Obstructions to Compositionality

Compositionality is at the heart of computer science and several other areas of applied category theory such as computational linguistics, categorical quantum mechanics, interpretable AI, dynamical systems, compositional game theory, and Petri nets. However, the meaning of the term seems to vary across the many different applications. This work contributes to understanding, and in particular qualifying, different kinds of compositionality. Formally, we introduce invariants of categories that we call zeroth and first homotopy posets, generalising in a precise sense the $pi_0$ and $pi_1$ of a groupoid. These posets can be used to obtain a qualitative description of how far an object is from being terminal and a morphism is from being iso. In the context of applied category theory, this formal machinery gives us a way to qualitatively describe the “failures of compositionality”, seen as failures of certain (op)lax functors to be strong, by classifying obstructions to the (op)laxators being isomorphisms. Failure of compositionality, for example for the interpretation of a categorical syntax in a semantic universe, can both be a bad thing and a good thing, which we illustrate by respective examples in graph theory and quantum theory.

The debate over QKD: A rebuttal to the NSA’s objections

A recent publication by the NSA assessing the usability of quantum cryptography has generated significant attention, concluding that this technology is not recommended for use. Here, we reply to this criticism and argue that some of the points raised are unjustified, whereas others are problematic now but can be expected to be resolved in the foreseeable future.

Light-cone thermodynamics: purification of the Minkowski vacuum

We explicitly express the Minkowski vacuum of a massless scalar field in terms of the particle notion associated with suitable spherical conformal killing fields. These fields are orthogonal to the light wavefronts originating from a sphere with a radius of $r_H$ in flat spacetime: a bifurcate conformal killing horizon that exhibits semiclassical features similar to those of black hole horizons and Cauchy horizons of non-extremal spherically symmetric black holes. Our result highlights the quantum aspects of this analogy and extends the well-known decomposition of the Minkowski vacuum in terms of Rindler modes, which are associated with the boost Killing field normal to a pair of null planes in Minkowski spacetime (the basis of the Unruh effect). While some features of our result have been established by Kay and Wald’s theorems in the 90s — on quantum field theory in stationary spacetimes with bifurcate Killing horizons — the added value we provide here lies in the explicit expression of the vacuum.

Painlevé-Gullstrand coordinates discontinuity in the quantum Oppenheimer-Snyder model

A metric that describes a collapsing star and the surrounding black hole geometry accounting for quantum gravity effects has been derived independently by different research groups. There is consensus regarding this metric up until the star reaches its minimum radius, but there is disagreement about what happens past this event. The discrepancy stems from the appearance of a discontinuity in the Hamiltonian evolution of the metric components in Painlev’e-Gullstrand coordinates. Here we show that the continuous geometry that describes this phenomenon is represented by a discontinuous metric when written in these coordinates. The discontinuity disappears by changing coordinates. The discontinuity found in the Hamiltonian approach can therefore be interpreted as a coordinate effect. The geometry continues regularly into an expanding white hole phase, without the occurrence of a shock wave caused by a physical discontinuity.

Local Inaccessibility of Random Classical Information : Conditional Nonlocality demands Entanglement

Discrimination of quantum states under local operations and classical communication (LOCC) is an intriguing question in the context of local retrieval of classical information, encoded in the multipartite quantum systems. All the local quantum state discrimination premises, considered so far, mimic a basic communication set-up, where the spatially separated decoding devices are independent of any additional input. Here, exploring a generalized communication scenario we introduce a framework for input-dependent local quantum state discrimination, which we call local random authentication (LRA). Referring to the term nonlocality, often used to indicate the impossibility of local state discrimination, we coin the term conditional nonlocality for the impossibility associated with the task LRA. We report that conditional nonlocality necessitates the presence of entangled states in the ensemble, a feature absent from erstwhile nonlocality arguments based on local state discrimination. Conversely, all the states in a complete basis set being entangled implies conditional nonlocality. However, the impossibility of LRA also exhibits more conditional nonlocality with less entanglement. The relation between the possibility of LRA and local state discrimination for sets of multipartite quantum states, both in the perfect and conclusive cases, has also been established. The results highlight a completely new aspect of the interplay between the security of information in a network and quantum entanglement under the LOCC paradigm.

Discreteness Unravels the Black Hole Information Puzzle: Insights from a Quantum Gravity Toy Model

The black hole information puzzle can be resolved if two conditions are met. Firstly, if the information of what falls inside a black hole remains encoded in degrees of freedom that persist after the black hole completely evaporates. These degrees of freedom should be capable of purifying the information. Secondly, if these purifying degrees of freedom do not significantly contribute to the system’s energy, as the macroscopic mass of the initial black hole has been radiated away as Hawking radiation to infinity. The presence of microscopic degrees of freedom at the Planck scale provides a natural mechanism for achieving these two conditions without running into the problem of the large pair-creation probabilities of standard remnant scenarios. In the context of Hawking radiation, the first condition implies that correlations between the ‘in’ and ‘out’ Hawking partner particles need to be transferred to correlations between the microscopic degrees of freedom and the ‘out’ partners in the radiation. This transfer occurs dynamically when the ‘in’ partners reach the singularity inside the black hole, entering the UV regime of quantum gravity where the interaction with the microscopic degrees of freedom becomes strong. The second condition suggests that the conventional notion of the vacuum’s uniqueness in quantum field theory should fail when considering the full quantum gravity degrees of freedom. In this paper, we demonstrate both key aspects of this mechanism using a solvable toy model of a quantum black hole inspired by loop quantum gravity.

The Generative Programs Framework

Recently there has been significant interest in using causal modelling techniques to understand the structure of physical theories. However, the notion of `causation’ is limiting – insisting that a physical theory must involve causal structure already places significant constraints on the form that theory may take. Thus in this paper, we aim to set out a more general structural framework. We argue that any quantitative physical theory can be represented in the form of a generative program, i.e. a list of instructions showing how to generate the empirical data; the information-processing structure associated with this program can be represented by a directed acyclic graph (DAG). We suggest that these graphs can be interpreted as encoding relations of `ontological priority,’ and that ontological priority is a suitable generalisation of causation which applies even to theories that don’t have a natural causal structure. We discuss some applications of our framework to philosophical questions about realism, operationalism, free will, locality and fine-tuning.

Which theories have a measurement problem?

It is shown that any theory that has certain properties has a measurement problem, in the sense that it makes predictions that are incompatible with measurement outcomes being absolute (that is, unique and non-relational). These properties are Bell Nonlocality, Information Preservation, and Local Dynamics. The result is extended by deriving Local Dynamics from No Superluminal Influences, Separable Dynamics, and Consistent Embeddings. As well as explaining why the existing Wigner’s-friend-inspired no-go theorems hold for quantum theory, these results also shed light on whether a future theory of physics might overcome the measurement problem. In particular, they suggest the possibility of a theory in which absoluteness is maintained, but without rejecting relativity theory (as in Bohm theory) or embracing objective collapses (as in GRW theory).

A no-go theorem for absolute observed events without inequalities or modal logic

This paper builds on no-go theorems to the effect that quantum theory is inconsistent with observations being absolute; that is, unique and non-relative. Unlike the existing no-go results, the one introduced here is based on a theory-independent absoluteness assumption, and there is no need to assume the validity of standard probability theory or of modal logic. The contradiction is derived by assuming that quantum theory applies in any inertial reference frame; accordingly, the result also illuminates a tension between special relativity and absoluteness.

Quantum Causal Inference with Extremely Light Touch

We consider the quantum version of inferring the causal relation between events. There has been recent progress towards identifying minimal interventions and observations needed. We here show, by means of constructing an explicit scheme, that quantum observations alone are sufficient for quantum causal inference for the case of a bipartite quantum system with measurements at two times. Our scheme involves the derivation of a closed-form expression for the space-time pseudo-density matrix associated with many times and qubits. This matrix can be determined by coarse-grained quantum observations alone. We show that from this matrix one can infer the causal structure via the sign of a particular function called a causal monotone. Our results show that for quantum processes one can infer the causal structure solely from correlations between observations at different times.