Quantum Foundations

I will give a general method for producing a process theory of local spacetime events and higher-order transformations from any base process theory of first-order maps. This process theory models events as intervention-context pairs, uniting the local actions by agents with the structure of the spacetime around them. I will show how this theory is richer than a standard process theory by permitting additional ways of composing agents beyond the usual tensor product, thereby capturing various strengths of possible spatio-temporal correlations. I will also explain the connection between these compositions and the logic "system BV".

Recent efforts to formulate a unified, causally neutral approach to quantum theory have highlighted the need for a framework treating spatial and temporal correlations on an equal footing. Building on this motivation, we propose operationally inspired axioms for quantum states over time, demonstrating that, unlike earlier approaches, these axioms yield a unique quantum state over time that is valid across both bipartite and multipartite spacetime scenarios. In particular, we show that the Fullwood-Parzygnat state over time uniquely satisfies these axioms, thus unifying bipartite temporal correlations and extending seamlessly to any number of temporal points. In particular, we identify two simple assumptions—linearity in the initial state and a quantum analog of conditionability—that single out a multipartite extension of bipartite quantum states over time, giving rise to a canonical generalization of Kirkwood-Dirac type quasi-probability distributions. This result provides a new characterization of quantum Markovianity, advancing our understanding of quantum correlations across both space and time.

the no-cloning theorem is an essential result in quantum information on top of which many quantum cryptography protocols are built. In this talk we examine the cloning question in the context of classical mechanics/Hamiltonian mechanics. We find the answer is quite subtle: whether a mechanical system can be cloned depends on the topological structure of its phase space. In particular, for a system to be clonable, its phase space must be contractible. This means certain systems (e.g. particle moving on a line) is clonable, while others (e.g. the simple pendulum) cannot be cloned. We explain the idea of the proof, which uses tools from algebraic topology (homotopy groups and Whitehead’s theorem). Finally we discuss the physical interpretations of this result: how do we reconcile this theorem with the experience that generally speaking, classical information is clonable? Can we use this no-cloning theorem to build secure communication protocols in classical systems instead of quantum ones?

To justify the existence of measurements that can not be performed jointly on quantum systems, Heisenberg put forward a heuristic argument, involving the famous gamma-ray microscope Gedankenexperiment, based on the existence of measurements that irreversibly alter the physical system on which they act. Today, the impossibility of jointly measuring some physical quantities, termed measurement incompatibility, and irreversible disturbance, namely the existence of operations that irreversibly alter the system on which they act, are understood to be distinct but related features of quantum mechanics. In our work, we formally characterized the relationship between these two properties, showing that measurement incompatibility implies irreversible disturbance, though the converse is false. The counterexamples are two toy theories: Minimal Classical Theory and Minimal Strongly Causal Bilocal Classical Theory. These two are distinct as counterexamples because the latter allows for classical conditioning. Our research followed an operational approach exploiting the framework of Operational Probabilistic Theories. In particular, it required the development of two new classes of operational theories: Minimal Operational Probabilistic Theories and Minimal Strongly Causal Operational Probabilistic Theories. These theories are characterized by a restricted set of dynamics, limited to the minimal set consistent with the set of states. In Minimal Strongly Causal Operational Probabilistic Theories, classical conditioning is also allowed.

The standard perspective on subsystems in quantum theory is a bottom-up, compositional one: one starts with individual "small" systems, viewed as primary, and composes them together to form larger systems. The top-down, decompositional perspective goes the other way, starting with a "large" system and asking what it means to partition it into smaller parts. In this talk, I will 1/ argue that the adoption of the top-down perspective is the key to progress in several current areas of foundational research; and 2/ present an integrated mathematical framework for partitions into three or more subsystems, using sub-C* algebras. Concerning the first item, I will explain how the top-down perspective becomes crucial whenever the way in which a quantum system is partitioned into smaller subsystems is not unique, but might depend on the physical situation at hand. I will display how that precise feature lies at the heart of a flurry of current hot foundational topics, such as quantum causal models, Wigner's friend scenarios, superselection rules, quantum reference frames, and debates over the implementability of the quantum switch. Concerning the second item, I will argue that partitions in (finite-dimensional) quantum theory can be naturally pinned down using sub-C* algebras. Building on simple illustrative examples, I will discuss the often-overlooked existence of non-factor C*-algebras, and how it leads to numerous subtleties -- in particular a generic failure of local tomography. I will introduce a sound framework for quantum partitions that overcomes these challenges; it is the first top-down framework that allows to consider three or more subsystems. Finally, as a display of this framework's technical power, I will briefly present how its application to quantum causal modelling unlocked the proof that all 1D quantum cellular automata admit causal decompositions.

(This is joint work with Octave Mestoudjian and Pablo Arrighi. This talk is complementary to my Causalworlds 2024 presentation, which will focus on the issue of causal decompositions.)

In this talk, I shall recall the nonlocality transitivity problem, which concerns the possibility of inferring the Bell-nonlocality of certain marginals in a multipartite scenario based on other given marginals. Then, I explain how considering this problem has led to a more general class of problems known as resource marginal problems (RMPs). More precisely, RMPs concern the possibility of having a resource-free target subsystem compatible with a given collection of marginal density matrices. We briefly discuss how a resource theory for a collection of marginal density matrices naturally arises from any given RMP and present some general features of such a theory. After that, we focus on a special case of RMPs known as the entanglement transitivity problems and explain how our progress on this problem has led to progress in the original nonlocality transitivity problem.

While quantum correlations between two spacelike-separated systems are fully encoded by the bipartite density operator associated with the joint system, what operator encodes quantum correlations across space and time? I will describe the general theory of such "quantum states over time" as well as a canonical example that encodes the expectation values of certain observables measured sequentially in time. The latter extends the theory of pseudo-density matrices to arbitrary dimensions, not necessarily restricted to multi-qubit systems. In addition, quantum states over time admit a natural proposal for a general-purpose quantum Bayes' rule. Our results specialize to many well-studied examples, such as the state-update rule, the two-state vector formalism and weak values, and the Petz recovery map. This talk is based on joint work with James Fullwood and the two papers: arXiv: 2212.08088 [quant-ph] and 2405.17555 [quant-ph].

Bayesian causal structure learning aims to learn a posterior distribution over directed acyclic graphs (DAGs), and the mechanisms that define the relationship between parent and child variables. By taking a Bayesian approach, it is possible to reason about the uncertainty of the causal model. The notion of modelling the uncertainty over models is particularly crucial for causal structure learning since the model could be unidentifiable when given only a finite amount of observational data. In this paper, we introduce a novel method to jointly learn the structure and mechanisms of the causal model using Variational Bayes, which we call Variational Bayes-DAG-GFlowNet (VBG). We extend the method of Bayesian causal structure learning using GFlowNets to learn not only the posterior distribution over the structure, but also the parameters of a linear-Gaussian model. Our results on simulated data suggest that VBG is competitive against several baselines in modelling the posterior over DAGs and mechanisms, while offering several advantages over existing methods, including the guarantee to sample acyclic graphs, and the flexibility to generalize to non-linear causal mechanisms.

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For continuous-variable systems, the negativities in the s-parametrized family of quasi-probability representations on a classical phase space establish a sort of hierarchy of non-classility measures. The coherent states, by design, display no negativity for any value of -1≤s≤1, meaning that sampling from the quantum probability distribution resulting from any measurement of a coherent state can be classically simulated, placing the coherent states as the most classical states according to this particular choice of phase space.

In this talk, I will describe how to construct s-ordered quasi-probability representations for finite-dimensional quantum systems when the phase space is equipped with more general group symmetries, focusing on the fermionic SO(2n) symmetry. Along the way, I will comment on an obstruction to an analogue of Hudson's theorem, namely that the only pure states that have positive s=0 Wigner functions are Gaussian states, and a possible remedy by giving up linearity in the phase-space correspondence.

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A Bell scenario can be conceptualized as a "communication" scenario with zero rounds of communication between parties, i.e., although each party can receive a system from its environment on which it can implement a measurement, it cannot send out any system to another party. Under this constraint, there is a strict hierarchy of correlation sets, namely, classical, quantum, and non-signalling. However, without any constraints on the number of communication rounds between the parties, they can realize arbitrary correlations by exchanging only classical systems. We consider a multipartite scenario where the parties can engage in at most a single round of communication, i.e., each party is allowed to receive a system once, implement any local intervention on it, and send out the resulting system once. Taking our cue from Bell nonlocality in the "zero rounds" scenario, we propose a notion of nonclassicality---termed antinomicity---for correlations in scenarios with a single round of communication. Similar to the zero rounds case, we establish a strict hierarchy of correlation sets classified by their antinomicity in single-round communication scenarios. Since we do not assume a global causal order between the parties, antinomicity serves as a notion of nonclassicality in the presence of indefinite causal order (as witnessed by causal inequality violations). A key contribution of this work is an explicit antinomicity witness that goes beyond causal inequalities, inspired by a modification of the Guess Your Neighbour's Input (GYNI) game that we term the Guess Your Neighbour's Input or NOT (GYNIN) game. Time permitting, I will speculate on why antinomicity is a strong notion of nonclassicality by interpreting it as an example of fine-tuning in classical models of indefinite causality.This is based on joint work with Ognyan Oreshkov, arXiv:2307.02565.

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Information-theoretic insights have proven fruitful in many areas of quantum physics. But can the fundamental dynamics of quantum systems be derived from purely information-theoretic principles, without resorting to Hilbert space structures such as unitary evolution and self-adjoint observables? Here we provide a model where the dynamics originates from a condition of informational non-equilibrium, the deviation of the system’s state from a reference state associated to a field of identically prepared systems. Combining this idea with three basic information-theoretic principles, we derive a notion of energy that captures the main features of energy in quantum theory: it is observable, bounded from below, invariant under time-evolution, in one-to-one correspondence with the generator of the dynamics, and quantitatively related to the speed of state changes. Our results provide an information-theoretic reconstruction of the Mandelstam-Tamm bound on the speed of quantum evolutions, establishing a bridge between dynamical and information-theoretic notions.

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