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Fortschritte der Physik

Impossibility of Determining the Unknown Quantum Wavefunction of a Single System: Quantum Non-Demolition Measurements, Measurements without Entanglement and Adiabatic Measurements

Orly Alter, Yoshihisa Xamamoto

Published Nov 1, 1998• 3 citations
Publisher

Abstract

We establish that the information which can be obtained in the measurement of a single system about the unknown quantum wavefunction of the system is limited to estimates of the expectation values of the measured observables, with the estimate errors satisfying the uncertainty principle. Only the fully a-priori known wavefunction could be determined exactly. We consider specifically measurement schemes which may induce minimal change in the measured system, such as quantum non-demolition measurements, measurements without entanglement and adiabatic measurements.

DOI
10.1002/(sici)1521-3978(199811)46:6/8<817::aid-prop817>3.0.co;2-y

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•May 19, 2021•0 citations
We establish that the information which can be obtained in the measurement of a single system about the unknown quantum wavefunction of the system is limited to estimates of the expectation values of the measured observables, with the estimate errors satisfying the uncertainty principle. Only the fully a-priori known wavefunction could be determined exactly. We consider specifically measurement schemes which may induce minimal change in the measured system, such as quantum non-demolition measurements, measurements without entanglement and adiabatic measurements.
Preface List of Illustrations Open Fundamental Questions of Limits to Information in the Quantum Measurement of a Single System Impossibility of Determining the Unknown Quantum Wavefunction of a Single System Quantum Nondemolition (QND) Measurements of a Single System Measurements without Entanglement of a Single Quantum System Adiabatic Measurements of a Single Quantum System Quantum Zeno Effect of a Single System Fundamental Quantum Limit to External Force Detection via Monitoring a Single Harmonic Oscillator (or Free Mass) Established Limits to Information in the Quantum Measurement of a Single System Bibliography Index.
Suppose that the Hamiltonian acting on a quantum system is unknown and one wants to determine which is the Hamiltonian. We show that, in general, this requires a time $\ensuremath{\Delta}t$ that obeys the uncertainty relation $\ensuremath{\Delta}t\ensuremath{\Delta}H\ensuremath{\gtrsim}1,$ where $\ensuremath{\Delta}H$ is a measure of how accurately the unknown Hamiltonian must be estimated. We apply this result to the problem of measuring the energy of an unknown quantum state. It has been previously shown that if the Hamiltonian is known, then the energy can, in principle, be measured with arbitrarily large precision in an arbitrarily short time. On the other hand, we show that if the Hamiltonian is not known then an energy measurement necessarily takes a minimum time $\ensuremath{\Delta}t$ which obeys the uncertainty relation $\ensuremath{\Delta}t\ensuremath{\Delta}E\ensuremath{\gtrsim}1,$ where $\ensuremath{\Delta}E$ is the precision of the energy measurement. Several examples are studied to address the question of whether it is possible to saturate these uncertainty relations. Their interpretation is discussed in detail.
According to Heisenberg’s uncertainty principle, measurement of a quantum observable introduces noise to this observable and thus limits the available precision of measurement. Quantum non-demolition measurements are designed to circumvent this limitation and have been demonstrated in detecting the photon flux of classical light beam. Quantum non-demolition measurement of a single photon is the ultimate goal because it is of great interest in fundamental physics and also a powerful tool for applications in quantum information processing. This chapter presents a brief introduction of the history and a review of the progress in quantum non-demolition measurement of light. In particular, a detailed description is presented for two works toward cavity-free schemes of quantum non-demolition measurement of single photons. Afterward, an outlook of the future in this direction is given.
Knowing and guessing, these are two essential epistemological pillars in the theory of quantum-mechanical measurement. As formulated quantum mechanics is a statistical theory. In general, a priori unknown states can be completely determined only when measurements on infinite ensembles of identically prepared quantum systems are performed. But how one can estimate (guess) quantum state when just incomplete data are available (known)? What is the most reliable estimation based on a given measured data? What is the optimal measurement providing only a finite number of identically prepared quantum objects are available? These are some of the questions we address. We present several schemes for a reconstruction of states of quantum systems from measured data: (1) We show how the maximum entropy (MaxEnt) principle can be efficiently used for an estimation of quantum states on incomplete observation levels. (2) We show how Bayesian inference can be used for reconstruction of quantum states when only a finite number of identically prepared systems are measured. (3) We describe the optimal generalized measurement of a finite number of identically prepared quantum systems which results in the estimation of a quantum state with the highest fidelity.
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 XXX and momentum PPP 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 Xopt{X}_{opt}Xopt​ and Popt{P}_{opt}Popt​ of XXX and PPP. These improved estimates are not only statistically closer to XXX and PPP 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.
Quantum uncertainty is a well-known property of quantum mechanics that states the impossibility of predicting measurement outcomes of multiple incompatible observables simultaneously. In contrast, the uncertainty in the classical domain comes from the lack of information about the exact state of the system. One may naturally ask, whether the quantum uncertainty is indeed a fully intrinsic property of the quantum theory, or whether similar to the classical domain lack of knowledge about specific parts of the physical system might be the source of this uncertainty. This question has been addressed in [New J. Phys.19 023038 (2017)] where the authors argue that in the entropic formulation of the uncertainty principle that can be illustrated using the so-called, guessing games, indeed such lack of information has a significant contribution to the arising quantum uncertainty. Here we investigate this issue experimentally by implementing the corresponding two-dimensional and three-dimensional guessing games. Our results confirm that within the guessing-game framework, the quantum uncertainty to a large extent relies on the fact that quantum information determining the key properties of the game is stored in the degrees of freedom that remain inaccessible to the guessing party.