arXiv subjects

Prakash Panangaden

Publications and source records attributed to Prakash Panangaden.

At least 55 records · Page 3Linked to original sources

arXiv2014

Proceedings 9th Workshop on Quantum Physics and Logic

This volume contains the proceedings of the ninth workshop on Quantum Physics and Logic (QPL2012) which took place in Brussels from the 10th to the 12th of October 2012. QPL2012 brought together researchers working on mathematical foundations of quantum physics, quantum computing, and spatio-temporal causal structures. The particular focus was on the use of logical tools, ordered algebraic and category-theoretic structures, formal languages, semantical techniques, and other computer science methods for the study of physical behaviour in general.

quant-ph↗

arXiv2013

An Exact Expression for Photon Polarization in Kerr Geometry

We analyze the transformation of the polarization of a photon propagating along an arbitrary null geodesic in Kerr geometry. The motivation comes from the problem of an observer trying to communicate quantum information to another observer in Kerr spacetime by transmitting polarized photons. It is essential that the observers understand the relationship between their frames of reference and also know how the photon's polarization transforms as it travels through Kerr spacetime. Existing methods to calculate the rotation of the photon polarization (Faraday rotation) depend on choices of coordinate systems, are algebraically complex and yield results only in the weak-field limit. We give a closed-form expression for a parallel propagated frame along an arbitrary null geodesic using Killing-Yano theory, and thereby solve the problem of parallel transport of the polarization vector in an intrinsic, geometrically-motivated fashion. The symmetries of Kerr geometry are utilized to obtain a remarkably compact expression for the geometrically induced phase of the photon's polarization. We show that this phase vanishes on the equatorial plane and the axis of symmetry.

math-ph↗

arXiv2012

Metrics for Finite Markov Decision Processes

We present metrics for measuring the similarity of states in a finite Markov decision process (MDP). The formulation of our metrics is based on the notion of bisimulation for MDPs, with an aim towards solving discounted infinite horizon reinforcement learning tasks. Such metrics can be used to aggregate states, as well as to better structure other value function approximators (e.g., memory-based or nearest-neighbor approximators). We provide bounds that relate our metric distances to the optimal values of states in the given MDP.

cs.AI↗

arXiv2012

Metrics for Markov Decision Processes with Infinite State Spaces

We present metrics for measuring state similarity in Markov decision processes (MDPs) with infinitely many states, including MDPs with continuous state spaces. Such metrics provide a stable quantitative analogue of the notion of bisimulation for MDPs, and are suitable for use in MDP approximation. We show that the optimal value function associated with a discounted infinite horizon planning task varies continuously with respect to our metric distances.

cs.AI↗

arXiv2012

Methods for computing state similarity in Markov Decision Processes

A popular approach to solving large probabilistic systems relies on aggregating states based on a measure of similarity. Many approaches in the literature are heuristic. A number of recent methods rely instead on metrics based on the notion of bisimulation, or behavioral equivalence between states (Givan et al, 2001, 2003; Ferns et al, 2004). An integral component of such metrics is the Kantorovich metric between probability distributions. However, while this metric enables many satisfying theoretical properties, it is costly to compute in practice. In this paper, we use techniques from network optimization and statistical sampling to overcome this problem. We obtain in this manner a variety of distance functions for MDP state aggregation, which differ in the tradeoff between time and space complexity, as well as the quality of the aggregation. We provide an empirical evaluation of these trade-offs.

cs.AI↗

arXiv2011

Quantum Communication in Rindler Spacetime

A state that an inertial observer in Minkowski space perceives to be the vacuum will appear to an accelerating observer to be a thermal bath of radiation. We study the impact of this Davies-Fulling-Unruh noise on communication, particularly quantum communication from an inertial sender to an accelerating observer and private communication between two inertial observers in the presence of an accelerating eavesdropper. In both cases, we establish compact, tractable formulas for the associated communication capacities assuming encodings that allow a single excitation in one of a fixed number of modes per use of the communications channel. Our contributions include a rigorous presentation of the general theory of the private quantum capacity as well as a detailed analysis of the structure of these channels, including their group-theoretic properties and a proof that they are conjugate degradable. Connections between the Unruh channel and optical amplifiers are also discussed.

quant-ph↗

arXiv2010

Proceedings Sixth Workshop on Developments in Computational Models: Causality, Computation, and Physics

DCM 2010 provides a forum for ideas about new computing means and models, with a particular emphasis in 2010 on computational and causal models related to physics and biology. We believe that bringing together different approaches - in a community with the strong foundational background characteristic of FLoC - results in inspirational cross-boundary exchanges, and innovative further research. Day two of this pre-FLoC 2010 workshop is given over to physics and quantum related computation. The content of day one is more typical of previous DCM workshops - covering a full spectrum of topics related to the development of new computational models or new features for traditional computational models. DCM 2010 was designed to foster interactions, and provide a forum for presenting new ideas and work in progress. It is also intended to enable newcomers to learn about current research in this area.

quant-ph↗

arXiv2009

Classifying all mutually unbiased bases in Rel

Finding all the mutually unbiased bases in various dimensions is a problem of fundamental interest in quantum information theory and pure mathematics. The general problem formulated in finite-dimensional Hilbert spaces is open. In the categorical approach to quantum mechanics one can find examples of categories which behave ``like'' the category of finite-dimensional Hilbert spaces in various ways but are subtly different. One such category is the category of sets and relations, $\mathbf{Rel}$. One can formulate the concept of mutually unbiased bases here as well. In this note we classify all the mutually unbiased bases in this category by relating it to a standard question in combinatorics.

quant-ph↗

arXiv2009

Private information via the Unruh effect

In a relativistic theory of quantum information, the possible presence of horizons is a complicating feature placing restrictions on the transmission and retrieval of information. We consider two inertial participants communicating via a noiseless qubit channel in the presence of a uniformly accelerated eavesdropper. Owing to the Unruh effect, the eavesdropper's view of any encoded information is noisy, a feature the two inertial participants can exploit to achieve perfectly secure quantum communication. We show that the associated private quantum capacity is equal to the entanglement-assisted quantum capacity for the channel to the eavesdropper's environment, which we evaluate for all accelerations.

quant-ph↗

arXiv2007

The Measurement Calculus

Measurement-based quantum computation has emerged from the physics community as a new approach to quantum computation where the notion of measurement is the main driving force of computation. This is in contrast with the more traditional circuit model which is based on unitary operations. Among measurement-based quantum computation methods, the recently introduced one-way quantum computer stands out as fundamental. We develop a rigorous mathematical model underlying the one-way quantum computer and present a concrete syntax and operational semantics for programs, which we call patterns, and an algebra of these patterns derived from a denotational semantics. More importantly, we present a calculus for reasoning locally and compositionally about these patterns. We present a rewrite theory and prove a general standardization theorem which allows all patterns to be put in a semantically equivalent standard form. Standardization has far-reaching consequences: a new physical architecture based on performing all the entanglement in the beginning, parallelization by exposing the dependency structure of measurements and expressiveness theorems. Furthermore we formalize several other measurement-based models: Teleportation, Phase and Pauli models and present compositional embeddings of them into and from the one-way model. This allows us to transfer all the theory we develop for the one-way model to these models. This shows that the framework we have developed has a general impact on measurement-based computation and is not just particular to the one-way quantum computer.

quant-ph↗

arXiv2006

Approximate reasoning for real-time probabilistic processes

We develop a pseudo-metric analogue of bisimulation for generalized semi-Markov processes. The kernel of this pseudo-metric corresponds to bisimulation; thus we have extended bisimulation for continuous-time probabilistic processes to a much broader class of distributions than exponential distributions. This pseudo-metric gives a useful handle on approximate reasoning in the presence of numerical information -- such as probabilities and time -- in the model. We give a fixed point characterization of the pseudo-metric. This makes available coinductive reasoning principles for reasoning about distances. We demonstrate that our approach is insensitive to potentially ad hoc articulations of distance by showing that it is intrinsic to an underlying uniformity. We provide a logical characterization of this uniformity using a real-valued modal logic. We show that several quantitative properties of interest are continuous with respect to the pseudo-metric. Thus, if two processes are metrically close, then observable quantitative properties of interest are indeed close.

cs.LO↗

arXiv2006

The Computational Power of the W and GHZ states

It is well understood that the use of quantum entanglement significantly enhances the computational power of systems. Much of the attention has focused on Bell states and their multipartite generalizations. However, in the multipartite case it is known that there are several inequivalent classes of states, such as those represented by the W-state and the GHZ-state. Our main contribution is a demonstration of the special computational power of these states in the context of paradigmatic problems from classical distributed computing. Concretely, we show that the W-state is the only pure state that can be used to exactly solve the problem of leader election in anonymous quantum networks. Similarly we show that the GHZ-state is the only one that can be used to solve the problem of distributed consensus when no classical post-processing is considered. These results generalize to a family of W- and GHZ-like states. At the heart of the proofs of these impossibility results lie symmetry arguments.

quant-ph↗

arXiv2006

Quantum Weakest Preconditions

We develop a notion of predicate transformer and, in particular, the weakest precondition, appropriate for quantum computation. We show that there is a Stone-type duality between the usual state-transformer semantics and the weakest precondition semantics. Rather than trying to reduce quantum computation to probabilistic programming we develop a notion that is directly taken from concepts used in quantum computation. The proof that weakest preconditions exist for completely positive maps follows immediately from the Kraus representation theorem. As an example we give the semantics of Selinger's language in terms of our weakest preconditions. We also cover some specific situations and exhibit an interesting link with stabilizers.

quant-ph↗

arXiv2006

Reasoning about quantum knowledge

We construct a formal framework for investigating epistemic and temporal notions in the context of distributed quantum computation. While we rely on structures developed earlier, we stress that our notion of quantum knowledge makes sense more generally in any agent-based model for distributed quantum systems. Several arguments are given to support our view that an agent's possibility relation should not be based on the reduced density matrix, but rather on local classical states and local quantum operations. In this way, we are able to analyse distributed primitives such as superdense coding and teleportation, obtaining interesting conclusions as to how the knowledge of individual agents evolves. We show explicitly that the knowledge transfer in teleportation is essentially classical, in that eventually, the receiving agent knows that its state is equal to the initial state of the sender. The relevant epistemic statements for teleportation deal with this correlation rather than with the actual quantum state, which is unknown throughout the protocol.

quant-ph↗

arXiv2005

Distributed measurement-based quantum computation

We develop a formal model for distributed measurement-based quantum computations, adopting an agent-based view, such that computations are described locally where possible. Because the network quantum state is in general entangled, we need to model it as a global structure, reminiscent of global memory in classical agent systems. Local quantum computations are described as measurement patterns. Since measurement-based quantum computation is inherently distributed, this allows us to extend naturally several concepts of the measurement calculus, a formal model for such computations. Our goal is to define an assembly language, i.e. we assume that computations are well-defined and we do not concern ourselves with verification techniques. The operational semantics for systems of agents is given by a probabilistic transition system, and we define operational equivalence in a way that it corresponds to the notion of bisimilarity. With this in place, we prove that teleportation is bisimilar to a direct quantum channel, and this also within the context of larger networks.

quant-ph↗

arXiv2004

The Measurement Calculus

We propose a calculus of local equations over one-way computing patterns, which preserves interpretations, and allows the rewriting of any pattern to a standard form where entanglement is done first, then measurements, then local corrections. We infer from this that patterns with no dependencies, or using only Pauli measurements, can only realise unitaries belonging to the Clifford group.

quant-ph↗

arXiv2004

Robust and parsimonious realisations of unitaries in the one-way model

We present a new set of generators for unitary maps over \otimes^n(C^2) which differs from the traditional rotation-based generating set in that it uses a single-parameter family of 1-qubit unitaries J(a), together with a single 2-qubit unitary controlled-Z. Each generator is implementable in the one-way model using only two qubits, and this leads to both parsimonious and robust implementations of general unitaries. As an illustration, we give an implementation of the controlled-U family which uses only 14 qubits, and has a 2-colourable underlying entanglement graph (known to yield robust entangled states).

quant-ph↗

arXiv2004

Spacetime topology from causality

We prove that a globally hyperbolic spacetime with its causality relation is a bicontinuous poset whose interval topology is the manifold topology. This provides an abstract mathematical setting in which one can study causality independent of geometry and differentiable structure.

gr-qc↗