A View on Decoherence Via Master Equations

This paper addresses the discussion on probabilistic features of the concept of decoherence as it appeared in quantum physics. Given a Lindblad-type generator of an open system dynamics, we derive applicable criteria to characterize decoherent behaviour.Presented at the International Conference “Entanglement, Information & Noise”, Krzyżowa, Poland, June 14-20, 2004.Partially supported by FONDECYT grant 1030552.     

Description of Quantum Dynamics of Open Systems Based on Collision-Like Models

Master equations in the Lindblad form describe evolution of open quantum systems that are completely positive and simultaneously have a semigroup property. We analyze the possibility to derive this type of master equations from an intrinsically discrete dynamics that is modelled as a sequence of collisions between a given quantum system (a qubit) with particles that form the environment. In order to illustrate our approach we analyze in detail how the process of an exponential decay and the process of decoherence can be derived from a collision-like model in which particular collisions are described by SWAP and controlled-NOT interactions, respectively.Presented at the 36th Symposium on Mathematical Physics, “Open Systems & Quantum Information”, Toruń, Poland, June 9-12, 2004.     

Entanglement,Purity, and Information Entropies in Continuous Variable Systems

Quantum entanglement of pure states of a bipartite system is defined as the amount of local or marginal (i.e. referring to the subsystems) entropy. For mixed states this identification vanishes, since the global loss of information about the state makes it impossible to distinguish between quantum and classical correlations. Here we show how the joint knowledge of the global and marginal degrees of information of a quantum state, quantified by the purities or, in general, by information entropies, provides an accurate characterization of its entanglement. In particu-lar, for Gaussian states of continuous variable systems, we classify the entanglement of two-mode states according to their degree of total and partial mixedness, comparing the different roles played by the purity and the generalized p-entropies in quantifying the mixedness and bounding the entanglement. We prove the existence of strict upper and lower bounds on the entanglement and the existence of extremally (maximally and minimally) entangled states at fixed global and marginal degrees of information. This results allow for a powerful, operative method to measure mixed-state entanglement without the full tomographic reconstruction of the state. Finally, we briefly discuss the ongoing extension of our analysis to the quantification of multipartite entan-glement in highly symmetric Gaussian states of arbitrary 1 × N-mode partitions.Presented at the International Conference “Entanglement, Information & Noise”, Krzyżowa, Poland, June 14–20, 2004.     

Can quantum entanglement detection schemes improve search?

Quantum computation, in particular Grover’s algorithm, has aroused a great deal of interest since it allows for a quadratic speed-up to be obtained in search procedures. Classical search procedures for an N element database require at most O(N) time complexity. Grover’s algorithm is able to find a solution with high probability in ){O(\sqrt{N})} time through an amplitude amplification scheme. In this work we draw elements from both classical and quantum computation to develop an alternative search proposal based on quantum entanglement detection schemes. In 2002, Horodecki and Ekert proposed an efficient method for direct detection of quantum entanglement. Our proposition to quantum search combines quantum entanglement detection alongside entanglement inducing operators. The quantum search algorithm relies on measuring a quantum superposition after having applied a unitary evolution. We deviate from the standard method by focusing on fine-tuning a unitary operator in order to infer the solution with certainty. Our proposal sacrifices space for speed and depends on the mathematical properties of linear positive maps Λ which have not been operationally characterized. Whether such a Λ can be easily determined remains an open question.     

Spin-Based Quantum Information Processing in Magnetic Quantum Dots

We define the qubit as a pair of singlet and triplet states of two electrons in a He-type quantum dot (QD) placed in a diluted magnetic semiconductor (DMS) medium. The molecular field is here essential as it removes the degeneracy of the triplet state and strongly enhances the Zeeman splitting. Methods of qubit rotation as well as two-qubit operations are suggested. The system of a QD in a DMS is described in a way which allows an analysis of the decoherence due to spin waves in the DMS subsystem.on leave from Institute of Physics, Odessa UniversityPresented at the 36th Symposium on Mathematical Physics, “Open Systems & Quantum Information”, Toruń, Poland, June 9–12, 2004.     

On Reversible Combinatory Logic

The λ-calculus is destructive: its main computational mechanism – beta reduction – destroys the redex and makes it thus impossible to replay the computational steps. Recently, reversible computational models have been studied mainly in the context of quantum computation, as (without measurements) quantum physics is inherently reversible. However, reversibility also changes fundamentally the semantical framework in which classical computation has to be investigated. We describe an implementation of classical combinatory logic into a reversible calculus for which we present an algebraic model based on a generalisation of the notion of group.     

Quantum Erasure of Decoherence

We consider the classical algebra of observables that are diagonal in a given orthonormal basis, and define a complete decoherence process as a completely positive map that asymptotically converts any quantum observable into a diagonal one, while preserving the elements of the classical algebra. For quantum systems in dimension two and three any decoherence process can be undone by collecting classical information from the environment and using such an information to restore the initial system state. As a relevant example, we illustrate the quantum eraser of Scully et al. [Nature 351, 111 (1991)] as an example of environment-assisted correction, and present the generalization of the eraser setup for d-dimensional systems. Presented at the 38th Symposium on Mathematical Physics “Quantum Entanglement & Geometry”, Toruń, June 4–7, 2006.     

Non-Decomposable Quantum Dynamical Semigroups and Bound Entangled States

We use open quantum system techniques to construct one-parameter semigroups of positive maps and apply them to study the entanglement properties of a class of 16-dimensional density matrices, representing states of a 4 × 4 bipartite system.Presented at the International Conference “Entanglement, Information & Noise”, Krzyżowa, Poland, June 14–20, 2004.     

Fidelity of Gaussian Channels

A noisy Gaussian channel is defined as a channel in which an input field mode is subjected to random Gaussian displacements in phase space. We introduce the quantum fidelity of a Gaussian channel for pure and mixed input states, and we derive a universal scaling law of the fidelity for pure initial states. We also find the maximum fidelity of a Gaussian channel over all input states. Quantum cloning and continuous-variable teleportation are presented as physical examples of Gaussian channels to which the fidelity results can be applied.Presented at the 36th Symposium on Mathematical Physics, “Open Systems & Quantum Information”, Toruń, Poland, June 9–12, 2004.Supported in part by US O.ce of Naval Research Grant No. N00014–03–1–0426.Supported in part by KBN Grant No. PBZ–Min–008/P03/03     

Planning as incremental adaptation of a reactive system

The importance of solving the problem of integrating deliberative (“planning”) capabilities and reactive capabilities when building robust, ‘real-world’ robot systems is becoming widely accepted (Bresina and Drummond, 1990; Fraichard and Laugier, 1991; McDermott, 1991). This paper presents a solution to this problem: cast planning as the incremental adaptation of a reactive system to suit changes in goals or the environment. Our application domain is a manufacturing problem - robotic kitting. This paper represents an advance on existing work in two ways: It presents and formally examines an architecture that incorporates the benefits of a deliberative component without compromising the reactive component. Secondly, it provides the first set of performance statistics in the literature for this class of system. In our approach, the reactive system (the reactor) is a real-time system that continually interacts with the environment, and the planner is a separate and concurrent system that incrementally ‘tunes’ the behavior of the reactor to ensure that goals are achieved. We call this the planner-reactor approach. The reactor is described using a formal framework for representing flexible robot plans, the model (Lyons, 1990; Lyons and Arbib, 1989). Thus, the behavior of the reactor, and the rules by which the reactor can be modified, become open to mathematical analysis. We employ this to determine the constraints the planner must abide by to make safe adaptations and to ensure that incremental adaptations converge to a desired reactor. We discuss our current implementation of planner and reactor, work through an example from the kitting robot application, and present implementation results.     

Expressing Priorities and External Probabilities in Process Algebra via Mixed Open/Closed Systems

Defining operational semantics for a process algebra is often based either on labeled transition systems that account for interaction with a context or on the so-called reduction semantics: we assume to have a representation of the whole system and we compute unlabeled reduction transitions (leading to a distribution over states in the probabilistic case). In this paper we consider mixed models with states where the system is still open (towards interaction with a context) and states where the system is already closed. The idea is that (open) parts of a system “P” can be closed via an operator “P↑G” that turns already synchronized actions whose “handle” is specified inside “G” into prioritized reduction transitions (and, therefore, states performing them into closed states). We show that we can use the operator “P↑G” to express multi-level priorities and external probabilistic choices (by assigning weights to handles inside G), and that, by considering reduction transitions as the only unobservable τ transitions, the proposed technique is compatible, for process algebra with general recursion, with both standard (probabilistic) observational congruence and a notion of equivalence which aggregates reduction transitions in a (much more aggregating) trace based manner. We also observe that the trace-based aggregated transition system can be obtained directly in operational semantics and we present the “aggregating” semantics. Finally, we discuss how the open/closed approach can be used to also express discrete and continuous (exponential probabilistic) time and we show that, in such timed contexts, the trace-based equivalence can aggregate more with respect to traditional lumping based equivalences over Markov Chains.     

Dimension elevation formula for Chebyshevian blossoms

A given polynomial of degree less than or equal to n naturally “blossoms” into a function of n variables called its blossom. Considered as a polynomial function of degree less than or equal to (n+1) it “blossoms” into a “new” blossom which is now a function of (n+1) variables. A classical formula expresses any value of this new blossom as a strictly convex combination of (n+1) values of the initial one. We establish a similar formula for Chebyshevian blossoms.     

A Symmetric Lambda Calculus for Classical Program Extraction

We introduce aλ-calculus with symmetric reduction rules and “classical” types, i.e., types corresponding to formulas of classical propositional logic. The strong normalization property is proved to hold for such a calculus, as well as for its extension to a system equivalent to Peano arithmetic. A theorem on the shape of terms in normal form is also proved, making it possible to get recursive functions out of proofs ofΠ⁰₂formulas, i.e., those corresponding to program specifications.     

On the computational power of probabilistic and quantum branching program

In this paper, we show that one-qubit polynomial time computations are as powerful as NC¹ circuits. More generally, we define syntactic models for quantum and stochastic branching programs of bounded width and prove upper and lower bounds on their power. We show that any NC¹ language can be accepted exactly by a width-2 quantum branching program of polynomial length, in contrast to the classical case where width 5 is necessary unless NC¹ = ACC. This separates width-2 quantum programs from width-2 doubly stochastic programs as we show the latter cannot compute the middle bit of multiplication. Finally, we show that bounded-width quantum and stochastic programs can be simulated by classical programs of larger but bounded width, and thus are in NC¹. For read-once quantum branching programs (QBPs), we give a symmetric Boolean function which is computable by a read-once QBP with O (log n) width, but not by a deterministic read-once BP with o (n) width, or by a classical randomized read-once BP with o (n) width which is “stable” in the sense that its transitions depend on the value of the queried variable but do not vary from step to step. Finally, we present a general lower bound on the width of read-once QBPs, showing that our O (log n) upper bound for this symmetric function is almost tight.     

A Simple Proof of the Jamiołkowski Criterion for Complete Positivity of Linear Maps

We give a simple direct proof of the Jamiołkowski criterion to check whether a linear map between matrix algebras is completely positive or not. This proof is more accessible for physicists than other ones found in the literature and provides a systematic method to give any set of Kraus matrices of the Kraus decomposition.Presented at the 36th Symposium on Mathematical Physics, “Open Systems & Quantum Information”, Toruń, Poland, June 9-12, 2004.     

Efficiency in Quantum Key Distribution Protocols with Entangled Gaussian States

Quantum key distribution (QKD) refers to specific quantum strategies which permit the secure distribution of a secret key between two parties that wish to communicate secretly. Quantum cryptography has proven unconditionally secure in ideal scenarios and has been successfully implemented using quantum states with finite (discrete) as well as infinite (continuous) degrees of freedom. Here, we analyze the efficiency of QKD protocols that use as a resource entangled gaussian states and gaussian operations only. In this framework, it has already been shown that QKD is possible [1] but the issue of its efficiency has not been considered. We propose a figure of merit (the efficiency E) to quantify the number of classical correlated bits that can be used to distill a key from a sample of N entangled states. We relate the efficiency of the protocol to the entanglement and purity of the states shared between the parties. Presented at the 38th Symposium on Mathematical Physics “Quantum Entanglement & Geometry”, Toruń, June 4–7, 2006.