Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

On the Work Required for Selective Projective Quantum Measurement

Eiji Konishi

Published Oct 3, 2016• 0 citations
OpenAlex

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 $k_BT$ where $T$ is the thermodynamic temperature of the system $S$. Equivalently, an extra amount of work, $k_BT$, is required to be done by the system $M$.

Related papers

Work required for selective quantum measurement

Eiji Konishi

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 .

arXiv:1610.00757•Journal of Statistical Mechanics Theory and Experiment•Jun 1, 2018•4 citations

Thermodynamics of projective quantum measurements

Noam Erez A

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.

Physica Scripta•Nov 1, 2012•10 citations

Thermodynamics of quantum measurements

Noam Erez A

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: $W_{max}^{(non-)sel.}=ΔE_{meas}-T_{Bath}ΔS_{meas}^{(non-)sel.}$. 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 $W_{max}^{sel.}-W_{max}^{non-sel.}$ 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 $T_{Bath}ΔS_{meas}^{non-sel}$ 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.

arXiv:1011.1020•Nov 3, 2010•1 citation

Selective Entropy Based on Measurement, Conjugate Variables and Quantum Numbers

Francesco R. Ruggeri

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.

•Jun 9, 2023•0 citations

Relation between Quantum Thermodynamics and Classical Thermodynamics

Herbert W. Zimmermann

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.

Zeitschrift für Physikalische Chemie•Nov 24, 2010•15 citations

Measurements in Quantum Optics

U. Schilling

This thesis fathoms the possibilities and limitations of measurements in quantum optics. After a short introduction to the general aspects of measurements in quantum mechanics, it presents three examples of a measurement process in concrete applications to quantum optics. For one, while operators describing quantities of interest might have a simple theoretical form, it is often hard to actually implement a measurement of their expectation values experimentally. This situation is encountered in the first part which is devoted to the measurement of quantum statistical properties of a light beam. In contrast to classical optics, these statistics are not fully determined by first- and second-order correlations. Instead, higher-order correlations, usually described by non-Hermitian operators, are necessary to characterize the quantum state completely. A particular example is the N-photon Fock state, for which only a measurement of the Nth-order correlations allows for a full characterization. Here, we present a new method how to measure an important class of Nth-order correlations for arbitrary N in a single spatial mode with two polarizations with limited experimental resources. The second part is again centered around the realization of specific measurement operators for photon correlation measurements. However, in contrast to the previous chapter, these measurements are not an end in their own, but are designed to create specific quantum states as a consequence of the measurement process: A suitable measurement of photons emitted by a number of atoms allows to transfer the atoms into long-lived entangled states. We find that with this method it is possible to entangle two remote atomic qubits to an arbitrary and well-defined degree. Furthermore, we show how to generate two families of quantum states in an arbitrary number of remote atomic qubits: one family consists of all -- symmetric and non-symmetric -- total angular momentum eigenstates in N remote qubits, the other is a family of cluster states. It is also systematically investigated whether the technique of quantum state engineering by projective measurements allows for the creation of any arbitrary state. We find explicit solutions for the case of two and three qubits, while it is shown that no solution exist for N > 6. We conclude the section by proving that with some changes in the setup, this measurement-based quantum state engineering technique allows to create the same states among the scattered photons themselves rather than among the emitting atoms. The last part is devoted to the amount of information one is able to gain about a given system when performing a measurement. The corresponding limits of information are investigated at the example of the duality that arises in a two-way interferometer between the visibility of the interference pattern and the knowledge about the path of the interfering object. We show that there exist correlations between the phase difference the interfering object acquires on its way through the interferometer and the amount of which-way information retrievable from certain observables of the which-way detector. In this way, the which-way information becomes a phase-dependent quantity. In particular, we find that for certain values of the phase shift, the amount of extractable which-way information can be larger than allowed by phase-independent measurement. This property is put to use to find the optimal observable for the which-way detector readout in dependence on the phase of the interfering object. Following this strategy, we are able to gain more which-way information than previously thought possible.

•Oct 24, 2011•0 citations
In quantum mechanics, we define the measuring system MMM in a selective measurement by two conditions. Firstly, when we define the measured system SSS as the system in which the non-selective measurement part acts, MMM is independent from the measured system SSS as a quantum system in the sense that any time-dependent process in the total system S+MS+MS+M is divisible into parts for SSS and MMM. Secondly, when we can separate SSS and MMM from each other without changing the unitary equivalence class of the state of SSS from that obtained by the partial trace of MMM, the eigenstate selection in the selective measurement cannot be realized. In order for such a system MMM to exist, we show that in one selective measurement of an observable of a quantum system S0S_0S0​ of particles in SSS, there exists a negative entropy transfer from MMM to SSS that can be directly transformed into an amount of Helmholtz free energy of kBTk_BTkB​T where TTT is the thermodynamic temperature of the system SSS. Equivalently, an extra amount of work, kBTk_BTkB​T, is required to be done by the system MMM.
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 .
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.
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.W_{max}^{(non-)sel.}=ΔE_{meas}-T_{Bath}ΔS_{meas}^{(non-)sel.}Wmax(non−)sel.​=ΔEmeas​−TBath​ΔSmeas(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.W_{max}^{sel.}-W_{max}^{non-sel.}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−selT_{Bath}ΔS_{meas}^{non-sel}TBath​ΔSmeasnon−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.
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.
This thesis fathoms the possibilities and limitations of measurements in quantum optics. After a short introduction to the general aspects of measurements in quantum mechanics, it presents three examples of a measurement process in concrete applications to quantum optics. For one, while operators describing quantities of interest might have a simple theoretical form, it is often hard to actually implement a measurement of their expectation values experimentally. This situation is encountered in the first part which is devoted to the measurement of quantum statistical properties of a light beam. In contrast to classical optics, these statistics are not fully determined by first- and second-order correlations. Instead, higher-order correlations, usually described by non-Hermitian operators, are necessary to characterize the quantum state completely. A particular example is the N-photon Fock state, for which only a measurement of the Nth-order correlations allows for a full characterization. Here, we present a new method how to measure an important class of Nth-order correlations for arbitrary N in a single spatial mode with two polarizations with limited experimental resources. The second part is again centered around the realization of specific measurement operators for photon correlation measurements. However, in contrast to the previous chapter, these measurements are not an end in their own, but are designed to create specific quantum states as a consequence of the measurement process: A suitable measurement of photons emitted by a number of atoms allows to transfer the atoms into long-lived entangled states. We find that with this method it is possible to entangle two remote atomic qubits to an arbitrary and well-defined degree. Furthermore, we show how to generate two families of quantum states in an arbitrary number of remote atomic qubits: one family consists of all -- symmetric and non-symmetric -- total angular momentum eigenstates in N remote qubits, the other is a family of cluster states. It is also systematically investigated whether the technique of quantum state engineering by projective measurements allows for the creation of any arbitrary state. We find explicit solutions for the case of two and three qubits, while it is shown that no solution exist for N > 6. We conclude the section by proving that with some changes in the setup, this measurement-based quantum state engineering technique allows to create the same states among the scattered photons themselves rather than among the emitting atoms. The last part is devoted to the amount of information one is able to gain about a given system when performing a measurement. The corresponding limits of information are investigated at the example of the duality that arises in a two-way interferometer between the visibility of the interference pattern and the knowledge about the path of the interfering object. We show that there exist correlations between the phase difference the interfering object acquires on its way through the interferometer and the amount of which-way information retrievable from certain observables of the which-way detector. In this way, the which-way information becomes a phase-dependent quantity. In particular, we find that for certain values of the phase shift, the amount of extractable which-way information can be larger than allowed by phase-independent measurement. This property is put to use to find the optimal observable for the which-way detector readout in dependence on the phase of the interfering object. Following this strategy, we are able to gain more which-way information than previously thought possible.