

Bell's theorem proves that quantum theory is inconsistent with local physical models and, from the perspective of causal inference, it can be seen as the impossibility of providing a classical causal explanation to quantum correlations. Bell's theorem has propelled research in the foundations of quantum theory and quantum information science. In the last decade, the investigation of nonlocality has moved beyond Bell's theorem to consider more complicated causal structures allowing for communication between the parts and involving several independent sources which distribute shares of physical systems in a network. For the case of three observable variables, it is known that there are three non-trivial causal networks. Two of those, are known to give rise to quantum non-classicality: the instrumental and the triangle scenarios. In this talk, we introduce new tools to tackle the compatibility problem in the general framework of Bayesian networks and explore the remaining non-trivial network, which we call the Evans scenario. We do not solve its main open problem –whether quantum non-classical correlations can arise from it – but give a significant step in this direction by proving that post-quantum correlations, analogous to the Popescu-Rohrlich box, do violate the constraints imposed by a classical description of Evans causal structure.

Zoom Link: https://pitp.zoom.us/j/98549320714?pwd=QVllZXNpZTFqekI1ZVUrK3ZLdnRjZz09

No-go theorems (Bell, Kochen-Specker, …) formally show the departure of quantum theory from classical theory. These are formulated in the framework of ontological models and, if one accepts such framework, entail that quantum theory involves problematic (“fine-tuned”) properties. I will argue that the lesson to take from the no-go theorems is to abandon the framework of ontological models as the way to model reality. I will analyze what I believe to be the unnatural assumptions of such framework and I will propose a way to change it. The basic principle of the new notion of reality I propose is that for something to exist is for something to be recorded. I will motivate the principle and explore its consequences. In order to implement such proposal into a precise theory-independent mathematical framework I will make use of point-free topological spaces (locales). I will discuss why this new proposal should be promising for understanding quantum theory and I will present several open questions.

Zoom link:  https://pitp.zoom.us/j/91292006884?pwd=V2EzaEw5Z3NRUGd4cVdSRnlOOWFVZz09

Understanding gravity in the framework of quantum mechanics is one of the great challenges in modern physics. Along this line, a prime question is to find whether gravity is a quantum entity subject to the rules of quantum mechanics. It is fair to say that there are no feasible ideas yet to test the quantum coherent behaviour of gravity directly in a laboratory experiment. I will introduce an idea for such a test based on the principle that two objects cannot be entangled without a quantum mediator.  I will show that despite the weakness of gravity, the phase evolution induced by the gravitational interaction of two micron size test masses in adjacent matter-wave interferometers can detectably entangle them even when they are placed far apart enough to keep Casimir-Polder forces at bay. I will provide a prescription for witnessing this entanglement, which certifies gravity as a quantum coherent mediator, through simple correlation measurements between two spins: one embedded in each test mass. Fundamentally, the above entanglement is shown to certify the presence of non-zero off-diagonal terms in the coherent state basis of the gravitational field modes.

Zoom link:  https://pitp.zoom.us/j/99584743899?pwd=aHl1cVlpK29ZVDkrdFZyM01GemJJdz09

Nonclassicality, as witnessed by the incapacity of Classical Causal Theory (CCT) of explaining a system's behavior given its causal structure, come to be one of the hottest topics in Quantum Foundations over the last decades, a movement that was motivated both by its vast range of practical applications and by the powerful insights it provides about the rules of the quantum world. Among the many attempts at understanding/quantifying this phenomenon, we highlight the idea of inquiring how further would it be necessary to relax the causal structure associated with a given system in order to have its nonclassical behavior explained by CCT. More recently, we showed that the relaxation demanded to explain the behavior of a subset of variables in a given experiment may not be allowed by the embedding causal structure when considering the behavior of the remaining variables, which led to a new way of witnessing nonclassicality. In this seminar, we discuss a new way of quantifying this incompatibility and possible generalizations of this approach to different scenarios.

Zoom link:  https://pitp.zoom.us/j/96051313203?pwd=bFNhOEhkSVhXWk8yd1hVZWVVa0U4UT09

In this talk, I present the latest works on anyonic information theory and how it is linked to aspects of quantum foundations. First, the theory of 2+1 D non-abelian anyons will be introduced. The newly discovered notion of anyonic creation operators will be presented, as well as their use as local elements of reality within the Deutsch-Hayden interpretation of quantum mechanics. Lastly, I will show strange properties of anyonic entanglement that appear due to the lack of a tensor product structure, such as the different spectra of marginals in bipartite systems. This property makes the Von Neumann entropy a bad entanglement measure. I will explain the challenges of defining entanglement measures for anyonic systems and current approaches. 

Zoom link:  https://pitp.zoom.us/j/99863263804?pwd=MUhkYTBzcUlwTmJ0Z3F4aFo3Rkt6QT09

In spite of its immense importance in the present-day information technology, the foundational aspects of quantum theory (QT) remain still elusive. In particular, there is no such set of physically motivated axioms which can answer why Hilbert space formalism is the only natural choice to describe the microscopic world. Hence, to shed light on the unique formalism of QT, two different operational frameworks will be described in the primitive of various convex operational theories. The first one refers to a kinematical symmetry principle which would be proposed from the perspective of single copy state discrimination and it would be shown that this symmetry holds for both classical and QT – two successful descriptions of the physical world. On the other hand, studying a wide range of convex operational theories, namely the General Probabilistic Theories (GPTs) with polygonal state spaces, we observe the absence of such symmetry. Thus, the principle deserves its own importance to mark a sharp distinction between the physical and unphysical theories. Thereafter, a distributed computing scenario will be introduced for which the other convex theories except the QT turn out to be equivalent to the classical one even though the theories possess more exotic state and effect spaces. We have coined this particular operational framework as ‘Distributed computation with limited communication’ (DCLC). Furthermore, it will be shown that the distributed computational strength of quantum communication will be justified in terms of a stronger version of this task, namely the ‘Delayed choice distributed computation with limited communication’ (DC2LC). The proposed task thus provides a new approach to operationally single out quantum theory in the theory-space and hence promises a novel perspective towards the axiomatic derivation of Hilbert space quantum mechanics.

References:
Phys. Rev. A (Rapid)100, 060101 (2019)
Ann. Phys.(Berlin)2020,532, 2000334 (2020)
arXiv:2012.05781 [quant-ph](2020)

Zoom link:  https://pitp.zoom.us/j/92924188227?pwd=ODJYQXVoaUtzZmZIdFlmcUNIV3Rmdz09

In the context of irreversible dynamics, the meaning of the reverse of a physical evolution can be quite ambiguous. It is a standard choice to define the reverse process using Bayes' theorem, but, in general, this is not optimal with respect to the relative entropy of recovery. In this work we explore whether it is possible to characterise an optimal reverse map building from the concept of state retrieval maps. In doing so, we propose a set of  principles that state retrieval maps should satisfy. We find out that the Bayes inspired reverse is just one case in a whole class of possible choices, which can be optimised to give a map retrieving the initial state more precisely than the Bayes rule. Our analysis has the advantage of naturally extending to the quantum regime. In fact, we find a class of reverse transformations containing the Petz recovery map as a particular case, corroborating its interpretation as a quantum analogue of the Bayes retrieval.

Finally, we present numerical evidence showing that by adding a single extra axiom one can isolate for classical dynamics the usual reverse process derived from Bayes' theorem.

Zoom link:  https://pitp.zoom.us/j/93589286500?pwd=dkZuRzR0SlhVd1lPdGNOZWFYQWtRZz09

Non-normalizable quantum states are usually discarded as mathematical artefacts in quantum mechanics. However, such states naturally occur in quantum gravity as solutions to physical constraints. This suggests reconsidering the interpretation of such states. Some of the existing approaches to this question seek to redefine the inner product, but this arguably leads to further challenges. 

In this talk, I will propose an alternative interpretation of non-normalizable states using pilot-wave theory. First, I will argue that the basic conceptual structure of the theory contains a straightforward interpretation of these states. Second, to better understand such states, I will discuss non-normalizable states of the quantum harmonic oscillator from a pilot-wave perspective. I will show that, contrary to intuitions from orthodox quantum mechanics, the non-normalizable eigenstates and their superpositions are bound states in the sense that the pilot-wave velocity field vy→0 at large ±y. Third, I will introduce a new notion of equilibrium, named pilot-wave equilibrium, and use it to define physically-meaningful equilibrium densities for such states. I will show, via an H-theorem, that an arbitrary initial density with compact support relaxes to pilot-wave equilibrium at a coarse-grained level, under assumptions similar to those for relaxation to quantum equilibrium. I will conclude by discussing the implications for pilot-wave theory, quantum gravity and quantum foundations in general. 

Based on:

I. Sen. "Physical interpretation of non-normalizable harmonic oscillator states and relaxation to pilot-wave equilibrium" arXiv:2208.08945 (2022)

Zoom link:  https://pitp.zoom.us/j/93736627504?pwd=VGtxZE5rTFdnT1dqZlFRWTFvWlFQUT09