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Journal of Mathematical Physics

Consideration of the general state which minimizes the Nth order minimum uncertainty product

Robert Lynch, Harry A. Mavromatis

Published Feb 1, 1993• 5 citations
Publisher

Abstract

This paper considers the problem of whether mixed states can arise in the Nth order minimum uncertainty expression UN=<Ψ‖‖x‖N‖Ψ≳<Ψ‖‖p‖N‖Ψ≳. Using a method due to R. Jackiw [J. Math. Phys. 9, 339 (1968)], and finding the extremum of the appropriate functional, a set of equations is derived which must be satisfied by such a mixed state. It is easily shown that a sufficient condition that the equations have a solution is that the mixed state in fact reduces to a pure state. It does not appear possible to rigorously prove that such a reduction is necessary, although the authors present approximate calculations which make such a conclusion plausible. In the course of the approximate calculations generalized Nth order ‘‘squeezed state’’ values for UN are also obtained which are of some interest in and of themselves.

DOI
10.1063/1.530258

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This paper considers the problem of whether mixed states can arise in the Nth order minimum uncertainty expression UN=<Ψ‖‖x‖N‖Ψ≳<Ψ‖‖p‖N‖Ψ≳. Using a method due to R. Jackiw [J. Math. Phys. 9, 339 (1968)], and finding the extremum of the appropriate functional, a set of equations is derived which must be satisfied by such a mixed state. It is easily shown that a sufficient condition that the equations have a solution is that the mixed state in fact reduces to a pure state. It does not appear possible to rigorously prove that such a reduction is necessary, although the authors present approximate calculations which make such a conclusion plausible. In the course of the approximate calculations generalized Nth order ‘‘squeezed state’’ values for UN are also obtained which are of some interest in and of themselves.
This paper considers the problem of finding the quantum states that minimize the products of the (even) Nth-order fluctuation of two canonically conjugate operators. The problem is first attacked in an abstract form and an equation derived for the desired states. A consideration of the N=2 case then leads to a connection with the new concept of ‘‘squeezed states’’ of the electromagnetic field, and the usual exact solution involving the coherent state. The concept of ‘‘higher-order squeezing’’ is touched on to motivate the further discussion. The cases N=4, 6, and 8 are then taken up, and approximate solutions to them found via a new numerical technique. During these calculations, it is noted that only the first few terms in the expansion in number states figure in the solution; this observation is then exploited to find an approximate closed solution to the problem valid for all N.
The authors consider the problem of finding the quantum states that minimize the products of the 'absolute value' Nth-order fluctuation of two canonically conjugate operators. A scheme is applied to the problem, wherein the exact solution state is approximated by working in a subspace consisting of the ground state plus a finite number of higher-order states, using a variational technique. The solution is first obtained for the ground state, and this is subsequently compared to the solution assuming 2*2 and 3*3 spaces. For positive N they obtain a small, smooth decrease in the value of the fluctuation as the space increases and expect the 3*3 space results to be close to the exact Nth-order fluctuation. For negative N, however, they get an abrupt decrease in the fluctuation as the space increases.
In this article, we study the problem of comparing mixed quantum states: given nnn unknown mixed quantum states, can one determine whether they are identical or not with an unambiguous quantum measurement? We first study the universal comparison of mixed quantum states, and prove that this task is generally impossible to accomplish. Then, we focus on the unambiguous comparison of nnn mixed quantum states arbitrarily chosen from a set of kkk mixed quantum states. The condition for the existence of an unambiguous measurement operator which can produce a conclusive result when the unknown states are actually the same and the condition for the existence of an unambiguous measurement operator when the unknown states are actually different are studied independently. We derive a necessary and sufficient condition for the existence of the first measurement operator, and a necessary condition and two sufficient conditions for the second. Furthermore, we find that the sufficiency of the necessary condition for the second measurement operator has a simple and interesting dependence on nnn and kkk. At the end, a unified condition is obtained for the simultaneous existence of these two unambiguous measurement operators.
Hong and Mandel have recently suggested a scheme which allows generalization of ``squeezing'' to higher-order (even) moments of the electromagnetic field. The new criterion for squeezing is that the expectation value of the moment be less than that for the coherent state. In this paper it is explicitly shown that both electric-field quadrature components are squeezed to fourth order in a ``Jackiw state,'' thus proving that the coherent state does not minimize the fourth-order uncertainty product. The implications of this result for Hong and Mandel's proposal are briefly considered.
The latest results of density matrix theory are summarized in terms of analysis of off-diagonal components. For a given reduced density matrix (RDM) of arbitrary order, the full density matrix is explicitly constructed, from which the RDM is derived by means of the corresponding reduction procedure. Both the matrices satisfy all the necessary conditions, such antisymmetry, normalization, hermiticity, except that the non-negative definiteness of the full DM is not guaranteed. If an RDM satisfies the necessary inequalities with the same number of indices as it has itself, then, as proved here, there is an additive correction to the full DM matrix that turns it into a non-negative definite matrix leading to the same RDM and satisfying the other requirements. This establishes a fundamental fact that these inequalities are sufficient as well, providing the N-representability of the given RDM. The construction is based on the classification, introduced in the article, of the multi-index elements of the RDM and full DM according to their degree of off-diagonality. RDM elements with a predetermined degree of off-diagonality are expressed through full DM elements with the same off-diagonality. This shows that the conventional variational approach to calculating the energy of the system by convex programming methods under the conditions of N-representability (before and even after reaching the minimum energy) generally does not lead to constructing a mixed quantum state composed of solutions to the Schrödinger equation. Therefore, except for the energy, it does not guarantee a correct description of the other properties so long as the conditions of correctness for each particular property are not expressed in terms of the RDM and not incorporated into the specific conditions of N-representability (regardless of the requirement of energy minimization).