Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Mode displacements in multimode systems and applications to entanglement

Apostol Vourdas

Published Apr 26, 2005• 2 citations
Publisher

Abstract

A $d$-mode system is considered and the concepts of mode-position operator and mode-momentum operator are introduced. Their eigenvalues are integers in ${Z}_{d}$ and the mode phase space is ${Z}_{d}\ifmmode\times\else\texttimes\fi{}{Z}_{d}$. The exponentials of these operators are displacement operators in the mode phase space and they form a Heisenberg-Weyl group. The displacement operators perform a cyclic permutation of a state among the $d$ modes; and they also change its mode momentum (i.e., the rate with which this cyclic permutation takes place). In the case of entangled states the mode displacement operators permute the entanglement among the $d$ observers; and they also change the momentum with which the entanglement propagates.

DOI
10.1103/physreva.71.043821

Related papers

Mode displacements in multimode systems and applications to entanglement (11 pages)

Apostol Vourdas

Physical Review A•Jan 1, 2005•0 citations

Propagation of a quantum state in a continuum of coupled oscillators and applications to entanglement

Apostol Vourdas

The exponential Hilbert space approach is used to link quantum mechanics in a small system with d -dimensional Hilbert space , with the collective quantum behaviour in a large system comprised of d coupled oscillators with Hilbert space H . In the large system, the expectation value of the mode position operator describes the location of a quantum state within the chain of d oscillators, and the expectation value of the mode momentum operator describes the change of the mode position with time. A continuum of modes ( d → ∞ ) is also considered and in this case these operators obey the commutation relation . A consequence of this is that uncertainties in the location of a state and in its momentum obey an uncertainty relation. Displacements and squeezing in the mode phase space are discussed. For a certain Hamiltonian, the obey equations of motion which are very similar to those of a harmonic oscillator. If the system is in an entangled state, the formalism can be used to quantify concepts like the location of entanglement, and the speed with which the entanglement propagates.

Journal of Physics A Mathematical and General•Oct 26, 2005•0 citations

Modewise entanglement of Gaussian states

Alonso Botero, Benni Reznik

We address the decomposition of a multimode pure Gaussian state with respect to a bipartite division of the modes. For any such division the state can always be expressed as a product state involving entangled two-mode squeezed states and single-mode local states at each side. The character of entanglement of the state can therefore be understood modewise; that is, a given mode on one side is entangled with only one corresponding mode of the other, and therefore the total bipartite entanglement is the sum of the modewise entanglement. This decomposition is generally not applicable to all mixed Gaussian states. However, the result can be extended to a special family of ``isotropic'' states, characterized by a phase space covariance matrix with a completely degenerate symplectic spectrum.

arXiv:quant-ph/0209026•Physical Review A•May 27, 2003•103 citations

Scaling up multi-mode entanglement generated by mode swapping

Yuki Kodama, Holger F. Hofmann

Entanglement between two local multi-photon multi-mode systems can be generated by swapping a pair of modes between the two local multi-mode systems. In this presentation, we consider possible strategies for the efficient generation of entanglement between systems with three or more modes. It is shown that the choice of photon number inputs in the local systems adds a new degree of freedom to the non-local interference effects observed in the output photon number statistics.

quant-pharXiv:2609.15040•Sep 15, 2026

On the exponential form of the displacement operator for different systems

Václav Potoček, Stephen Mark Barnett

The family of displacement operators , a central concept in the theory of coherent states of a quantum mechanical harmonic oscillator, has been successfully generalized to systems of quantized, cyclic or finite position coordinates. However, out of the plethora of mutually equivalent expressions for the displacement operators valid in the continuous case, only few are directly applicable in the other systems of interest. The aim of this paper is to strengthen the analogy between the different cases by identifying the root cause of the issues accompanying the straightforward generalization of certain important expressions and, more importantly, offering alternative ones of general validity. Ultimately we arrive at an algorithm allowing one to express any displacement operator as an exponential of a pure imaginary multiple of a generalized 'quadrature' observable that is not obtained by a linear combination of position and momentum observables but rather by a shear transform of one of them in the system's phase space.

Physica Scripta•May 21, 2015•12 citations

Quantum systems with finite Hilbert space: Galois fields in quantum mechanics

Apostol Vourdas

A 'Galois quantum system' in which the position and momentum take values in the Galois field GF( p ℓ ) is considered. It is comprised of ℓ-component systems which are coupled in a particular way and is described by a certain class of Hamiltonians. Displacements in the GF( p ℓ ) × GF( p ℓ ) phase space and the corresponding Heisenberg–Weyl group are studied. Symplectic transformations are shown to form the Sp(2, GF( p ℓ )) group. Wigner and Weyl functions are defined and their properties are studied. Frobenius symmetries, which are based on Frobenius automorphisms in the theory of Galois fields, are a unique feature of these systems (for ℓ ⩾ 2). If they commute with the Hamiltonian, there are constants of motion which are discussed. An analytic representation in the ℓ-sheeted complex plane provides an elegant formalism that embodies the properties of Frobenius transformations. The difference between a Galois quantum system and other finite quantum systems where the position and momentum take values in the ring is discussed.

Journal of Physics A Mathematical and Theoretical•Aug 1, 2007•83 citations
A ddd-mode system is considered and the concepts of mode-position operator and mode-momentum operator are introduced. Their eigenvalues are integers in Zd{Z}_{d}Zd​ and the mode phase space is Zd\ifmmode×\else\texttimes\fiZd{Z}_{d}\ifmmode\times\else\texttimes\fi{}{Z}_{d}Zd​\ifmmode×\else\texttimes\fiZd​. The exponentials of these operators are displacement operators in the mode phase space and they form a Heisenberg-Weyl group. The displacement operators perform a cyclic permutation of a state among the ddd modes; and they also change its mode momentum (i.e., the rate with which this cyclic permutation takes place). In the case of entangled states the mode displacement operators permute the entanglement among the ddd observers; and they also change the momentum with which the entanglement propagates.
The exponential Hilbert space approach is used to link quantum mechanics in a small system with d -dimensional Hilbert space , with the collective quantum behaviour in a large system comprised of d coupled oscillators with Hilbert space H . In the large system, the expectation value of the mode position operator describes the location of a quantum state within the chain of d oscillators, and the expectation value of the mode momentum operator describes the change of the mode position with time. A continuum of modes ( d → ∞ ) is also considered and in this case these operators obey the commutation relation . A consequence of this is that uncertainties in the location of a state and in its momentum obey an uncertainty relation. Displacements and squeezing in the mode phase space are discussed. For a certain Hamiltonian, the obey equations of motion which are very similar to those of a harmonic oscillator. If the system is in an entangled state, the formalism can be used to quantify concepts like the location of entanglement, and the speed with which the entanglement propagates.
We address the decomposition of a multimode pure Gaussian state with respect to a bipartite division of the modes. For any such division the state can always be expressed as a product state involving entangled two-mode squeezed states and single-mode local states at each side. The character of entanglement of the state can therefore be understood modewise; that is, a given mode on one side is entangled with only one corresponding mode of the other, and therefore the total bipartite entanglement is the sum of the modewise entanglement. This decomposition is generally not applicable to all mixed Gaussian states. However, the result can be extended to a special family of ``isotropic'' states, characterized by a phase space covariance matrix with a completely degenerate symplectic spectrum.
The family of displacement operators , a central concept in the theory of coherent states of a quantum mechanical harmonic oscillator, has been successfully generalized to systems of quantized, cyclic or finite position coordinates. However, out of the plethora of mutually equivalent expressions for the displacement operators valid in the continuous case, only few are directly applicable in the other systems of interest. The aim of this paper is to strengthen the analogy between the different cases by identifying the root cause of the issues accompanying the straightforward generalization of certain important expressions and, more importantly, offering alternative ones of general validity. Ultimately we arrive at an algorithm allowing one to express any displacement operator as an exponential of a pure imaginary multiple of a generalized 'quadrature' observable that is not obtained by a linear combination of position and momentum observables but rather by a shear transform of one of them in the system's phase space.
A 'Galois quantum system' in which the position and momentum take values in the Galois field GF( p ℓ ) is considered. It is comprised of ℓ-component systems which are coupled in a particular way and is described by a certain class of Hamiltonians. Displacements in the GF( p ℓ ) × GF( p ℓ ) phase space and the corresponding Heisenberg–Weyl group are studied. Symplectic transformations are shown to form the Sp(2, GF( p ℓ )) group. Wigner and Weyl functions are defined and their properties are studied. Frobenius symmetries, which are based on Frobenius automorphisms in the theory of Galois fields, are a unique feature of these systems (for ℓ ⩾ 2). If they commute with the Hamiltonian, there are constants of motion which are discussed. An analytic representation in the ℓ-sheeted complex plane provides an elegant formalism that embodies the properties of Frobenius transformations. The difference between a Galois quantum system and other finite quantum systems where the position and momentum take values in the ring is discussed.