C Muñoz

Discrete coherent and squeezed states of many-qudit systems

We consider the phase space for n identical qudits (each one of dimension d, with d a primer number) as a grid of d{sup n}xd{sup n} points and use the finite Galois field GF(d{sup n}) to label the corresponding axes. The associated displacement operators permit to define s-parametrized quasidistributions on this grid, with properties analogous to their continuous counterparts. These displacements allow also for the construction of finite coherent states, once a fiducial state is fixed. We take this reference as one eigenstate of the discrete Fourier transform and study the factorization properties of the resulting coherent states. We extend these ideas to include discrete squeezed states, and show their intriguing relation with entangled states of different qudits.

Coherent, isotropic and squeezed states in an N-qubit system

We discuss how to construct a sensible set of discrete coherent states for an N-qubit system and how to use them as a tool for the analysis and representation of states in the discrete phase space. Considering N qubits as a single system of dimension 2N, we analyze the concept of squeezing as a reduction of isotropic fluctuations below the minimum achievable level for factorized (but non-necessary symmetric under permutations) qubit states.

Symmetric discrete coherent states for n-qubits

We put forward a method of constructing discrete coherent states for n qubits. After establishing appropriate displacement operators, the coherent states appear as displaced versions of a fiducial vector that is fixed by imposing a number of natural symmetry requirements on its Q-function. Using these coherent states, we establish a partial order in the discrete phase space, which allows us to picture some n-qubit states as apparent distributions. We also analyze correlations in terms of sums of squared Q-functions.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.

SHARING OF D-DIMENSIONAL QUANTUM STATES

We propose a scheme for the deterministic sharing arbitrary qudit states among three distant parties and characterize the set of ideal quantum channels. We also show that the use of non-ideal quantum channels for quantum state sharing can be related to the problem of quantum state discrimination. This allows us to formulate a protocol which leads to perfect quantum state sharing with a finite success probability.

Discrete coherent states for n qubits

Discrete coherent states for a system of n qubits are introduced in terms of eigenstates of the finite Fourier transform. The properties of these states are pictured in phase space by resorting to the discrete Wigner function.

Graph states in phase space

The phase space for a system of n qubits is a discrete grid of 2n ? 2n points, whose axes are labeled in terms of the elements of the finite field to endow it with proper geometrical properties. We analyze the representation of graph states in that phase space, showing that these states can be identified with a class of nonsingular curves. We provide an algebraic representation of the most relevant quantum operations acting on these states and discuss the advantages of this approach.

Isotropic and squeezed fluctuations in an n-qubit system

We show that 2n discrete coherent states of an n-qubit system, generated by application of the discrete displacement operators to a symmetric fiducial state, have isotropic fluctuations, with n/2 ⩽ 〈ΔS2〉 ⩽ n, in a specific tangent plane, which in general is not orthogonal to the mean spin direction. This allows us to use them as reference states to define a discrete squeezing for non-symmetric n-qubit states. Examples of states with reduced fluctuations, obtained after application of XOR gates to correlate (partially entangle) qubits, are analyzed.

Discrete phase-space structures and Wigner functions for N qubits

We further elaborate on a phase-space picture for a system of N qubits and explore the structures compatible with the notion of unbiasedness. These consist of bundles of discrete curves satisfying certain additional properties and different entanglement properties. We discuss the construction of discrete covariant Wigner functions for these bundles and provide several illuminating examples.