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Roger Balian

Publications and source records attributed to Roger Balian.

13 recordsLinked to original sources
arXiv2024

Teaching ideal quantum measurement, from dynamics to interpretation

We present a graduate course on ideal measurements, analyzed as dynamical processes of interaction between the tested system S and an apparatus A, described by quantum statistical mechanics. The apparatus A=M+B involves a macroscopic measuring device M and a bath B. The requirements for ideality of the measurement allow us to specify the Hamiltonian of the isolated compound system S+M+B. The resulting dynamical equations may be solved for simple models. Conservation laws are shown to entail two independent relaxation mechanisms: truncation and registration. Approximations, justified by the large size of M and of B, are needed. The final density matrix $\hat{\cal D}(t_f)$ of S+A has an equilibrium form. It describes globally the outcome of a large set of runs of the measurement. The measurement problem, i.e., extracting physical properties of individual runs from $\hat{\cal D}(t_f)$, then arises due to the ambiguity of its splitting into parts associated with subsets of runs. To deal with this ambiguity, we postulate that each run ends up with a distinct pointer value $A_i$ of the macroscopic M. This is compatible with the principles of quantum mechanics. Born's rule then arises from the conservation law for the tested observable; it expresses the frequency of occurrence of the final indications $A_i$ of M in terms of the initial state of S. Von Neumann's reduction amounts to updating of information due to selection of $A_i$. We advocate the terms $q$-probabilities and $q$-correlations when analyzing measurements of non-commuting observables. These ideas may be adapted to different types of courses.

quant-ph↗
arXiv2016

A sub-ensemble theory of ideal quantum measurement processes

To elucidate ideal measurements, one must explain how individual events emerge from quantum theory which deals with statistical ensembles, and how different may end up with different final states. This so-called "measurement problem" is tackled with two guidelines. On the one hand, the dynamics of the macroscopic apparatus A coupled to the tested system S is described mathematically within a standard quantum formalism, where "q-probablities" remain devoid of interpretation. On the other hand, interpretative principles are introduced to account for the expected features of ideal measurements. Most of the five principles, which relate the quantum formalism to physical reality, are straightforward and refer to macroscopic variables. The process can be identified with a relaxation of S+A to thermodynamic equilibrium, not only for a large ensemble $\scriptE$ of runs but even for its sub-ensembles. The different mechanisms of quantum statistical dynamics that ensure the relaxation are exhibited. The additional information provided by the sub-ensembles remove Schrödinger's quantum ambiguity of the final density operator for $\scriptE$ which hinders its direct interpretation, and bring out a commutative behaviour of the pointer observable at the final time. The latter property supports the introduction of a last principle, needed to switch from the statistical ensembles and sub-ensembles described by quantum theory to individual experimental events. It amounts to identify some formal "q-probabilities" with ordinary frequencies, but only those which refer to the final pointer indications. The desired properties of ideal measurements, in particular the uniqueness of the result for each individual run and von Neumann's reduction, are thereby recovered with economic interpretations. The status of Born's rule involving both A and S is re-evaluated, and contextuality is made obvious.

quant-ph↗
arXiv2014

Lectures on dynamical models for quantum measurements

In textbooks, ideal quantum measurements are described in terms of the tested system only by the collapse postulate and Born's rule. This level of description offers a rather flexible position for the interpretation of quantum mechanics. Here we analyse an ideal measurement as a process of interaction between the tested system S and an apparatus A, so as to derive the properties postulated in textbooks. We thus consider within standard quantum mechanics the measurement of a quantum spin component $\hat s_z$ by an apparatus A, being a magnet coupled to a bath. We first consider the evolution of the density operator of S+A describing a large set of runs of the measurement process. The approach describes the disappearance of the off-diagonal terms ("truncation") of the density matrix as a physical effect due to A, while the registration of the outcome has classical features due to the large size of the pointer variable, the magnetisation. A quantum ambiguity implies that the density matrix at the final time can be decomposed on many bases, not only the one of the measurement. This quantum oddity prevents to connect individual outcomes to measurements, a difficulty known as the "measurement problem". It is shown that it is circumvented by the apparatus as well, since the evolution in a small time interval erases all decompositions, except the one on the measurement basis. Once one can derive the outcome of individual events from quantum theory, the so-called "collapse of the wave function" or the "reduction of the state" appears as the result of a selection of runs among the original large set. Hence nothing more than standard quantum mechanics is needed to explain features of measurements. The employed statistical formulation is advocated for the teaching of quantum theory.

quant-ph↗
arXiv2013

Understanding quantum measurement from the solution of dynamical models

The quantum measurement problem, understanding why a unique outcome is obtained in each individual experiment, is tackled by solving models. After an introduction we review the many dynamical models proposed over the years. A flexible and rather realistic model is introduced, describing the measurement of the $z$-component of a spin through interaction with a magnetic memory simulated by a Curie--Weiss magnet, including $N \gg1$ spins weakly coupled to a phonon bath. Initially prepared in a metastable paramagnetic state, it may transit to its up or down ferromagnetic state, triggered by its coupling with the tested spin, so that its magnetization acts as a pointer. A detailed solution of the dynamical equations is worked out. Conditions are found, which ensure that the process satisfies the features of ideal measurements. Various imperfections are discussed, as well as attempts of incompatible measurements. The first steps consist in the solution of the Hamiltonian dynamics for the spin-apparatus density matrix $D(t)$. On a longer time scale, the trend towards equilibrium of the magnet produces a final state $D(t_{\rm f})$ that involves correlations between the system and the indications of the pointer, thus ensuring registration. A difficulty lies in a quantum ambiguity: There exist many incompatible decompositions of the density matrix $\scriptD(t_{\rm f})$. This difficulty is overcome by dynamics due to suitable interactions within the apparatus. Any subset of runs thus reaches over a brief delay a stable state which satisfies the same hierarchic property as in classical probability theory. Standard quantum statistical mechanics alone appears sufficient to explain the occurrence of a unique answer in each run. Finally, pedagogical exercises are proposed while the statistical interpretation is promoted for teaching. [Abridged]

quant-ph↗
arXiv2007

The Quantum Measurement Process: Lessons from an Exactly Solvable Model

The measurement of a spin-$\half$ is modeled by coupling it to an apparatus, that consists of an Ising magnetic dot coupled to a phonon bath. Features of quantum measurements are derived from the dynamical solution of the measurement, regarded as a process of quantum statistical mechanics. Schrödinger cat terms involving both the system and the apparatus, die out very quickly, while the registration is a process taking the apparatus from its initially metastable state to one of its stable final states. The occurrence of Born probabilities can be inferred at the macroscopic level, by looking at the pointer alone. Apparent non-unitary behavior of the measurement process is explained by the arisal of small many particle correlations, that characterize relaxation.

quant-ph↗
arXiv2005

Phase transitions and quantum measurements

In a quantum measurement, a coupling $g$ between the system S and the apparatus A triggers the establishment of correlations, which provide statistical information about S. Robust registration requires A to be macroscopic, and a dynamical symmetry breaking of A governed by S allows the absence of any bias. Phase transitions are thus a paradigm for quantum measurement apparatuses, with the order parameter as pointer variable. The coupling $g$ behaves as the source of symmetry breaking. The exact solution of a model where S is a single spin and A a magnetic dot (consisting of $N$ interacting spins and a phonon thermal bath) exhibits the reduction of the state as a relaxation process of the off-diagonal elements of S+A, rapid due to the large size of $N$. The registration of the diagonal elements involves a slower relaxation from the initial paramagnetic state of A to either one of its ferromagnetic states. If $g$ is too weak, the measurement fails due to a ``Buridan's ass'' effect. The probability distribution for the magnetization then develops not one but two narrow peaks at the ferromagnetic values. During its evolution it goes through wide shapes extending between these values.

quant-ph↗
arXiv2004

Dynamics of a quantum measurement

We work out an exactly solvable hamiltonian model which retains all the features of realistic quantum measurements. In order to use an interaction process involving a system and an apparatus as a measurement, it is necessary that the apparatus is macroscopic. This implies to treat it with quantum statistical mechanics. The relevant time scales of the process are exhibited. It begins with a very rapid disappearance of the off-diagonal blocks of the overall density matrix of the tested system and the apparatus. Possible recurrences are hindered by the large size of the latter. On a much larger time scale the apparatus registers the outcome: Correlations are established between the final values of the pointer and the initial diagonal blocks of the density matrix of the tested system. We thus derive Born's rule and von Neumann's reduction of the state from the dynamical process.

quant-ph↗
arXiv2004

Geometry of the Casimir Effect

When the vacuum is partitioned by material boundaries with arbitrary shape, one can define the zero-point energy and the free energy of the electromagnetic waves in it: this can be done, independently of the nature of the boundaries, in the limit that they become perfect conductors, provided their curvature is finite. The first examples we consider are Casimir's original configuration of parallel plates, and the experimental situation of a sphere in front of a plate. For arbitrary geometries, we give an explicit expression for the zero-point energy and the free energy in terms of an integral kernel acting on the boundaries; it can be expanded in a convergent series interpreted as a succession of an even number of scatterings of a wave. The quantum and thermal fluctuations of vacuum then appear as a purely geometric property. The Casimir effect thus defined exists only owing to the electromagnetic nature of the field. It does not exist for thin foils with sharp folds, but Casimir forces between solid wedges are finite. We work out various applications: low temperature, high temperature where wrinkling constraints appear, stability of a plane foil, transfer of energy from one side of a curved boundary to the other, forces between distant conductors, special shapes (parallel plates, sphere, cylinder, honeycomb).

quant-ph↗
arXiv2004

The quantum measurement process in an exactly solvable model

An exactly solvable model for a quantum measurement is discussed which is governed by hamiltonian quantum dynamics. The $z$-component $\hat s_z$ of a spin-1/2 is measured with an apparatus, which itself consists of magnet coupled to a bath. The initial state of the magnet is a metastable paramagnet, while the bath starts in a thermal, gibbsian state. Conditions are such that the act of measurement drives the magnet in the up or down ferromagnetic state according to the sign of $s_z$ of the tested spin. The quantum measurement goes in two steps. On a timescale $1/\sqrt{N}$ the off-diagonal elements of the spin's density matrix vanish due to a unitary evolution of the tested spin and the $N$ apparatus spins; on a larger but still short timescale this is made definite by the bath. Then the system is in a `classical' state, having a diagonal density matrix. The registration of that state is a quantum process which can already be understood from classical statistical mechanics. The von Neumann collapse and the Born rule are derived rather than postulated.

cond-mat.stat-mech↗
arXiv2002

Curie-Weiss model of the quantum measurement process

A hamiltonian model is solved, which satisfies all requirements for a realistic ideal quantum measurement. The system S is a spin-$\half$, whose $z$-component is measured through coupling with an apparatus A=M+B, consisting of a magnet $\RM$ formed by a set of $N\gg 1$ spins with quartic infinite-range Ising interactions, and a phonon bath $\RB$ at temperature $T$. Initially A is in a metastable paramagnetic phase. The process involves several time-scales. Without being much affected, A first acts on S, whose state collapses in a very brief time. The mechanism differs from the usual decoherence. Soon after its irreversibility is achieved. Finally the field induced by S on M, which may take two opposite values with probabilities given by Born's rule, drives A into its up or down ferromagnetic phase. The overall final state involves the expected correlations between the result registered in M and the state of S. The measurement is thus accounted for by standard quantum statistical mechanics and its specific features arise from the macroscopic size of the apparatus.

cond-mat.stat-mech↗
arXiv2002

Thomson's formulation of the second law: an exact theorem and limits of its validity

Thomson's formulation of the second law - no work can be extracted from a system coupled to a bath through a cyclic process - is believed to be a fundamental principle of nature. For the equilibrium situation a simple proof is presented, valid for macroscopic sources of work. Thomson's formulation gets limited when the source of work is mesoscopic, i.e. when its number of degrees of freedom is large but finite. Here work-extraction from a single equilibrium thermal bath is possible when its temperature is large enough. This result is illustrated by means of exactly solvable models. Finally we consider the Clausius principle: heat goes from high to low temperature. A theorem and some simple consequences for this statement are pointed out.

cond-mat.stat-mech↗
arXiv2001

Quantum measurement as driven phase transition: An exactly solvable model

A model of quantum measurement is proposed, which aims to describe statistical mechanical aspects of this phenomenon, starting from a purely Hamiltonian formulation. The macroscopic measurement apparatus is modeled as an ideal Bose gas, the order parameter of which, that is, the amplitude of the condensate, is the pointer variable. It is shown that properties of irreversibility and ergodicity breaking, which are inherent in the model apparatus, ensure the appearance of definite results of the measurement, and provide a dynamical realization of wave-function reduction or collapse. The measurement process takes place in two steps: First, the reduction of the state of the tested system occurs over a time of order $\hbar/(TN^{1/4})$, where $T$ is the temperature of the apparatus, and $N$ is the number of its degrees of freedom. This decoherence process is governed by the apparatus-system interaction. During the second step classical correlations are established between the apparatus and the tested system over the much longer time-scale of equilibration of the apparatus. The influence of the parameters of the model on non-ideality of the measurement is discussed. Schrödinger kittens, EPR setups and information transfer are analyzed.

cond-mat.stat-mech↗