arXiv:1702.02070 · Journal of Physics A Mathematical and Theoretical

Number and phase: complementarity and joint measurement uncertainties

PEKKA J. LAHTI, Juha-Pekka Pellonpää, Jussi Schultz

Published Aug 3, 2017• 3 citations

Abstract

Abstract We show that number and canonical phase (of a single mode optical field) are complementary observables. We also bound the measurement uncertainty region for their approximate joint measurements.

DOI: 10.1088/1751-8121/aa83bc

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arXiv:1702.02070 · Journal of Physics A Mathematical and Theoretical

Number and phase: complementarity and joint measurement uncertainties

PEKKA J. LAHTI, Juha-Pekka Pellonpää, Jussi Schultz

Published Aug 3, 2017• 3 citations

arXiv HTML PDF

Abstract

DOI: 10.1088/1751-8121/aa83bc

Related papers

Number-phase uncertainty relations: Verification by balanced homodyne measurement

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arXiv:quant-ph/9704037Physical Review AMar 1, 199814 citations

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Abstract The problem of approximate joint measurement of complementary observables, like position and momentum, and the relevance of the uncertainty relations to that question were at the heart of the investigations of Paul Busch into the foundations of quantum mechanics. A good third of his published work dealt with this and closely related questions. This paper is an attempt to survey some of the steps taken in that research starting with Paul’s first papers on the subject matter in the early 1980’s and reaching its height in recent years.Abstract The problem of approximate joint measurement of complementary observables, like position and momentum, and the relevance of the uncertainty relations to that question were at the heart of the investigations of Paul Busch into the foundations of quantum mechanics. A good third of his published work dealt with this and closely related questions. This paper is an attempt to survey some of the steps taken in that research starting with Paul’s first papers on the subject matter in the early 1980’s and reaching its height in recent years.

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The principle of complementarity is quantified in two ways: by a universal uncertainty relation valid for arbitrary joint estimates of any two observables from a given measurement setup and by a general uncertainty relation valid for the optimal estimates of the same two observables when the state of the system prior to measurement is known. A formula is given for the optimal estimate of any given observable, based on arbitrary measurement data and prior information about the state of the system, which generalizes and provides a more robust interpretation of previous formulas for ``local expectations'' and ``weak values'' of quantum observables. As an example, the canonical joint measurement of position $X$ and momentum $P$ corresponds to measuring the commuting operators ${X}_{J}=X+{X}^{\ensuremath{'}}$ and ${P}_{J}=P\ensuremath{-}{P}^{\ensuremath{'}}$, where the primed variables refer to an auxiliary system in a minimum-uncertainty state. It is well known that $\ensuremath{\Delta}{X}_{J}\ensuremath{\Delta}{P}_{J}\ensuremath{\geqslant}\ensuremath{\hbar}$. Here it is shown that given the same physical experimental setup and knowledge of the system density operator prior to measurement, one can make improved joint estimates ${X}_{opt}$ and ${P}_{opt}$ of $X$ and $P$ . These improved estimates are not only statistically closer to $X$ and $P$ but further satisfy $\ensuremath{\Delta}{X}_{opt}\ensuremath{\Delta}{P}_{opt}\ensuremath{\geqslant}\ensuremath{\hbar}∕4$, where equality can be achieved in certain cases. Thus one can do up to four times better than the standard lower bound (where the latter corresponds to the limit of no prior information). Other applications include the heterodyne detection of orthogonal quadratures of a single-mode optical field and joint measurements based on Einstein-Podolsky-Rosen correlations.The principle of complementarity is quantified in two ways: by a universal uncertainty relation valid for arbitrary joint estimates of any two observables from a given measurement setup and by a general uncertainty relation valid for the optimal estimates of the same two observables when the state of the system prior to measurement is known. A formula is given for the optimal estimate of any given observable, based on arbitrary measurement data and prior information about the state of the system, which generalizes and provides a more robust interpretation of previous formulas for ``local expectations'' and ``weak values'' of quantum observables. As an example, the canonical joint measurement of position $X$ and momentum $P$ corresponds to measuring the commuting operators ${X}_{J}=X+{X}^{\ensuremath{'}}$ and ${P}_{J}=P\ensuremath{-}{P}^{\ensuremath{'}}$, where the primed variables refer to an auxiliary system in a minimum-uncertainty state. It is well known that $\ensuremath{\Delta}{X}_{J}\ensuremath{\Delta}{P}_{J}\ensuremath{\geqslant}\ensuremath{\hbar}$. Here it is shown that given the same physical experimental setup and knowledge of the system density operator prior to measurement, one can make improved joint estimates ${X}_{opt}$ and ${P}_{opt}$ of $X$ and $P$. These improved estimates are not only statistically closer to $X$ and $P$ but further satisfy $\ensuremath{\Delta}{X}_{opt}\ensuremath{\Delta}{P}_{opt}\ensuremath{\geqslant}\ensuremath{\hbar}∕4$, where equality can be achieved in certain cases. Thus one can do up to four times better than the standard lower bound (where the latter corresponds to the limit of no prior information). Other applications include the heterodyne detection of orthogonal quadratures of a single-mode optical field and joint measurements based on Einstein-Podolsky-Rosen correlations.

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On various joint measurements of position and momentum observables in quantum theory

Paul Busch, PEKKA J. LAHTI

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fieldsApr 15, 198465 citations

X

and momentum

P

corresponds to measuring the commuting operators ${X}_{J}=X+{X}^{\ensuremath{'}}$ and ${P}_{J}=P\ensuremath{-}{P}^{\ensuremath{'}}$, where the primed variables refer to an auxiliary system in a minimum-uncertainty state. It is well known that $\ensuremath{\Delta}{X}_{J}\ensuremath{\Delta}{P}_{J}\ensuremath{\geqslant}\ensuremath{\hbar}$. Here it is shown that given the same physical experimental setup and knowledge of the system density operator prior to measurement, one can make improved joint estimates

{X}_{opt}

and

{P}_{opt}

X

and

P

. These improved estimates are not only statistically closer to

X

and

P

but further satisfy $\ensuremath{\Delta}{X}_{opt}\ensuremath{\Delta}{P}_{opt}\ensuremath{\geqslant}\ensuremath{\hbar}∕4$, where equality can be achieved in certain cases. Thus one can do up to four times better than the standard lower bound (where the latter corresponds to the limit of no prior information). Other applications include the heterodyne detection of orthogonal quadratures of a single-mode optical field and joint measurements based on Einstein-Podolsky-Rosen correlations.