There are heuristic arguments proposing that the accuracy of monitoring position of a free mass $m$ is limited by the standard quantum limit (SQL): ${\ensuremath{\sigma}}^{2}(X(t))\ensuremath{\ge}{\ensuremath{\sigma}}^{2}(X(0))+({t}^{2}/{m}^{2}){\ensuremath{\sigma}}^{2}(P(0))\ensuremath{\ge}\ensuremath{\hbar}t/m$, where ${\ensuremath{\sigma}}^{2}(X(t))$ and ${\ensuremath{\sigma}}^{2}(P(t))$ denote variances of the Heisenberg representation position and momentum operators. Yuen [Phys. Rev. Lett. 51, 719 (1983)] discovered that there are contractive states for which this result is incorrect. Here I prove universally valid rigorous quantum limits (RQL), viz. rigorous upper and lower bounds on ${\ensuremath{\sigma}}^{2}(X(t))$ in terms of ${\ensuremath{\sigma}}^{2}(X(0))$ and ${\ensuremath{\sigma}}^{2}(P(0))$, given by Eq. (12) for a free mass and by Eq. (36) for an oscillator. I also obtain the maximally contractive and maximally expanding states which saturate the RQL, and use the contractive states to set up an Ozawa-type measurement theory with accuracies respecting the RQL but beating the standard quantum limit. The contractive states for oscillators improve on the Schr\"odinger coherent states of constant variance and may be useful for gravitational wave detection and optical communication.There are heuristic arguments proposing that the accuracy of monitoring position of a free mass $m$ is limited by the standard quantum limit (SQL): ${\ensuremath{\sigma}}^{2}(X(t))\ensuremath{\ge}{\ensuremath{\sigma}}^{2}(X(0))+({t}^{2}/{m}^{2}){\ensuremath{\sigma}}^{2}(P(0))\ensuremath{\ge}\ensuremath{\hbar}t/m$, where ${\ensuremath{\sigma}}^{2}(X(t))$ and ${\ensuremath{\sigma}}^{2}(P(t))$ denote variances of the Heisenberg representation position and momentum operators. Yuen [Phys. Rev. Lett. 51, 719 (1983)] discovered that there are contractive states for which this result is incorrect. Here I prove universally valid rigorous quantum limits (RQL), viz. rigorous upper and lower bounds on ${\ensuremath{\sigma}}^{2}(X(t))$ in terms of ${\ensuremath{\sigma}}^{2}(X(0))$ and ${\ensuremath{\sigma}}^{2}(P(0))$, given by Eq. (12) for a free mass and by Eq. (36) for an oscillator. I also obtain the maximally contractive and maximally expanding states which saturate the RQL, and use the contractive states to set up an Ozawa-type measurement theory with accuracies respecting the RQL but beating the standard quantum limit. The contractive states for oscillators improve on the Schr\"odinger coherent states of constant variance and may be useful for gravitational wave detection and optical communication.

arXiv:1711.10261Physical Review AMar 13, 20181 citation

Monitoring the states of single quantum systems.

Kevin Garapo

Jan 1, 20120 citations

Contractive States and the Standard Quantum Limit for Monitoring Free-Mass Positions

Horace P. Yuen

The familiar minimum-uncertainty wave packets for masses are generalized in analogy with the two-photon coherent states of the radiation field. The free evolution of a subclass of these states, the contractive states, leads to a narrowing of the position uncertainty in contrast with the usual minimum-uncertainty wave packets. As a consequence the standard quantum limit for monitoring the positions of a free mass can be breached. Further implications on quantum nondemolition measurements are discussed.The familiar minimum-uncertainty wave packets for masses are generalized in analogy with the two-photon coherent states of the radiation field. The free evolution of a subclass of these states, the contractive states, leads to a narrowing of the position uncertainty in contrast with the usual minimum-uncertainty wave packets. As a consequence the standard quantum limit for monitoring the positions of a free mass can be breached. Further implications on quantum nondemolition measurements are discussed.

Physical Review LettersAug 29, 1983177 citations

Quantum States for Quantum Measurements

O. Tapia

Advances in quantum chemistryJan 1, 201110 citations

Quantum States and Quantum Measurements

Dan Cristian Marinescu, Gabriela M. Marinescu

Jan 1, 20050 citations

Comment on "Contractive States and the Standard Quantum Limit for Monitoring Free-Mass Positions"

Robert Lynch

Physical Review LettersMay 7, 198416 citations

There are heuristic arguments proposing that the accuracy of monitoring position of a free mass

m

is limited by the standard quantum limit (SQL): ${\ensuremath{\sigma}}^{2}(X(t))\ensuremath{\ge}{\ensuremath{\sigma}}^{2}(X(0))+({t}^{2}/{m}^{2}){\ensuremath{\sigma}}^{2}(P(0))\ensuremath{\ge}\ensuremath{\hbar}t/m$, where ${\ensuremath{\sigma}}^{2}(X(t))$ and ${\ensuremath{\sigma}}^{2}(P(t))$ denote variances of the Heisenberg representation position and momentum operators. Yuen [Phys. Rev. Lett. 51, 719 (1983)] discovered that there are contractive states for which this result is incorrect. Here I prove universally valid rigorous quantum limits (RQL), viz. rigorous upper and lower bounds on ${\ensuremath{\sigma}}^{2}(X(t))$ in terms of ${\ensuremath{\sigma}}^{2}(X(0))$ and ${\ensuremath{\sigma}}^{2}(P(0))$, given by Eq. (12) for a free mass and by Eq. (36) for an oscillator. I also obtain the maximally contractive and maximally expanding states which saturate the RQL, and use the contractive states to set up an Ozawa-type measurement theory with accuracies respecting the RQL but beating the standard quantum limit. The contractive states for oscillators improve on the Schr\"odinger coherent states of constant variance and may be useful for gravitational wave detection and optical communication.