An alternative approach for deriving a positive P-representation

Jarosław K. Korbicz, J. I. Cirac, Jan Wehr +1 more

A state of a quantum system can be regarded as classical (quantum) with respect to measurements of a set of canonical observables if and only if there exists (does not exist) a well defined, positive phase-space distribution, the so called Glauber-Sudarshan P representation. We derive a family of classicality criteria that requires that the averages of positive functions calculated using P representation must be positive. For polynomial functions, these criteria are related to Hilbert's 17th problem, and have physical meaning of generalized squeezing conditions; alternatively, they may be interpreted as nonclassicality witnesses. We show that every generic nonclassical state can be detected by a polynomial that is a sum-of-squares of other polynomials. We introduce a very natural hierarchy of states regarding their degree of quantumness, which we relate to the minimal degree of a sum-of-squares polynomial that detects them.

Peter Leyland

Crispin W. Gardiner, P. D. Drummond

Luis F. Barragán-Gil, R. Walser

A first-order partial differential equation is derived whose solution enables us to find straightforwardly the off-diagonal matrix elements in the position representation of the harmonic oscillator density operator. This approach constitutes an alternative to techniques that require advanced knowledge of mathematical and quantum mechanical results.

Hong-Yi Fan, Yue Fan, Junhua Chen

Based on the newly constructed overcompleteness relation of the nonlinear coherent states (NLCSs) and corresponding 𝒫-representation we derive: (1) a new antinormally ordered expansion of density matrices; and (2) the corresponding positive 𝒫-representation. Our derivation employs the technique of integration within an ordered product of nonlinear Bose operator functions. A relationship between the positive 𝒫-representation and the generalized characteristic function in the context of NLCSs is also obtained.