Searcharxiv⌕ Search

arXiv subjects

Ian T. Durham

Publications and source records attributed to Ian T. Durham.

29 records · Page 2Linked to original sources
arXiv2009

A category theoretic approach to asymptotic quantum channel approximation and Birkhoff's Theorem

Birkhoff's Theorem states that doubly stochastic matrices are convex combinations of permutation matrices. Quantum mechanically these matrices are doubly stochastic channels, i.e. they are completely positive maps preserving both the trace and the identity. We expect these channels to be convex combinations of unitary channels and yet it is known that some channels cannot be written that way. Recent work has suggested that $n$ copies of a single channel might approximate a mixture (convex combination) of unitaries. In this paper we show that $n(n+1)/2$ copies of a symmetric unital quantum channel may be arbitrarily-well approximated by a mixture (convex combination) of unitarily implemented channels. In addition, we prove that any extremal properties of a channel are preserved over $n$ (and thus $n(n+1)/2$) copies. The result has the potential to be completely generalized to include non-symmetric channels.

quant-ph↗
arXiv2008

Quantum communication on closed time-like curves

We define a new quantity we call a ctcbit that provides a means for quantifying a qubit on a closed time-like curve (CTC) as a shared resource. We describe a simple protocol for the sharing of information that is similar to quantum teleportation but does not require an entangled particle pair or ebit. While there is the appearance that the given resource is free, we employ a non-Hausdorff topology to prevent any limitless information exchanges. Nonetheless, the resource does limit the type of information that may be transferred, though this limitation can be reduced through the use of weak measurements. While the reality of CTCs is highly speculative, the present paper provides a manner by which quantum informational methods may be employed to study such problems and may ultimately prove useful in studying quantum gravity.

quant-ph↗
arXiv2008

The non-conditional nature of the Cerf-Adami inequalities and implications for thermodynamics

We show that the Cerf-Adami inequalities do not necessarily depend on conditional entropies nor any reference to Markov chains. While the latter are not explicit in the original form, they are often implied in certain derivations. We also show that these inequalities are intimately related to at least one interpretation of the second law of thermodynamics. The combination of these results provides added insight into why some quantum systems violate the Cerf-Adami inequalities thereby improving our understanding of the quantum-classical boundary. As a result we suggest that the second law may serve as some type of boundary condition on classical knowledge.

quant-ph↗
arXiv2007

Are entropic Bell inequalities implied by the second law?

Bell's inequalities, in the form given by Cerf and Adami, are derived from the combination of the second law of thermodynamics and the Markov postulate. Violations of these inequalities are discussed in terms of the mixing characteristics of the density operators. A quantum mechanical bound for a system of three qubits, two of which are entangled, is found and is shown to lead to strong subadditivity in one case. Finally, in interpreting the results, the role the uncertainty relations play in the definition of entropy and the second law is discussed, demonstrating a link to Bell's inequalities. Additionally the second law is shown to be dependent on the type of measurement in quantum systems. Specifically, a more general statement of the second law appropriate to both quantum and classical systems is suggested.

quant-ph↗
arXiv2006

Limitations on entropic Bell inequalities

The derivation of Bell inequalities in terms of quantum statistical (thermodynamic) entropies is considered. Inequalities of the Wigner form are derived but shown to be extremely limiting in their applicability due to the nature of the density matrices involved. This also helps to identify a limitation in the Cerf-Adami inequalities.

quant-ph↗
arXiv2005

Correlated observables in single-particle systems and field theoretic interpretations

Bell-type experiments that test correlated observables typically involve measurements of spin or polarization on multi-particle systems in singlet states. These observables are all non-commuting and satisfy an uncertainty relation. Theoretically, the non-commuting nature should be independent of whether the singlet state consists of multiple particles or a single particle. Recent experiments in single neutron interferometry have in fact demonstrated this. In addition, if Bell-type inequalities can be found for experiments involving spin and polarization, the same should be true for experiments involving other non-commuting observables such as position and momentum as in the original EPR paper. As such, an experiment is proposed to measure (quantum mechanically) position and momentum for a single oscillator as a means for deriving a Bell-type inequality for these correlated observables. The experiment, if realizable, would shed light on the basic nature of matter, perhaps pointing to some form of self-entanglement, and would also help to further elucidate a possible mechanism behind the Heisenberg uncertainty principle. Violation of these inequalities would, in fact, offer yet another confirmation of the principle.

quant-ph↗
arXiv2005

Quantum Puzzles in the Set Theoretic Metaworld of Heisenberg, Clauser, and Horne

This paper follows up on a recent pre-print (Durham [2005]) by first deriving a set theoretic relationship between the generalized uncertainty relations and the Clauser-Horne inequalities. The physical, metaphysical, and metamathematical implications and problems are then explored. The discussion builds on previous work by Pitowsky [1994] and suggests that there is a fundamental problem in quantum correlation that could potentially lead to a paradox. It leaves open the question of whether the problem is in experiment, theory, or phenomena.

quant-ph↗
arXiv2003

Eddington & Uncertainty

Sir Arthur Eddington is considered one of the greatest astrophysicist of the twentieth century and yet he gained a stigma when, in the 1930s, he embarked on a quest to develop a unified theory of gravity and quantum mechanics. His attempts ultimately proved fruitless and he was unfortunately partially shunned by some physicists in the latter portion of his career. In addition some historians have been less than kind to him regarding this portion of his work. However, detailed analysis of how this work got started shows that Eddington's theories were not as outlandish as they are often purported to be. His entire theory rested on the use of quantum mechanical methods of uncertainty in the reference frames of relativity. Though the work was ultimately not fruitful, in hindsight it did foreshadow several later results in physics and his methods were definitely rigorous. In addition, his philosophy regarding determinism and uncertainty was actually fairly orthodox for his time. This work begins by looking at Eddington's life and philosophy and uses this as a basis to explore his work with uncertainty.

physics.hist-ph↗
arXiv2000

A Historical Perspective on the Topology and Physics of Hyperspace

This paper examines the history of the development of dimensional analysis, from Euclidean geometry to hyperspace. This is largely a secondary source work intended to summarize, as briefly as possible, with enough mathematical rigor, the mathematical developments in dimensional analysis over the past 2550 years. It is intended to stimulate further research and interest in the topic while also presenting a few original thoughts on certain aspects of the subject. In particular, the paper focuses on the development of Euclidean geometry, the intense study of the parallel axiom in the 1800's, the development of time as a dimension leading to Kaluza-Klein theory, and finally the explosion of work in the combined areas of string theory, M-theory, and F-theory in the past 30 years.

physics.hist-ph↗