Heisenberg-Weyl Observables: Bloch vectors in phase space

Abstract

We introduce a Hermitian generalization of Pauli matrices to higher dimensions which is based on Heisenberg-Weyl operators. The complete set of Heisenberg-Weyl observables allows us to identify a real-valued Bloch vector for an arbitrary density operator in discrete phase space, with a smooth transition to infinite dimensions. Furthermore, we derive bounds on the sum of expectation values of any set of anti-commuting observables. Such bounds can be used in entanglement detection and we show that Heisenberg-Weyl observables provide a first non-trivial example beyond the dichotomic case.PACS numbers: 07.10. Cm, 03.65.Ta, 03.65.Ud Introduction. The Bloch representation is a cornerstone of analyzing the characteristics of quantum systems. It was first introduced for two-level systems by Bloch [1] and has since been used in a wide variety of settings (for comprehensive reviews consult [2][3][4]). It is usually defined via a decomposition of the density matrix into a complete operator basis. Defining quantum states via the expectation values of a complete set of measurements gives a very practical account of their properties. Apart from being intuitive this approach gives convenient solutions for Hamiltonian evolutions (see, e.g., [5]) and has found many applications in entanglement theory [6][7][8][9][10][11][12].However, there is not a unique Bloch decomposition of a given quantum state; this fact may favor a particular representation over another for certain tasks. The canonical choice for a complete basis of observables is usually given by the so-called generalized Gell-Mann matrices, generators of the special unitary group [SU(d)]. Being a natural choice for higher-dimensional spin representations they have been extensively used in parametrizations of corresponding density matrices [4,13] and in entanglement detection. Other choices, such as the HeisenbergWeyl (HW) operators, and the non-Hermitian generalization of the 1/2-spin Pauli operators, have also been explored [3,[14][15][16][17][18]. While having some convenient properties, they are unitary, but not Hermitian matrices. Thus the associated Bloch vector itself has imaginary entries that do not correspond to expectation values of physical observables. This makes both the theoretical description and experimental realization more cumbersome and therefore requires more effort to identify the relevant parameters.In this article we introduce a Hermitian Bloch-basis derived from HW-operators. It conveniently combines multiple desirable properties of Bloch vector parametrizations, allowing a smooth transition to the infinite dimensional limit. We first explore properties such as (anti-) commutativity. We then proceed with the derivation of an inequality which bounds sums of anti-commuting observables with which we show in exemplary cases how one can construct powerful criteria for entanglement detection in this new basis. Finally we present a scheme for practical experimental acquisition through a Ramseytype measurement.Phase-space displacements. We start our a...

History states of systems and operators

“…a fourth order expansion of (35) and (41) leads to (46) is verified to be non-negative and of fourth order in t f , being proportional to the fluctuation of (∆H) 2 . The latter vanishes just for the geodesic evolution, where ∆H = ∆H min = φ t f σ y and hence (∆H min ) 4 = (∆H min ) 2 2 = φ 4 /t 4 f , implying E 2 (S, T ) = E min 2 (S, T ).…”

“…Eq. (36) provides a good approximation to (35) if (36) are essentially measures of the spread of |S 0 over distinct energy eigenstates. For small t f such that |E k − E k ′ |t f ≪ 1 ∀ k, k ′ , a second order expansion shows they are proportional to the energy fluctuation in…”

Entanglement certification from theory to experiment

“…Determining the coefficients of the Bloch decomposition requires the measurement of all d 2 − 1 local Bloch vector elements in every possible combination. Two anti-commuting operators, however, are already sufficient for constructing entanglement witnesses in bipartite [119] and multipartite systems [120,121]. Post-selecting coincidence counts in a single-outcome scenario (i.e., filtering) requires every possible projection on a tomographically complete set of states, i.e., d n measurement settings per single basis ( * * * although it is possible to detect entanglement without knowing any density matrix element [122].…”

Monogamy of correlations and entropy inequalities in the Bloch picture

“…In large dimensions there is some freedom in the choice of l i with the canonical choices being either the generalized Gell-Mann [30,37] matrices or the (non-hermitian) Heisenberg-Weyl [38] matrices, alongside more unusual choices like the Heisenberg-Weyl observables [39]. The single party case can be naturally extended to multi-partite systems by a tensor product construction.…”