The consensus regarding quantum measurements rests on two statements: (i) von Neumann’s standard quantum measurement theory leaves undetermined the basis in which observables are measured, and (ii) the environmental decoherence of the measuring device (the “meter”) unambiguously determines the measuring (“pointer”) basis. The latter statement means that the environment monitors (measures) selected observables of the meter and (indirectly) of the system. Equivalently, a measured quantum state must end up in one of the “pointer states” that persist in the presence of the environment. We find that, unless we restrict ourselves to projective measurements, decoherence does not necessarily determine the pointer basis of the meter. Namely, generalized measurements commonly allow the observer to choose from a multitude of alternative pointer bases that provide the same information on the observables, regardless of decoherence. By contrast, the measured observable does not depend on the pointer basis, whether in the presence or in the absence of decoherence. These results grant further support to our notion of Quantum Lamarckism, whereby the observer’s choices play an indispensable role in quantum mechanics.
The quantum theory of indirect measurements in physical systems is studied. The example of an indirect measurement of an observable represented by a self-adjoint operator
N with finite spectrum is analysed in detail. The Hamiltonian generating the time evolution of the system in the absence of direct measurements is assumed to be given by the sum of a term commuting with
N and a small perturbation not commuting with
N. The system is subject to repeated direct (projective) measurements using a single instrument whose action on the state of the system commutes with
N. If the Hamiltonian commutes with the observable
N (i.e., if the perturbation vanishes) the state of the system approaches an eigenstate of
N, as the number of direct measurements tends to
∞. If the perturbation term in the Hamiltonian does \textit{not} commute with
N the system exhibits "jumps" between different eigenstates of
N. We determine the rate of these jumps to leading order in the strength of the perturbation and show that if time is re-scaled appropriately a maximum likelihood estimate of
N approaches a Markovian jump process on the spectrum of
N, as the strength of the perturbation tends to
0.
A solution to the second measurement problem, determining what prior microscopic properties can be inferred from measurement outcomes (``pointer positions''), is worked out for projective and generalized (POVM) measurements, using consistent histories. The result supports the idea that equipment properly designed and calibrated reveals the properties it was designed to measure. Applications include Einstein's hemisphere and Wheeler's delayed choice paradoxes, and a method for analyzing weak measurements without recourse to weak values. Quantum measurements are noncontextual in the original sense employed by Bell and Mermin: if $[A,B]=[A,C]=0,\phantom{\rule{0.16em}{0ex}}[B,C]\ensuremath{\ne}0$, the outcome of an
A measurement does not depend on whether it is measured with
B or with
C. An application to Bohm's model of the Einstein-Podolsky-Rosen situation suggests that a faulty understanding of quantum measurements is at the root of this paradox.
We report and give an alternative derivation of some results on a model for a quantum measurement studied in [1]. The measured microscopic system is coupled to the position of a macroscopic pointer, which itself interacts with its environment via its momentum. The entanglement between the system and the pointer produced by their mutual interaction is simultaneous with the decoherence of distinct pointer readings resulting from leakage of information to the environment. After a discussion on the various time scales in the model we calculate the matrix elements of the system-pointer density operator between eigenstates of the measured observable with distinct eigenvalues. In general, the decay with time of these coherences is neither exponential nor gaussian. We determine the decoherence (decay) time in terms of the strength of the system-pointer and pointer-environment couplings. This decoherence time does not depend upon the details of the pointer-bath coupling as soon as it is smaller than the bath correlation time (non-Markov regime). In contrast, in the Markov regime it depends strongly on whether this coupling is Ohmic or super-Ohmic.
As per the projection postulate, the evolution of the quantum state of a system upon measurement results in state update. In this work we investigate the characterization of updated states of multipartite entangled qubit states with nondestructive weak measurements, involving weakly coupling a pointer to each qubit while considering a relativistic setup involving different reference frames. As is well known, the updated state at intermediate times is frame dependent and outcome randomness, intrinsic to projective measurements, prevents any information from being acquired on the updated state. Here we will see that when weak measurements are implemented there is instead an interplay between randomness and interference, depending on the number of qubits, so as to maintain consistency between descriptions in arbitrary reference frames. Analytical results are given for a small or infinite number of qubits, while for a finite number of qubits we resort to Monte Carlo sampling in order to simulate numerically the results of weak measurements. As a by-product, we show how a single-shot measurement of the pointers' positions allows us to make quantitative predictions on the outcomes that can be obtained for any observable measured on a distant qubit of the multipartite state.
We propose a general theoretical approach to quantum measurements based on the path (histories) summation technique. For a given dynamical variable A, the Schrödinger state of a system in a Hilbert space of arbitrary dimensionality is decomposed into a set of substates, each of which corresponds to a particular detailed history of the system. The coherence between the substates may then be destroyed by meter(s) to a degree determined by the nature and the accuracy of the measurement(s) which may be of von Neumann, finite-time or continuous type. Transformations between the histories obtained for non-commuting variables and construction of simultaneous histories for non-commuting observables are discussed. Important cases of a particle described by Feynman paths in the coordinate space and a qubit in a two dimensional Hilbert space are studied in some detail.