Blind encoding into qudits

We consider the problem of encoding classical information into unknown qudit states belonging to any basis, of a maximal set of mutually unbiased bases, by one party and then decoding by another party who has perfect knowledge of the basis. Working with qudits of prime dimensions, we point out a no-go theorem that forbids ‘shift’ operations on arbitrary unknown states. We then provide the necessary conditions for reliable encoding/decoding.     

Randomized Entangled Mixed States from Phase States

We construct randomized entangled mixed states by using the formalism of phase states for d-dimensional systems (qudits). The randomized entangled mixed states are a special kind of mixed states that exhibit genuine multipartite correlation. Such states are obtained by the application of randomized entangling operators to an arbitrary pair of qudits of a multiqudit system. The study of the entanglement of randomized mixed states is of great importance in quantum computation since any experimental implementation of entangled states in a realistic environment can be made by imperfect entangling gates. We give a brief review of some necessary background about unitary phase operators and phase states of a multi-qudit system. Evolved density matrices arise when qudits of the multi-qudit system interact via a Hamiltonian of Heisenberg type. The randomized entangled states associated with evolved density matrices are derived via the action of an entangling operator on a pair of two qudits {i, j} of the multi-qudit system with some probability p. The randomized entangled mixed states for bipartite, tripartite and multipartite systems are explicitly expressed and their Kraus decomposition properties are discussed.

     

Geometrical derivation of a new ground state formula for the n-electron Friedel resonance model

The n-electron ground state of the Friedel resonance model can be written as a single Slater determinant of n s-electrons plus d-electron-s-hole companion. This new formula is derived geometrically in the Hilbert space. The derivation uses the fact that a n-electron Slater determinant, built from N band states, corresponds to a n-dimensional subspace in the N-dimensional Hilbert space. Received: 4 November 1997 / Accepted: 19 November 1997     

Group-theoretical approach to the construction of bases in 2<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>-dimensional Hilbert space

We propose a systematic procedure to construct all the possible bases with definite factorization structure in 2 ⁿ -dimensional Hilbert space and discuss an algorithm for the determination of basis separability. The results are applied for classification of bases for an n-qubit system.     

Bases for qudits from a nonstandard approach to <Emphasis Type="Italic">SU</Emphasis>(2)

Bases of finite-dimensional Hilbert spaces (in dimension d) of relevance for quantum information and quantum computation are constructed from angular momentum theory and su(2) Lie algebraic methods. We report on a formula for deriving in one step the (1 + p)p qupits (i.e., qudits with d = p a prime integer) of a complete set of 1 + p mutually unbiased bases in C ^p . Repeated application of the formula can be used for generating mutually unbiased bases in C ^d with d = p ^e (e ≥ 2) a power of a prime integer. A connection between mutually unbiased bases and the unitary group SU(d) is briefly discussed in the case d = p ^e .     

Entanglement swapping with pure orbital angular momentum Bell-state analysis

We investigate a framework of an orbital angular momentum (OAM) entanglement swapping in a multi-dimensional Hilbert space with the spin angular momentum (SAM)-based orbital angular momentum (OAM) Bell-sate analysis. By the implementations of entanglement swapping with the SAM and OAM Bell-state measurements subsequently, the OAM entanglement states (qudits) are generated and then transferred between photons in multi-dimensional Hilbert space in a point-to-point fashion. In the proposed scheme, two pairs of the SAM-based OAM hybrid entanglement photons are deployed to conduct the successive SAM and OAM Bell-state measurements. It provides an alternative technique to transfer pure OAM Bell-states in qudits, which illustrates a possible experimental approach for devising a full repeater in a complex quantum computation network where entanglement swapping serves as a critical constituent.     

Standard form of qudit stabilizer groups

We investigate stabilizer codes with carrier qudits of equal dimension D, an arbitrary integer greater than 1. We prove that there is a direct relation between the dimension of a qudit stabilizer code and the size of its corresponding stabilizer, and this implies that the code and its stabilizer are dual to each other. We also show that any qudit stabilizer can be put in a canonical, or standard, form using a series of Clifford gates, and we provide an explicit efficient algorithm for doing this. Our work generalizes known results that were valid only for prime dimensional systems and may be useful in constructing efficient encoding/decoding quantum circuits for qudit stabilizer codes and better qudit quantum error correcting codes.     

Riesz Transforms on Deformed Fock Spaces

The Fock Von Neumann algebra , equipped with its canonical trace τ, is spanned by n hermitian operators acting on a Hilbert Fock space some commutation relations between and are defined by the n×n hermitian matrix A. We define a Riesz transform , where is the number operator, ∇^′ is aninner derivation (unbounded in general) and . Let 1<p<∞. We prove that is equivalent to for every with null trace, with constants which do not depend on n. Received: 24 November 1998 / Accepted: 2 March 1999     

Distillation protocols for mixed states of multilevel qubits and the quantum renormalization group

We study several properties of distillation protocols to purify multilevel qubit states (qudits) when applied to a certain family of initial mixed bipartite states. We find that it is possible to use qudits states to increase the stability region obtained with the flow equations to distill qubits. In particular, for qutrits we get the phase diagram of the distillation process with a rich structure of fixed points. We investigate the large-D limit of qudits protocols and find an analytical solution in the continuum limit. The general solution of the distillation recursion relations is presented in an appendix. We stress the notion of weight amplification for distillation protocols as opposed to the quantum amplitude amplification that appears in the Grover algorithm. Likewise, we investigate the relations between quantum distillation and quantum renormalization processes.Received: 23 April 2003, Published online: 12 August 2003PACS: 03.67.-a Quantum information - 03.67.Lx Quantum computation     

Unbounded Violations of Bipartite Bell Inequalities via Operator Space Theory

In this work we show that bipartite quantum states with local Hilbert space dimension n can violate a Bell inequality by a factor of order ${{\rm \Omega} \left(\frac{\sqrt{n}}{\log^2n} \right)}In this work we show that bipartite quantum states with local Hilbert space dimension n can violate a Bell inequality by a factor of order W (\frac?nlog²n ){{\rm \Omega} \left(\frac{\sqrt{n}}{\log^2n} \right)} when observables with n possible outcomes are used. A central tool in the analysis is a close relation between this problem and operator space theory and, in particular, the very recent noncommutative L _p embedding theory.     

Distributions on Minkowski space and their connection with analytic representations of the conformal group

Unitary analytic representations of the conformal group are relized on Hilbert spaces of holomoprhic or antiholomorphic functions over a tube domain in complex Minkowski space. The distributional boundary values of these functions are tempered distributions on real Minkowski space. The representations are characterized by an integral scale dimension labeln and two spin labelsj ₁ andj ₂. The connection between the dimensionn and the degree of singularity of the tempered distribution is investigated. We propose an application to inclusive reactions of elementary particles.     

Representation of solutions of the Burgers equation using functional integrals

In the paper, a representation of a solution of the Burgers equation in ℝ ⁿ is obtained by using integrals with respect to the Wiener measure on the space of trajectories in ℝ ⁿ . The Burgers equation is considered in a rigged Hilbert space. It is proved that, in the infinite-dimensional case, there is an analog of the Cole-Hopf transformation relating the Burgers equation and an analog of the heat equation with respect to measures. The Feynman-Kac formula for the heat equation (with potential) with respect to measures in a rigged Hilbert space is obtained.     

Entanglement witnesses for d ⊗ d systems and new classes of entangled qudit states

We provide a new class of entanglement witnesses for d ⊗ d systems (two qudits). Our construction generalizes the one proposed recently by Jafarizadeh et al. for d = 3 and d = 4 on the basis of semidefinite linear programming. Moreover, we provide a new class of PPT entangled states detected by our witnesses which generalizes well known family of states constructed by Horodecki et al. for d = 3.     

Security of quantum key distribution using d-level systems

We consider two quantum cryptographic schemes relying on encoding the key into qudits, i.e., quantum states in a d-dimensional Hilbert space. The first cryptosystem uses two mutually unbiased bases (thereby extending the BB84 scheme), while the second exploits all d+1 available such bases (extending the six-state protocol for qubits). We derive the information gained by a potential eavesdropper applying a cloning-based individual attack, along with an upper bound on the error rate that ensures unconditional security against coherent attacks.     

The cauchy problem for non-linear Klein-Gordon equations

We consider in ⁿ⁺¹,n2, the non-linear Klein-Gordon equation. We prove for such an equation that there is a neighbourhood of zero in a Hilbert space of initial conditions for which the Cauchy problem has global solutions and on which there is asymptotic completeness. The inverse of the wave operator linearizes the non-linear equation. If, moreover, the equation is manifestly Poincaré covariant then the non-linear representation of the Poincaré Lie algebra, associated with the non-linear Klein-Gordon equation is integrated to a non-linear representation of the Poincaré group on an invariant neighbourhood of zero in the Hilbert space. This representation is linearized by the inverse of the wave operator. The Hilbert space is, in both cases, the closure of the space of the differentiable vectors for the linear representation of the Poincaré group, associated with the Klein-Gordon equation, with respect to a norm defined by the representation of the enveloping algebra.     

Equations, State, and Lattices of Infinite-Dimensional Hilbert Spaces

We provide several new result on quantum state space, on the lattice of subspacesof an infinite-dimensional Hilbert space, and on infinite-dimensional Hilbert spaceequations as well as on connections between them. In particular, we obtainan n-variable generalized orthoarguesian equation which holds in anyinfinite-dimensional Hilbert space. Then we strengthen Godowski's equationsas well ass the orthomodularity hold. We also prove that all six- and four-variableorthoarguesian equation presented in the literature can be reduced to newfour- and three-variable ones, respectively, and that Mayet's examples follow fromGodowski's equations. To make a breakthrough in testing these massive equations,we designed several novel algorithms for generating Greechie diagrams with anarbitrary number of blocks and atoms (currently testing with up to 50) and forautomated checking of equations on them. A way of obtaining complexinfinite-dimensional Hilbert space from the Hilbert lattice equipped with several additionalconditions and without invoking the notion of state is presented. Possiblerepercussions of the results on quantum computing problems are discussed.     

Quantization of bounded domains

We consider the quantization of a complex manifold endowed with the Bergman form following the ideas of Cahen, Gutt and Rawnsley. In particular we give a geometric interpretation for the quantization to be regular in terms of the Hilbert space of square integrable holomorphic n-forms on M and the Hilbert space of holomorphic n-forms on M bounded with respect to the Liouville element.     

Two-way and one-way quantum cryptography protocols

We present two quantum cryptography protocols. The first one generalizes the concept of the two-way deterministic protocol to work with qudits in prime d-dimensional system where d is odd. The second protocol makes use of the tomographically complete set construction for odd d-dimensional systems where d = p₁p₂ to modify the BB84 protocol to work with qudits of such systems. The securities of the two protocols are analyzed according to the intercept and resend attack.     

Hidden Measurements,Hidden Variables and the Volume Representation of Transition Probabilities

We construct, for any finite dimension n, a new hidden measurement model for quantum mechanics based on representing quantum transition probabilities by the volume of regions in projective Hilbert space. For n=2 our model is equivalent to the Aerts sphere model and serves as a generalization of it for dimensions n . 3 We also show how to construct a hidden variables scheme based on hidden measurements and we discuss how joint distributions arise in our hidden variables scheme and their relationship with the results of Fine [J. Math. Phys. 23 1306 (1982)].