arXiv subjects

James Fullwood

Publications and source records attributed to James Fullwood.

At least 37 records · Page 2Linked to original sources

arXiv2024

General Communication Enhancement via the Quantum Switch

Recent studies have shown that quantum information may be effectively transmitted by a finite collection of completely depolarizing channels in a coherent superposition of different orders, via an operation known as the quantum $\tt SWITCH$. Such results are quite remarkable, as completely depolarizing channels taken in isolation and in a definite order can only output white noise. For general channels however, little is known about the potential communication enhancement provided by the quantum $\tt SWITCH$. In this Letter, we define an easily computable quantity $\mathcal{P}_n$ associated with the quantum ${\tt SWITCH}$ of $n$ copies of a fixed channel, and we conjecture that $\mathcal{P}_n>0$ is both a necessary and sufficient condition for communication enhancement via the quantum $\tt SWITCH$. In support of our conjecture, we derive a simple analytic expression for the classical capacity of the quantum $\tt SWITCH$ of $n$ copies of an arbitrary Pauli channel in terms of the quantity $\mathcal{P}_n$, which we then use to show that our conjecture indeed holds in the space of all Pauli channels. Utilizing such results, we then formulate a communication protocol involving the quantum $\tt SWITCH$ which enhances the private capacity of the BB84 channel.

quant-ph↗

arXiv2023

General covariance for quantum states over time

The theory of quantum states over time provides an approach to the dynamics of quantum information which is in direct analogy with spacetime and its relation to classical dynamics. In this work, we further such an analogy by formulating a notion of general covariance for the theory of quantum states over time. We then associate a canonical state over time with a density operator which is to evolve under a sequence of quantum processes modeled by completely positive trace-preserving (CPTP) maps, and we show that such a canonical state over time satisfies such a notion of covariance. We also show that the dynamical quantum Bayes' rule transforms covariantly with respect to states over time, and we conclude with a discussion of what it means for a physical law to be generally covariant when formulated in terms of quantum states over time.

quant-ph↗

arXiv2023

Virtual quantum broadcasting

The quantum no-broadcasting theorem states that it is impossible to produce perfect copies of an arbitrary quantum state, even if the copies are allowed to be correlated. Here we show that, although quantum broadcasting cannot be achieved by any physical process, it can be achieved by a virtual process, described by a Hermitian-preserving trace-preserving map. This virtual process is canonical: it is the only map that broadcasts all quantum states, is covariant under unitary evolution, is invariant under permutations of the copies, and reduces to the classical broadcasting map when subjected to decoherence. We show that the optimal physical approximation to the canonical broadcasting map is the optimal universal quantum cloning, and we also show that virtual broadcasting can be achieved by a virtual measure-and-prepare protocol, where a virtual measurement is performed, and, depending on the outcomes, two copies of a virtual quantum state are generated. Finally, we use canonical virtual broadcasting to prove a uniqueness result for quantum states over time.

quant-ph↗

arXiv2023

On dynamical measures of quantum information

In this work, we use the theory of quantum states over time to define an entropy $S(ρ,\mathcal{E})$ associated with quantum processes $(ρ,\mathcal{E})$, where $ρ$ is a state and $\mathcal{E}$ is a quantum channel responsible for the dynamical evolution of $ρ$. The entropy $S(ρ,\mathcal{E})$ is a generalization of the von Neumann entropy in the sense that $S(ρ,\mathrm{id})=S(ρ)$ (where $\mathrm{id}$ denotes the identity channel), and is a dynamical analogue of the quantum joint entropy for bipartite states. Such an entropy is then used to define dynamical formulations of the quantum conditional entropy and quantum mutual information, and we show such information measures satisfy many desirable properties, such as a quantum entropic Bayes' rule. We also use our entropy function to quantify the information loss/gain associated with the dynamical evolution of quantum systems, which enables us to formulate a precise notion of information conservation for quantum processes.

quant-ph↗

arXiv2023

From time-reversal symmetry to quantum Bayes' rules

Bayes' rule $\mathbb{P}(B|A)\mathbb{P}(A)=\mathbb{P}(A|B)\mathbb{P}(B)$ is one of the simplest yet most profound, ubiquitous, and far-reaching results of classical probability theory, with applications in any field utilizing statistical inference. Many attempts have been made to extend this rule to quantum systems, the significance of which we are only beginning to understand. In this work, we develop a systematic framework for defining Bayes' rule in the quantum setting, and we show that a vast majority of the proposed quantum Bayes' rules appearing in the literature are all instances of our definition. Moreover, our Bayes' rule is based upon a simple relationship between the notions of state over time and a time-reversal symmetry map, both of which are introduced here.

quant-ph↗

arXiv2022

An axiomatic characterization of mutual information

We characterize mutual information as the unique map on ordered pairs of random variables satisfying a set of axioms similar to those of Faddeev's characterization of the Shannon entropy. There is a new axiom in our characterization however which has no analogue for Shannon entropy, based on the notion of a Markov triangle, which may be thought of as a composition of communication channels for which conditional entropy acts functorially. Our proofs are coordinate-free in the sense that no logarithms appear in our calculations.

cs.IT↗

arXiv2022

On quantum states over time

In 2017, D. Horsman, C. Heunen, M. Pusey, J. Barrett, and R. Spekkens proved that there is no physically reasonable assignment that takes a quantum channel and an initial state and produces a joint state on the tensor product of the input and output spaces. The interpretation was that there is a clear distinction between space and time in the quantum setting that is not visible classically, where in the latter, one can freely use Bayes' theorem to go between joint states and marginals with noisy channels. In this paper, we prove that there actually is such a physically reasonable assignment, bypassing the no-go result of Horsman et al., and we illustrate that this is achievable by restricting the domain of their assignment to a domain which represents the given data more faithfully.

quant-ph↗

arXiv2022

On analytic groupoid cardinality

Groupoids graded by the groupoid of bijections between finite sets admit generating functions which encode the groupoid cardinalities of their graded components. As suggested in the work of Baez and Dolan, we use analytic continuation of such generating functions to define a complex-valued cardinality for groupoids whose usual groupoid cardinality diverges. The complex nature of such a cardinality invariant is shown to reflect a recursion of structure which we refer to as `nested equivalence'.

math.CT↗

arXiv2022

On a Lefschetz-type phenomenon for elliptic Calabi--Yaus

We consider 18 families of elliptic Calabi--Yaus which arise in constructing $F$-theory compactifications of string vacua, and show in each case that the upper Hodge diamond of a crepant resolution of the associated Weierstrass model coincides with the upper Hodge diamond of the (blown up) projective bundle in which the crepant resolution is naturally embedded. Such results are unexpected, as each crepant resolution we consider does \emph{not} satisfy the hypotheses of the Lefschetz hyperplane theorem. In light of such findings, we suspect that all ellipitic Calabi--Yaus satisfy such a `Lefschetz-type phenomenon'.

math.AG↗

arXiv2021

The information loss of a stochastic map

We provide a stochastic extension of the Baez-Fritz-Leinster characterization of the Shannon information loss associated with a measure-preserving function. This recovers the conditional entropy and a closely related information-theoretic measure that we call conditional information loss. Although not functorial, these information measures are semi-functorial, a concept we introduce that is definable in any Markov category. We also introduce the notion of an entropic Bayes' rule for information measures, and we provide a characterization of conditional entropy in terms of this rule.

cs.IT↗

arXiv2021

On a 2-relative entropy

We construct a 2-categorical extension of the relative entropy functor of Baez and Fritz, and show that our construction is functorial with respect to vertical morphisms. Moreover, we show such a `2-relative entropy' satisfies natural 2-categorial analogues of convex linearity, vanishing under optimal hypotheses, and lower semicontinuity. While relative entropy is a relative measure of information between probability distributions, we view our construction as a relative measure of information between channels.

cs.IT↗

arXiv2020

Orientifold limits of singular $F$-theory vacua

We construct global orientifold limits of singular $F$-theory vacua whose associated gauge groups are SO(3), SO(5), SO(6), $F_4$, SU(4), and Spin(7). For each limit we show a universal tadpole relation is satisfied, which is a homological identity whose dimension-zero component encodes the matching of the D3 charge between each $F$-theory compactification and its orientifold limit. While for smooth $F$-theory compactifications which admit global orientifold limits the contribution to the associated universal tadpole relation comes from its Chern class, we show that for all singular $F$-theory compactifications under consideration, the contribution to the universal tadpole relation comes from its \emph{stringy} Chern class.

hep-th↗

arXiv2019

On integral structure types

We introduce integral structure types as a categorical analogue of virtual combinatorial species. Integral structure types then categorify power series with possibly negative coefficients in the same way that combinatorial species categorify power series with non-negative rational coefficients. The notion of an operator on combinatorial species naturally extends to integral structure types, and in light of their `negativity' we define the notion of the commutator of two operators on integral structure types. We then extend integral structure types to the setting of stuff types as introduced by Baez and Dolan, and then conclude by using integral structure types to give a combinatorial description for Chern classes of projective hypersurfaces.

math.CO↗

arXiv2019

On the Euler characteristic of a relative hypersurface

We derive a general formula for the Euler characteristic of a fibration of projective hypersurfaces in terms of invariants of an arbitrary base variety. When the general fiber is an elliptic curve, such formulas have appeared in the physics literature in the context of calculating D-brane charge for M-/F-theory and type-IIB compactifications of string vacua. While there are various methods for computing Euler characteristics of algebraic varieties, we prove a base-independent pushforward formula which reduces the computation of the Euler characteristic of relative hypersurfaces to simple algebraic manipulations of rational expressions determined by its divisor class in a projective bundle. We illustrate our methods by applying them to an explicit family of relative hypersurfaces whose fibers are of arbitrary dimension and degree.

math.AG↗

arXiv2018

Stringy Hirzebruch classes of Weierstrass fibrations

A Weierstrass fibration is an elliptic fibration $Y\to B$ whose total space $Y$ may be given by a global Weierstrass equation in a $\mathbb{P}^2$-bundle over $B$. In this note, we compute stringy Hirzebruch classes of singular Weierstrass fibrations associated with constructing non-Abelian gauge theories in $F$-theory. For each Weierstrass fibration $Y\to B$ we then derive a generating function $χ^{\text{str}}_y(Y;t)$, whose degree-$d$ coefficient encodes the stringy $χ_y$-genus of $Y\to B$ over an unspecified base of dimension $d$, solely in terms of invariants of the base. To facilitate our computations, we prove a formula for general characteristic classes of blowups along (possibly singular) complete intersections.

math.AG↗

arXiv2018

On the Sen limit squared

We introduce a class of F-theory vacua which may be viewed as a specialization of the so-called $E_6$ fibration, and construct a weak coupling limit associated with such vacua which we view as the `square' of the Sen limit. We find that while Sen's limit is naturally viewed as an orientifold theory, the universal tadpole relation which equates the D3 charge between the associated F-theory compactification and the limit we construct suggests that perhaps the limiting theory is in fact an oriented theory compactified on the base of the F-theory elliptic fibration.

hep-th↗

arXiv2016

On CSM classes via Chern-Fulton classes of f-schemes

The Chern-Fulton class is a generalization of Chern class to the realm of arbitrary embeddable schemes. While Chern-Fulton classes are sensitive to non-reduced scheme structure, they are not sensitive to possible singularities of the underlying support, thus at first glance are not interesting from a singularity theory viewpoint. However, we introduce a class of formal objects which we think of as `fractional schemes', or f-schemes for short, and then show that when one broadens the domain of Chern-Fulton classes to f-schemes one may indeed recover singularity invariants of varieties and schemes. More precisely, we show that for almost all global complete intersections, their Chern-Schwartz-Macpherson classes may be computed via Chern-Fulton classes of f-schemes.

math.AG↗

arXiv2016

Towards a simple characterization of the Chern-Schwartz-MacPherson class

For a large class of possibly singular complete intersections we prove a formula for their Chern-Schwartz-MacPherson classes in terms of a single blowup along a scheme supported on the singular loci of such varieties. In the hypersurface case our formula recovers a formula of Aluffi proven in 1996. As our formula is in no way tailored to the complete intersection hypothesis, we conjecture that it holds for all closed subschemes of a smooth variety. If in fact true, such a formula would provide a simple characterization of the Chern-Schwartz-MacPherson class which does not depend on a resolution of singularities. We also show that our formula may be suitably interpreted as the Chern-Fulton class of a scheme-like object which we refer to as an `$\mathfrak{f}$-scheme'.

math.AG↗