Abstract In quantum mechanics, we define the measuring system M in a selective measurement by two conditions. Firstly, when we define the measured system S as the system in which the non-selective measurement part acts, M is independent from the measured system S as a quantum system in the sense that any time-dependent process in the total system S + M is divisible into parts for S and M . Secondly, when we can separate S and M from each other without changing the unitary equivalence class of the state of S from that obtained by the partial trace of M , the eigenstate selection in the selective measurement cannot be realized. In order for such a system M to exist, we show that in one selective measurement of an observable of a quantum system S 0 of particles in S , there exists a negative entropy transfer from M to S that can be directly transformed into an amount of Helmholtz free energy of where T is the thermodynamic temperature of the system S . Equivalently, an extra amount of work, , is required to be done by the system M .
Abstract In quantum mechanics, we define the measuring system M in a selective measurement by two conditions. Firstly, when we define the measured system S as the system in which the non-selective measurement part acts, M is independent from the measured system S as a quantum system in the sense that any time-dependent process in the total system S + M is divisible into parts for S and M . Secondly, when we can separate S and M from each other without changing the unitary equivalence class of the state of S from that obtained by the partial trace of M , the eigenstate selection in the selective measurement cannot be realized. In order for such a system M to exist, we show that in one selective measurement of an observable of a quantum system S 0 of particles in S , there exists a negative entropy transfer from M to S that can be directly transformed into an amount of Helmholtz free energy of where T is the thermodynamic temperature of the system S . Equivalently, an extra amount of work, , is required to be done by the system M .
In quantum mechanics, we define the measuring system M in a selective measurement by two conditions. Firstly, when we define the measured system S as the system in which the non-selective measurement part acts, M is independent from the measured system S as a quantum system in the sense that any time-dependent process in the total system S+M is divisible into parts for S and M. Secondly, when we can separate S and M from each other without changing the unitary equivalence class of the state of S from that obtained by the partial trace of M, the eigenstate selection in the selective measurement cannot be realized. In order for such a system M to exist, we show that in one selective measurement of an observable of a quantum system S0 of particles in S, there exists a negative entropy transfer from M to S that can be directly transformed into an amount of Helmholtz free energy of kBT where T is the thermodynamic temperature of the system S. Equivalently, an extra amount of work, kBT, is required to be done by the system M.
Quantum measurement of a system can change its mean energy, as well as entropy. A selective measurement (classical or quantum) can be used as a "Maxwell's demon" to power a single-temperature heat engine, by decreasing the entropy. Quantum mechanically, so can a non-selective measurement, despite increasing the entropy of a thermal state. The maximal amount of work extractable following the measurement is given by the change in free energy: Wmax(non−)sel.=ΔEmeas−TBathΔSmeas(non−)sel.. This follows from the "generalized 2nd law for nonequilibrium initial state" [Hasegawa et. al, PLA (2010)], of which an elementary reduction to the standard law is given here. It is shown that Wmaxsel.−Wmaxnon−sel. equals the work required to reset the memory of the measuring device, and that no such resetting is needed in the non-selective case. Consequently, a single-bath engine powered by either kind of measurement works at a net loss of TBathΔSmeasnon−sel per cycle. By replacing the measurement by a reversible "premeasurement" and allowing a work source to couple to the system and memory, the cycle can be made completely reversible.
Quantum measurement of a system can change its mean energy as well as entropy. A selective measurement (classical or quantum) can be used as a 'Maxwell's demon' to power a single-temperature heat engine by decreasing the entropy. Quantum mechanically, so can a non-selective measurement, despite increasing the entropy of a thermal state. The maximal amount of work extractable following the measurement is given by the change in free energy: W (non-)sel max = Δ E meas − T Bath Δ S (non-)sel meas . This follows from the 'generalized 2nd law for nonequilibrium initial state' (Hasegawa et al 2010 Phys. Lett. A 374 1001–4), an elementary reduction of which to the standard law is given here. It is shown that W sel max − W non-sel max is equal to the work required for resetting the memory of the measuring device and that no such resetting is needed in the non-selective case. Consequently, a single-bath engine powered by either kind of measurement works at a net loss of T Bath Δ S non-sel meas per cycle. By replacing the measurement by a reversible 'pre-measurement' and allowing a work source to couple to the system and memory , the cycle can be rendered completely reversible.
The rather new field of quantum thermodynamics intends to derive thermodynamics from quantum mechanics, as well as to study thermodynamical behavior of quantum systems. It turns out that, indeed, thermalization can be obtained even for very small quantum systems due to the interaction with an appropriate environment. The consideration of an ensemble, as in statistical mechanics, is not necessary. However, the total system - consisting of system and environment - in quantum thermodynamics is usually considered to be closed. In particular, there exists no interaction with an external observer. Since quantum measurements typically influence the measured system, one may wonder which influence such an observation will have in the quantum thermodynamical setting. In this thesis, we study this influence of measurements in quantum thermodynamical systems. It turns out that various concepts known from statistical mechanics arise from such a "quantum thermodynamics under observation", although they do not have any meaning in the isolated quantum thermodynamical setting. The model discussed in the first part consists of a two-level system coupled to a large spin environment, the magnetization of which is measured periodically. This leads to quasi-classical trajectories for the system and - as in statistical mechanics - thermalization and temperature only have a meaning in terms of an ensemble average or a long-time average. Calculating these averages, we prove the ergodicity of the system. Interestingly, the entropy of the obtained attractor state depends on the time between the measurements and can be shown to be connected with the subjective lack of knowledge of the observer. In case of optimal measurements, the ensemble averaged state of the system under periodic measurements equals the state of the system in the quantum thermodynamical model without measurements. Introducing a detuning between system and environment, we find that the system can be prepared in almost arbitrary thermal states due to the periodic measurements. Even the creation of inversion becomes possible. As shown in numerical simulations, our results, which were analytically obtained for large environments, are even applicable to describe the behavior of the system coupled to small environments consisting of only a few spins. Besides measurements of the environment, we also study direct projective measurements of the system, as well as measurements with finite duration, obtaining that the dynamics of the system is almost independent of the concrete realization of the measurements. In the second part of the thesis, we discuss the meaning of temperature fluctuations in quantum thermodynamical systems. In statistical mechanics, there exist different approaches to the problem of temperature fluctuations, which lead to the well-known temperature fluctuation formula. The nature and operational accessibility of these fluctuations, however, remains rather unclear. We argue that according to quantum thermodynamics a system relaxes into a quasi-stationary thermal state and therefore the temperature does not fluctuate. However, since temperature is not an observable, it can only be determined indirectly. For modular systems, we show that the fluctuations of the temperature estimate obtained from a single energy measurement, indeed, are given by the temperature fluctuation formula derived in statistical mechanics, but only within a certain temperature range. The validity range of this formula increases with increasing size of the system. Thus, one can conclude that these "temperature fluctuations" can be interpreted as representing a fundamental limit for the instantaneous determination of temperature.
In classical mechanics, one assumes one may measure precise momentum and x positions. One may argue, however, that a measurement involves changing the momentum of a particle (i.e. an impulse hit) and that such a change does not occur either in an instant of time or at a point x. For example, a classical particle in a potential V(x) changes its velocity in a region dx, but at the same time is said to have a velocity of v(x) to leading order as dx shrinks to zero. One may argue that dx does not shrink to zero and may not be tiny with respect to an overall length scale. Given that an interaction involves a particle and a potential or another particle, one may argue that a finite dx should be associated with both. Thus a particle with constant momentum should have a tiny dx associated with it, possibly a function of p, but at the same time there is a constant probability for a measurement to take place at any x point.Thus we argue that there are two sets of probability schemes, one for dx and another for a larger length scale and constant probability. This should point to two entropy schemes, each associated with the type of measurement done and its analysis. For example, a two slit system may be used to measure the interaction probability. This gives rise to an interference probability. At the same time, many two slit system may be used to measure the center of mass position of the particle, ignoring the interference pattern. This gives rise to P(x)=1/L and a completely different Shannon’s entropy. Given that there is a dx probability scheme for a constant p associated with interactions, an associated entropy should appear when such an interaction occurs, but this now involves breaking the constant p and considering the associated dx probabilities. Thus there is both momentum and spatial entropy combined as in two slit interference. In the case of a bound state, there are numerous exp(ipx) eigenstates (due to interactions with V(x)), suggesting both internal p and x entropies, but overall one has a single energy eigenstate. In terms of energy one has a fixed state (although exp(-iEn t) suggests there are frequency pockets in time pointing towards a kind of time entropy. If one considers only interactions which change a particle from one En to another, a particle in a single state at T=temperature=0 has entropy 0 (even though the state itself has momentum and spatial entropy). Thus it seems entropy is selective depending on the type of interactions present and what eigenvalues they change. In other words, classically one counts states when computing entropy, but in quantum mechanics one may count states by defining states in terms of quantum numbers. One may consider a bound state to be a single certain state with eigenvalue En and consider entropy being created if interactions (photon/phonon absorption/emission) break the single En. In such a case, one may ignore the breaking of exp(ipx)s within a bound state which implies momentum entropy as well as the spatial entropy linked with even a single exp(ipx). In other words, one may select various different kinds of probabilities linked with different eigenvalues and their measurements when calculating Shannon’s entropy.
Abstract Quantum thermodynamics describes dynamic processes by means of the operators of entropy production P and time t . P and t do not commute. It exists the non-vanishing t–P commutator [ t , P ]=ik. Here the Boltzmann constant k has the physical meaning of a quantum of entropy. The t–P commutator immediately leads us to the t–P uncertainty relation ΔtΔP≥k/2. Hence the observables t and P are not sharply defined simultaneously. Similar uncertainty relations can also be expected for other pairs of conjugate variables with products of the physical meaning of an entropy. The free energy F and the reciprocal temperature (1/T) are the respective conjugate variables of an isolated system of many particles, which leads us to the F–(1/T) uncertainty relation ǀΔFǁΔ(1/T)ǀ≥k/2. It can be traced back to the t–P uncertainty relation mentioned above. In this way the Helmholtz free energy F and the temperatur T are introduced into quantum thermodynamics. The uncertainties ǀΔFǀ→0 and ǀΔTǀ→0 are negligible at low temperatures T→0, and quantum thermodynamics turns into the time-independent classical thermodynamics. Against this the uncertainties ǀΔFǀ→∞ and ǀΔTǀ→∞ grow unlimited at high temperatures T→∞, and classical thermodynamics loses its sense. In the limit of one particle the uncertainties cannot be neglected even at low temperatures. However a detailed discussion shows that the free energy f of a single particle vanishes within the whole range of temperatures T. This defines the particle entropy σ=ε/T=ak. The dimensionless entropy number a connects the particle energy ε=akT with the temperature T. The entropy number a of a single (s) independent particle can be calculated with the extended, temperature-dependent Schrödinger equation A s φ=aφ. Here A s=−(Λ 2/4π)∇2 means the dimensionless entropy operator describing the entropy number a and thus the particle entropy σ=ak. Λ is the thermal de Broglie wave length. Finally we calculate by means of quantized particle entropies σ the internal energy E, the Helmholtz free energy F, the entropy S, the chemical potential μ, and the equation of state of an ideal gas of N monatomic free particles in full agreement with classical thermodynamics. We also calculate the partition function q=V/Λ 3 of a single free particle within the volume V. Here Λ 3 is a small volume element taking into account the wave–particle dualism of a single free particle of mass m at temperature T. Extension to a system of N free particles leads us to a simple geometrical model and to the conclusion that an ideal gas of independent particles becomes instable below a critical temperature T C. T C corresponds to the critical temperature T BE of Bose–Einstein condensation.