A limitation on simultaneous measurement of two arbitrary positive operator valued measures is discussed. In general, simultaneous measurement of two noncommutative observables is only approximately possible. Following Werner's formulation, we introduce a distance between observables to quantify an accuracy of measurement. We derive an inequality that relates the achievable accuracy with noncommutativity between two observables. As a byproduct a necessary condition for two positive operator valued measures to be simultaneously measurable is obtained.
At the recent QSCP XIX, the author claimed a procedure of using a scaled Fourier transform (the scaling being determined by the detailed interaction and particle mass for a harmonic oscillator) to achieve simultaneous resolution of position and momentum greater than the standard Heisenberg value of 1/2. The procedure is, in fact, invalid for quantum mechanics. The purpose of this paper is simply to give the correct analysis of the uncertainty product, thereby clarifying the error made.
We introduce a Hermitian generalization of Pauli matrices to higher dimensions which is based on Heisenberg-Weyl operators. The complete set of Heisenberg-Weyl observables allows us to identify a real-valued Bloch vector for an arbitrary density operator in discrete phase space, with a smooth transition to infinite dimensions. Furthermore, we derive bounds on the sum of expectation values of any set of anti-commuting observables. Such bounds can be used in entanglement detection and we show that Heisenberg-Weyl observables provide a first non-trivial example beyond the dichotomic case.
We have made the first experimental demonstration of the simultaneous minimum uncertainty product between two complementary observables for a two-state system (a qubit). A partially entangled two-photon state was used to perform such measurements. Each of the photons carries (partial) information of the initial state thus leaving a room for measurements of two complementary observables on every member in an ensemble.
We show that quantum entanglement and the Heisenberg uncertainty principle are inextricably connected. Toward this end, a complete set of commuting observables (CSCO) criterion for the entanglement is developed. Assuming (A1,A2,...) and (B1,B2,...) being two CSCO's for a given system, and C being the matrix, Cij = i [Bi,Aj], for each given row i (i=1,2,...) if at least one matrix element Cij (j=1,2,...) is nonzero, then for the simultaneous eigenstates |ψ)=|A1',A2',...) of (A1,A2,...), the simultaneous measurements of (B1,B2,...) are, in general,entangled. The only exception is when all the si…
The simultaneous measurement approach of Arthurs and Kelly has been a significant tool for the better understanding of the measurement process in quantum mechanics. This model considers a strong interaction Hamiltonian by discarding the free evolution part. In this work, we study the effect of the full dynamics -- taking into account the free Hamiltonian -- on the optimal limits of retrodictive and predictive accuracy of the simultaneous measurement process of position and momentum observables. To do that, we consider a minimum uncertainty Gaussian state as the system under inspection, which a…
Heisenberg's uncertainty principle was originally posed for the limit of the accuracy of simultaneous measurement of non-commuting observables as stating that canonically conjugate observables can be measured simultaneously only with the constraint that the product of their mean errors should be no less than a limit set by Planck's constant. However, Heisenberg with the subsequent completion by Kennard has long been credited only with a constraint for state preparation represented by the product of the standard deviations. Here, we show that Heisenberg actually proved the constraint for the ac…
The uncertainty relation, which displays an elementary property of quantum theory, was originally described by Heisenberg as the relation between error and disturbance. Ozawa presented a more rigorous expression of the uncertainty relation, which was later verified experimentally. Nevertheless, the operators corresponding to error and disturbance should be measurable in the identical state if we follow the presupposition of Heisenberg's thought experiment. In this letter, we discuss simultaneous measurability of error and disturbance and present a new inequality using error and disturbance in …
The quantum Fisher information matrix provides us with a tool to determine the precision, in any multiparametric estimation protocol, through quantum Cramér-Rao bound. In this work, we study simultaneous and individual estimation strategies using the density matrix vectorization method. Two special Heisenberg $XY$ models are considered. The first one concerns the anisotropic $XY$ model in which the temperature $T$ and the anisotropic parameter $γ$ are estimated. The second situation concerns the isotropic $XY$ model submitted to an external magnetic field $B$ in which the temperature and the m…
High-precision magnetic field measurement is an ubiquitous issue in physics and a critical task in metrology. Generally, magnetic field has DC and AC components and it is hard to extract both DC and AC components simultaneously. The conventional Ramsey interferometry can easily measure DC magnetic fields, while it becomes invalid for AC magnetic fields since the accumulated phases may average to zero. Here, we propose a scheme for simultaneous measurement of DC and AC magnetic fields by combining Ramsey interferometry and rapid periodic pulses. In our scheme, the interrogation stage is divided…
Algebra of quantum mechanics via classical phonons. II: Klein-Gordon dynamics, the Heisenberg formalism, the Dirac canonical commutation rule and the Poincare algebra through the continuous Poisson bracket formalism
The change of a quantum state can generally only be fully monitored through simultaneous measurements of two non-commuting observables X and Y spanning a phase space. A measurement device that is coupled to the thermal environment provides at a time a pair of values that have a minimal uncertainty product set by the Heisenberg uncertainty relation, which limits the precision of the monitoring. Here we report on an optical measurement setup that is able to monitor the time dependent change of the quantum state's displacement in phase space (< X (t)>; < Y (t)>) with an imprecision 10\,dB below t…
Reports on experiments recently performed in Vienna [Erhard et al, Nature Phys. 8, 185 (2012)] and Toronto [Rozema et al, Phys. Rev. Lett. 109, 100404 (2012)] include claims of a violation of Heisenberg's error-disturbance relation. In contrast, we have presented and proven a Heisenberg-type relation for joint measurements of position and momentum [Phys. Rev. Lett. 111, 160405 (2013)]. To resolve the apparent conflict, we formulate here a new general trade-off relation for errors in qubit measurements, using the same concepts as we did in the position-momentum case. We show that the combined e…
Heisenberg-scaling characterization of a two-channel optical network via two-port homodyne detection
School of Mathematics and Physics, University of Portsmouth（朴茨茅斯大学数学与物理学院）；Quantum Science and Technology Hub, University of Portsmouth（朴茨茅斯大学量子科学与技术中心）；Dipartimento di Fisica, Università di Bari & Politecnico di Bari（巴里大学与巴里理工大学物理系）；INFN, Sezione di Bari（意大利国家核物理研究所巴里分部）；Institute of Cosmology and Gravitation, University of Portsmouth（朴茨茅斯大学宇宙学与引力研究所）
We analyze the Heisenberg limit on phase estimation for Gaussian states. In the analysis, no reference to a phase operator is made. We prove that the squeezed vacuum state is the most sensitive for a given average photon number. We provide two adaptive local measurement schemes that attain the Heisenberg limit asymptotically. One of them is described by a positive operator-valued measure and its efficiency is exhaustively explored. We also study Gaussian measurement schemes based on phase quadrature measurements. We show that homodyne tomography of the appropriate quadrature attains the Heisen…
The time evolution of the entanglement of a pair of two spin qubits is investigated when the two qubits simultaneously couple to an environment of an anisotropic Heisenberg XXZ spin chain. The entanglement of the two spin qubits can be created and is a periodic function of the time if the magnetic field is greater than a critical value. If the two spin qubits are in the Bell state, the entanglement can be stored with relatively large value even when the magnetic field is large.
Using formal definitions for measurement precision ε and disturbance (measurement backaction) η, Ozawa [Phys. Rev. A 67, 042105 (2003)] has shown that Heisenberg's claimed relation between these quantities is false in general. Here we show that the quantities introduced by Ozawa can be determined experimentally, using no prior knowledge of the measurement under investigation --- both quantities correspond to the root-mean-squared difference given by a weak-valued probability distribution. We propose a simple three-qubit experiment which would illustrate the failure of Heisenberg's measurement-…
Heisenberg's uncertainty relation for measurement noise and disturbance states that any position measurement with noise epsilon brings the momentum disturbance not less than hbar/2epsilon. This relation holds only for restricted class of measuring apparatuses. Here, Heisenberg's uncertainty relation is generalized to a relation that holds for all the possible quantum measurements, from which conditions are obtained for measuring apparatuses to satisfy Heisenberg's relation. In particular, every apparatus with the noise and the disturbance statistically independent from the measured object is p…
Considering the simultaneous measurement of non-commuting observables, we define a geometric measure for the degree of non-commuting behavior of quantum measurements coming from the initial and final states of the measurements. The rationality of our geometric measure is demonstrated and the application of it is presented. The connection between our measure and Heisenberg's uncertainty principle is discussed as well. Our work deepens the understanding of quantum non-commuting measurement.
In order to explore the effect of external temperature $T$ in quantum correlation we compute thermal entanglement and thermal discord analytically in the Heisenberg $X$ $Y$ $Z$ model with Dzyaloshinskii-Moriya Interaction term ${\bm D} \cdot \left( {\bm σ}_1 \times {\bm σ}_2 \right)$. For the case of thermal entanglement it is shown that quantum phase transition occurs at $T = T_c$ due to sudden death phenomenon. For antiferromagnetic case the critical temperature $T_c$ increases with increasing $|{\bm D}|$. For ferromagnetic case, however, $T_c$ exhibits different behavior in the regions $|{\…
We investigate some spectral properties of time operators which are obtained through Canonical Commutation Relation (CCR) and Positive Operator Valued Measure (POVM) of quantum physics. In addition, we re-interpret the spectral properties of time operators from the standpoint of the Copenhagen Interpretation, especially, of W.Heisenberg.
In a previous paper we have presented a general scheme for the implementation of symmetric generalized measurements (POVMs) on a quantum computer. This scheme is based on representation theory of groups and methods to decompose matrices that intertwine two representations. We extend this scheme in such a way that the measurement is minimally disturbing, i.e., it changes the state vector \ketΨ of the system to \sqrtΠ \ketΨ where Πis the positive operator corresponding to the measured result. Using this method, we construct quantum circuits for measurements with Heisenberg-Weyl symmetry. A conti…
Incompatible observables can be approximated by compatible observables in joint measurement or measured sequentially, with constrained accuracy as implied by Heisenberg's original formulation of the uncertainty principle. Recently, Busch, Lahti, and Werner proposed inaccuracy trade-off relations based on statistical distances between probability distributions of measurement outcomes [Phys. Rev. Lett. 111, 160405 (2013); Phys. Rev. A 89, 012129 (2014)]. Here we reform their theoretical framework, derive an improved relation for qubit measurement, and perform an experimental test on a spin syste…
Physicists describe nature using mathematics as the natural language, and for quantum mechanics, it prefers to use complex numbers. However, whether complex numbers are really necessary for the theory has been debated ever since its birth. Recently, it has been shown that a three-party correlation created in entanglement swapping scenarios comprising independent states and measurements cannot be reproduced using only real numbers. Previous experiments have conceptually supported the predication, yet not satisfying the independent state preparations and measurements simultaneously. Here, we imp…
Continuous Variable Hamiltonian Learning at Heisenberg Limit via Displacement-Random Unitary Transformation
School of Stomatology, Peking University（北京大学口腔医学院）；Department of Chemistry and Biochemistry, University of California, Los Angeles（加州大学洛杉矶分校化学与生物化学系）；Department of Physics, Tsinghua University（清华大学物理系）；Institute of Advanced Study, Tsinghua University（清华大学高等研究院）
We experimentally demonstrate an atom number detector capable of simultaneous detection of two mesoscopic ensembles with single-atom resolution. Such a sensitivity is a prerequisite for quantum metrology at a precision approaching the Heisenberg limit. Our system is based on fluorescence detection of atoms in a novel hybrid trap in which a dipole barrier divides a magneto-optical trap into two separated wells. We introduce a noise model describing the various sources contributing to the measurement error and report a limit of up to 500 atoms for single-atom resolution in the atom number differ…
We discuss the Heisenberg limit in the multiparameter metrology within two different paradigms -- the one, where the measurement is repeated many times (so the Cramér-Rao bound is guaranteed to be asymptotically saturable) and the second one, where all the resources are allocated into one experimental realization (analyzed with the mimimax approach). We investigate the potential advantage of measuring all the parameter simultaneously compared to estimating them individually, while spending the same total amount of resources. We show that in general the existence of such an advantage, its magni…
By implicitly assuming that all measurements occur simultaneously, Bell's Theorem only applied to local theories that violated Heisenberg's Uncertainty Principle. By explicitly introducing time into our derivation of Bell's theorem, an extra term related to the time-ordering of actual measurements is found to augment (i.e. weaken) the upper bound of the inequality. Since the same locality assumptions hold for this rederivation as for the original, we conclude that only {\em classical} measurement-order independent local hidden variable theories are constrained by Bell's inequality; time depend…
What does it take for real-deterministic c-valued (i.e., classical, commuting) variables to comply with the Heisenberg uncertainty principle? Here, we construct a class of real-deterministic c-valued variables out of the weak values obtained via a non-perturbing weak measurement of quantum operators with a post-selection over a complete set of state vectors basis, which always satisfies the Kennard-Robertson-Schrödinger uncertainty relation. First, we introduce an auxiliary global random variable and couple it to the imaginary part of the weak value to transform the incompatibility between the…
The the over-complete eigenvector system of the operator Q^{-1}P (Q:position, P:momentum) which consists of the squeezed states |0; s, t > with various s and t are investigated from the viewpoint of the annihilation and creation relations related to the algebra su(1,1). We derive a positive operator-valued measure(POVM) for the simultaneous measurement between the self-adjoint and anti-self-adjoint parts of PQ^{-1}.
Atom interferometry tests of universality of free fall based on the differential measurement of two different atomic species provide a useful complement to those based on macroscopic masses. However, when striving for the highest possible sensitivities, gravity gradients pose a serious challenge. Indeed, the relative initial position and velocity for the two species need to be controlled with extremely high accuracy, which can be rather demanding in practice and whose verification may require rather long integration times. Furthermore, in highly sensitive configurations gravity gradients lead …
