It has been realized that the measurement problem of quantum mechanics is essentially the determinate-experience problem, and in order to solve the problem, the physical state representing the measurement result is required to be also the physical state on which the mental state of an observer supervenes. This necessitates a systematic analysis of the forms of psychophysical connection in the solutions to the measurement problem. In this paper, I propose a new, mentalistic formulation of the measurement problem which lays more stress on psychophysical connection. By this new formulation, it can be seen more clearly that the three main solutions to the measurement problem, namely Everett's theory, Bohm's theory and collapse theories, correspond to three different forms of psychophysical connection. I then analyze these forms of psychophysical connection. It is argued that the forms of psychophysical connection required by Everett's and Bohm's theories have potential problems, while an analysis of how the mental state of an observer supervenes on her wave function may help solve the structured tails problem of collapse theories.
The retrodiction of spin measurements along a set of different axes is revisited in detail. The problem is presented in two different pictures, a geometric and a general algebraic one. Explicit measurement operators that allow the retrodiction are given for the case of three and four axes. For the Vaidman-Aharanov-Albert case of three orthogonal axes the quantum network is constructed for two different initial Bell states.
The reduction paradigm of quantum interferometry and the objectivation problem in quantum measurements are reanalyzed. Both are shown to be amenable to straightforward mathematical treatment within "every-users" simple-minded quantum mechanics without reduction postulate etc., using only its probabilistic content.
By invoking quantum estimation theory we formulate bounds of errors in quantum measurement for arbitrary quantum states and observables in a finite-dimensional Hilbert space. We prove that the measurement errors of two observables satisfy Heisenberg's uncertainty relation, find the attainable bound, and provide a strategy to achieve it.
The Wigner-Araki-Yanase (WAY) theorem states a remarkable limitation to quantum mechanical measurements in the presence of additive conserved quantities. Discovered by Wigner in 1952, this limitation is known to induce constraints on the control of individual quantum systems in the context of information processing. It is therefore important to understand the precise conditions and scope of the WAY theorem. Here we elucidate its crucial assumptions, briefly review some generalizations, and show how a particular extension can already be obtained by a simple modification of the original proofs. …
We study the quantum-mechanical uncertainty relation originating from the successive measurement of two observables $\hat{A}$ and $\hat{B}$, with eigenvalues $a_n$ and $b_m$, respectively, performed on the same system. We use an extension of the von Neumann model of measurement, in which two probes interact with the same system proper at two successive times, so we can exhibit how the disturbing effect of the first interaction affects the second measurement. Detecting the statistical properties of the second {\em probe} variable $Q_2$ conditioned on the first {\em probe} measurement yielding $…
In this paper we extend the ladder proof of nonlocality without inequalities for two spin-half particles given by Boschi et al [PRL 79, 2755 (1997)] to the case in which the measurement settings of the apparatus measuring one of the particles are different from the measurement settings of the apparatus measuring the other particle. It is shown that, in any case, the proportion of particle pairs for which the contradiction with local realism goes through is maximized when the measurement settings are the same for each apparatus. Also we write down a Bell inequality for the experiment in questio…
It is shown that all the known uncertainty relations are the secondary consequences of Robertson's relation. The basic idea is to use the Heisenberg picture so that the time development of quantum mechanical operators incorporate the effects of the measurement interaction. A suitable use of triangle inequalities then gives rise to various forms of uncertainty relations. The assumptions of unbiased measurement and unbiased disturbance are important to simplify the resulting uncertainty relations and to give the familiar uncertainty relations such as a naive Heisenberg error-disturbance relation…
An interesting analogy between quantum entangled states and topological links was suggested by Aravind. In particular, he emphasized a connection between the Greenberger-Horne-Zeilinger (GHZ) state and the Borromean rings. However, he made the connection in a way that depends on the choice of measurement basis. We reconsider it in a basis-independent way by using the reduced density matrix.
Pearle (1970) gave an example of a local hidden variables model which exactly reproduced the singlet correlations of quantum theory, through the device of data-rejection: particles can fail to be detected in a way which depends on the hidden variables carried by the particles and on the measurement settings. If the experimenter computes correlations between measurement outcomes of particle pairs for which both particles are detected, he is actually looking at a subsample of particle pairs, determined by interaction involving both measurement settings and the hidden variables carried in the par…
After giving a panoramic view of the " text-book " interpretation of the new quantum mechanics, as a sequel to the old quantum theory, the conceptual basis of quantum theory since the Copenhagen Interpretation is reviewed in the context of various proposals since Einstein and Niels Bohr, designed to throw light on possible new facets bearing on its foundations, the key issues to its inherent "incompleteness" being A) measurement, and B) quantum non-locality. A related item on measurement, namely Quantum Zeno (as well as anti-Zeno) effect is also reviewed briefly. The inputs for the new facets …
Imposing restrictions on the Feynman paths of the monitored system has in the past been proposed as a universal model-free approach to continuous quantum measurements. Here we revisit this proposition, and demonstrate that a Gaussian restriction, resulting in a sequence of many highly inaccurate (weak) von Neumann measurements, is not sufficiently strong to ensure proximity between a readout and the Feynman paths along which the monitored system evolves. Rather, in the continuous limit, the variations of a typical readout become much larger than the separation between the eigenvalues of the me…
We provide an alternative view of the efficient classical simulatibility of fermionic linear optics in terms of Slater determinants. We investigate the generic effects of two-mode measurements on the Slater number of fermionic states. We argue that most such measurements are not capable (in conjunction with fermion linear optics) of an efficient exact implementation of universal quantum computation. Our arguments do not apply to the two-mode parity measurement, for which exact quantum computation becomes possible, see quant-ph/0401066.
It is by now well-recognised that the naïve application of the projection postulate on composite quantum systems can induce signalling between their constituent components, indicative of a breakdown of causality in a relativistic spacetime context. Here we introduce a necessary and sufficient condition for an ideal measurement of an observable to be non-signalling. As well as being particularly simple, it generalises previous no-signalling conditions in that it allows for degeneracies and can be applied to all bounded self-adjoint operators. The condition is used to establish that arbitrary su…
A semi-spectral Chebyshev method for solving numerically singular integral equations is presented and applied in the quarkonium bound-state problem in momentum space. The integrals containing both, logarithmic and Cauchy singular kernels, can be evaluated without subtractions by dedicated automatic quadratures. By introducing a Chebyshev mesh and using the Nystrom algorithm the singular integral equation is converted into an algebraic eigenvalue problem that can be solved by standard methods. The proposed scheme is very simple to use, is easy in programming and highly accurate.
The problem of bound states in delta potentials is revisited by means of Fourier transform approach. The problem in a simple delta potential sums up to solve an algebraic equation of degree one for the Fourier transform of the eigenfunction and the problem for more than one delta function also reveals itself to be a simple matter. Quite differently from direct methods, no knowledge about the jump discontinuity of the first derivative of the eigenfunction is required to determine the solution of the problem.
We revisit the algebraic description of shape invariance method in one-dimensional quantum mechanics. In this note we focus on four particular examples: the Kepler problem in flat space, the Kepler problem in spherical space, the Kepler problem in hyperbolic space, and the Rosen-Morse potential problem. Following the prescription given by Gangopadhyaya et al., we first introduce certain nonlinear algebraic systems. We then show that, if the model parameters are appropriately quantized, the bound-state problems can be solved solely by means of representation theory.
The WKB and instanton answers for the tunnel splitting of the ground state in a symmetric double well potential are both reduced to an expression involving only the functionals of the potential, without the need for solving any auxilliary problems. This formula is applied to simple model problems. The prefactor for the splitting in the text book by Landau and Lifshitz is amended so as to apply to the ground and low lying excited states.
