Some realizable joint measurements of complementary observables

Noncommuting quantum observables, if considered asunsharp observables, are simultaneously measurable. This fact is exemplified for complementary observables in two-dimensional state spaces. Two proposals of experimentally feasible joint measurements are presented for pairs of photon or neutron polarization observables and for path and interference observables in a photon split-beam experiment. A recent experiment proposed and performed by Mittelstaedt, Prieur, and Schieder in Cologne is interpreted as a partial version of the latter example.Work partly supported by the DFG, Bonn, and the BMFT, Bonn.Dedicated to Professor Ulrich Hauser on the occasion of his 60th birthday, November 1986.     

A straightforward introduction to continuous quantum measurement

We present a pedagogical treatment of the formalism of continuous quantum measurement. Our aim is to show the reader how the equations describing such measurements are derived and manipulated in a direct manner. We also give elementary background material for those new to measurement theory, and describe further various aspects of continuous measurements that should be helpful to those wanting to use such measurements in applications. Specifically, we use the simple and direct approach of generalized measurements to derive the stochastic master equation describing the continuous measurements of observables, give a tutorial on stochastic calculus, treat multiple observers and inefficient detection, examine a general form of the measurement master equation, and show how the master equation leads to information gain and disturbance. To conclude, we give a detailed treatment of imaging the resonance fluorescence from a single atom as a concrete example of how a continuous position measurement arises in a physical system.     

Some Remarks on Unsharp Quantum Measurements,Quantum Non-Demolition,and All That

The theory of unsharp quantum measurements is reviewed as a generalization of the von Neumann-Lüders measurement theory and applied to measurements of continuous quantities. A generalized notion of repeatability is proposed which is applicable even to intrinsically unsharp continuous observables. As an illustration a precise formulation of a quantum non-demolition (QND) measurement performed on a harmonic oscillator is given.     

Reducing backaction when measuring temporal correlations in quantum systems

Dynamic correlations of quantum observables are challenging to measure due to measurement backaction incurred at early times. Recent work [P. Uhrich et al., Phys. Rev. A 96, 022127 (2017)] has shown that ancilla-based noninvasive measurements are able to reduce this backaction, allowing for dynamic correlations of single-site spin observables to be measured. We generalise this result to correlations of arbitrary spin observables and extend the measurement protocol to simultaneous noninvasive measurements which allow for real and imaginary parts of correlations to be extracted from a single set of measurements. We use positive operator-valued measures to analyse the dynamics generated by the ancilla-based measurements. Using this framework we prove that special observables exist for which measurement backaction is of no concern, so that dynamic correlations of these can be obtained without making use of ancillas.     

Classical Versus Quantum Probability in Sequential Measurements

We demonstrate in this paper that the probabilities for sequential measurements have features very different from those of single-time measurements. First, they cannot be modelled by a classical stochastic process. Second, they are contextual, namely they depend strongly on the specific measurement scheme through which they are determined. We construct Positive-Operator-Valued measures (POVM) that provide such probabilities. For observables with continuous spectrum, the constructed POVMs depend strongly on the resolution of the measurement device, a conclusion that persists even if we consider a quantum mechanical measurement device or the presence of an environment. We then examine the same issues in alternative interpretations of quantum theory. We first show that multi-time probabilities cannot be naturally defined in terms of a frequency operator. We next prove that local hidden variable theories cannot reproduce the predictions of quantum theory for sequential measurements, even when the degrees of freedom of the measuring apparatus are taken into account. Bohmian mechanics, however, does not fall in this category. We finally examine an alternative proposal that sequential measurements can be modeled by a process that does not satisfy the Kolmogorov axioms of probability. This removes contextuality without introducing non-locality, but implies that the empirical probabilities cannot be always defined (the event frequencies do not converge). We argue that the predictions of this hypothesis are not ruled out by existing experimental results (examining in particular the “which way” experiments); they are, however, distinguishable in principle.     

The Measurement-Disturbance Relation and the Disturbance Trade-off Relation in Terms of Relative Entropy

We employ quantum relative entropy to establish the relation between the measurement uncertainty and its disturbance on a state in the presence (and absence) of quantum memory. For two incompatible observables, we present the measurement-disturbance relation and the disturbance trade-off relation. We find that without quantum memory the disturbance induced by the measurement is never less than the measurement uncertainty and with quantum memory they depend on the conditional entropy of the measured state. We also generalize these relations to the case with multiple measurements. These relations are demonstrated by two examples.     

Continuous quantum measurement with independent detector cross correlations

We investigate the advantages of using two independent, linear detectors for continuous quantum measurement. For single-shot measurement, the detection process may be quantum limited if the detectors are twins. For weak continuous measurement, cross correlations allow a violation of the Korotkov-Averin bound for the detector's signal-to-noise ratio. The joint weak measurement of noncommuting observables is also investigated, and we find the cross correlation changes sign as a function of frequency, reflecting a crossover from incoherent relaxation to coherent, out of phase oscillations. Our results are applied to a double quantum-dot charge qubit, simultaneously measured by two quantum point contacts.     

Towards a Resolution of Dilemma: Nonlocality or Nonobjectivity?

This paper discusses a possible resolution of the nonobjectivity-nonlocality dilemma in quantum mechanics in the light of experimental tests of the Bell inequality for two entangled photons and a Bell-like inequality for a single neutron. My conclusion is that these experiments show that quantum mechanics is nonobjective: that is, the values of physical observables cannot be assigned to a system before measurement. Bell’s assumption of nonlocality has to be rejected as having no direct experimental confirmation, at least thus far. I also consider the relationships between nonobjectivity and contextuality. Specifically, I analyze the impact of the Kochen-Specker theorem on the problem of contextuality of quantum observables. I argue that, just as von Neumann’s “no-go” theorem, the Kochen-Specker theorem is based on assumptions that do not correspond to the real physical situation. Finally, I present a theory of measurement based on a classical, purely wave model (pre-quantum classical statistical field theory), a model that reproduces quantum probabilities. In this model continuous fields are transformed into discrete clicks of detectors. While this model is classical, it is nonobjective. In this case, nonobjectivity is the result of the dependence of experimental outcomes on the context of measurement, in accordance with Bohr’s view.     

Quantum state reconstruction via continuous measurement

We present a new procedure for quantum state reconstruction based on weak continuous measurement of an ensemble average. By applying controlled evolution to the initial state, new information is continually mapped onto the measured observable. A Bayesian filter is then used to update the state estimate in accordance with the measurement record. This generalizes the standard paradigm for quantum tomography based on strong, destructive measurements on separate ensembles. This approach to state estimation induces minimal perturbation of the measured system, giving information about observables whose evolution cannot be described classically in real time and opening the door to new types of quantum feedback control.     

Uncertainty relations for noise and disturbance in generalized quantum measurements

Heisenberg’s uncertainty relation for measurement noise and disturbance is commonly understood to state that in any measurement the product of the position measurement noise and the momentum disturbance is not less than Planck’s constant divided by 4π. However, it has been shown in many ways that this relation holds only for a restricted class of measuring apparatuses in the most general formulation of measuring processes. Here, Heisenberg’s uncertainty relation is generalized to a relation that holds for all the possible quantum measurements, from which rigorous 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 proven to satisfy Heisenberg’s relation. For this purpose, all the possible quantum measurements are characterized by naturally acceptable axioms. Then, a mathematical notion of the distance between probability operator valued measures and observables is introduced and the basic properties are explored. Based on this notion, the measurement noise and disturbance are naturally defined for any quantum measurements in a model independent formulation. Under this formulation, various relations for noise and disturbance are also derived for apparatuses with independent noise, independent disturbance, unbiased noise, and unbiased disturbance as well as noiseless apparatuses and nondisturbing apparatuses. Two models of position measurements are also discussed in the light of the new uncertainty relations to show that Heisenberg’s relation can be violated even by approximately repeatable position measurements.     

Uncertainty relation of successive measurements based on Wigner–Yanase skew information

Wigner-Yanase skew information could quantify the quantum uncertainty of the observables that are not commuting with a conserved quantity.We present the uncertainty principle for two successive projective measurements in terms of Wigner-Yanase skew information based on a single quantum system.It could capture the incompatibility of the observables,i.e.the lower bound can be nontrivial for the observables that are incompatible with the state of the quanaim system.Furthermore,the lower bound is also constrained by the quantum Fisher information.In addition,we find the complementarity relation between the uncertainties of the observable which operated on the quantum state and the other observable that performed on the post-measured quantum state and the uncertainties formed by the non-degenerate quantum observables performed on the quantum state,respectively.     

An approach to measurement

We present a new approach to measurement theory. Our definition of measurement is motivated by direct laboratory procedures as they are carried out in practice. The theory is developed within the quantum logic framework. This work clarifies an important problem in the quantum logic approach; namely, where the Hilbert space comes from. We consider the relationship between measurements and observables, and present a Hilbert space embedding theorem. We conclude with a discussion of charge systems.     

WHEN DOES A MEASUREMENT OR EVENT OCCUR?

Within quantum mechanics, a complete set of commutting observables can be found which describe the attributes of a system at a given time. However, the correct way to describe attributes of a system in time is still an open question. We discuss the difficulties in extending the standard approach of quantum mechanics to describe attributes of a system in time. We find that measuring when an event occurred and measuring that it occurred, are complimentary in Bohr's sense. To exemplify the differences between measurements at a given time and in time, we will compare Rovelli's recent proposal (quant-ph/9802020), to determine “at what time does a measurement occurred” with another model of a continuous measurement in time. Rovelli's scheme answers the question “has the measurement already occurred at a certain time?”, but does not answer to the more difficult question: “when did the measurement occur?” We also discuss the use of the probability current to measure the time at which a particle arrives to a certain location.     

Undecidability and the Problem of Outcomes in Quantum Measurements

We argue that it is fundamentally impossible to recover information about quantum superpositions when a quantum system has interacted with a sufficiently large number of degrees of freedom of the environment. This is due to the fact that gravity imposes fundamental limitations on how accurate measurements can be. This leads to the notion of undecidability: there is no way to tell, due to fundamental limitations, if a quantum system evolved unitarily or suffered wavefunction collapse. This in turn provides a solution to the problem of outcomes in quantum measurement by providing a sharp criterion for defining when an event has taken place. We analyze in detail in examples two situations in which in principle one could recover information about quantum coherence: (a) “revivals” of coherence in the interaction of a system with the measurement apparatus and the environment and (b) the measurement of global observables of the system plus apparatus plus environment. We show in the examples that the fundamental limitations due to gravity and quantum mechanics in measurement prevent both revivals from occurring and the measurement of global observables. It can therefore be argued that the emerging picture provides a complete resolution to the measurement problem in quantum mechanics.     

On the measurability of quantum correlation functions

The concept of correlation function is widely used in classical statistical mechanics to characterize how two or more variables depend on each other. In quantum mechanics, on the other hand, there are observables that cannot be measured at the same time; the so-called incompatible observables. This prospect imposes a limitation on the definition of a quantum analog for the correlation function in terms of a sequence of measurements. Here, based on the notion of sequential weak measurements, we circumvent this limitation by introducing a framework to measure general quantum correlation functions, in principle, independently of the state of the system and the operators involved. To illustrate, we propose an experimental configuration to obtain explicitly the quantum correlation function between two Pauli operators, in which the input state is an arbitrary mixed qubit state encoded on the polarization of photons.     

Transport of quantum noise through random media

We present an experimental study of the propagation of quantum noise in a multiple scattering random medium. Both static and dynamic scattering measurements are performed: the total transmission of noise is related to the mean free path for scattering, while the noise frequency correlation function determines the diffusion constant. The quantum noise observables are found to scale markedly differently with scattering parameters compared to classical noise observables. The measurements are explained with a full quantum model of multiple scattering.     

Entanglement, Joint Measurement, and State Reduction

Entanglement is perhaps the most important new feature of the quantum world. It is expressed in quantum theory by the joint measurement formula. We prove the formula for projection valued observables from a plausible assumption, which for spacelike separated measurements is a consequence of causality. State reduction is simply a way to express the joint measurement formula after one measurement has been made, and its result known.