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A Brief Introduction to Observational Entropy

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Abstract

In the past several years, observational entropy has been developed as both a (time-dependent) quantum generalization of Boltzmann entropy, and as a rather general framework to encompass classical and quantum equilibrium and non-equilibrium coarse-grained entropy. In this paper we review the construction, interpretation, most important properties, and some applications of this framework. The treatment is self-contained and relatively pedagogical, aimed at a broad class of researchers.

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Notes

  1. For example how does a system in a pure state have both zero von Neumann entropy and nonzero thermodynamic entropy? Does black hole entropy correspond to entanglement, or coarse-graining? If information is preserved in a closed system, and entropy is information, how does entropy increase? etc.

  2. A generalization of Boltzmann entropy to quantum systems was first proposed by von Neumann citing personal discussion with Eugene Wigner [9]. He did this after expressing dissatisfaction with the von Neumann entropy as a proper measure of thermodynamic entropy, since it is “computed from the perspective of an observer who can carry out all measurements that are possible in principle, i.e., regardless of whether they are macroscopic (for example, there every pure state has entropy 0, only mixtures have entropies greater than 0!).” Since then the concept, also called “coarse-grained” entropy has appeared in literature both in quantum [10,11,12,13] and classical [11, 14,15,16,17,18] systems, but has been studied systematically only very recently.

  3. Some authors [19] simply use the term “property” synonymously with “macrostate,” which is a useful conceptualization.

  4. For a single coarse-graining, we can also define the coarse-grained density matrix \({\hat{\rho }}_{\mathrm {cg}}=\sum _i p_i ({\hat{P}_i}/{V_i})\), and define \(S_{{{\mathcal {C}}}}\equiv S_{\mathrm {vN}}({\hat{\rho }}_{\mathrm {cg}})\). This type of definition is common in literature [11, 13]. However, for multiple coarse-grainings that do not commute, writing observational entropy like this is not possible.

  5. Entropy of an observable \(S_{{\mathcal {C}}}^{\mathrm {O}}\equiv S_{\mathrm {Sh}}(p_i)\) is sometimes also called entropy of partition  [24,25,26,27].

  6. Even more general definition would involve generalized measurements (POVMs) which are defined by a trace-preserving (\(\sum _i\hat{K}_i^\dag \hat{K}_i=\hat{I}\)) set of Kraus operators \({{\mathcal {C}}}=\{\hat{K}_i\}\), which defines \(p_{{\varvec{i}}}=\mathrm {tr}[\hat{K}_{i_n}\cdots \hat{K}_{i_1}{\hat{\rho }}\hat{K}_{i_1}^\dag \cdots \hat{K}_{i_n}^\dag ]\) and \(V_{{\varvec{i}}}=\mathrm {tr}[\hat{K}_{i_n}\cdots \hat{K}_{i_1}\hat{K}_{i_1}^\dag \cdots \hat{K}_{i_n}^\dag ]\). Properties (5) and (6) still hold [41].

  7. One can think about Eqs. (5) and (6) in combination, and ask whether performing more measurements will always lead to the minimal uncertainty given by the von Neumann entropy. Closer analysis reveals that this is not always possible: performing a measurement that does not commute with the density matrix might irreversibly destroy some information. And when the state of the system is finally projected onto a pure state, the observational entropy is set—no additional coarse-graining will decrease it further. This also shows that initial measurements are more important than those performed later, because the later ones can uncover only information which has not been destroyed by those preceding them [2].

  8. Note that \(V_{{\varvec{i}}}/\dim {\mathcal {H}}=\mathrm {tr}[\hat{P}_{i_n}\cdots \hat{P}_{i_1}\frac{\hat{I}}{\dim {\mathcal {H}}}\hat{P}_{i_1}\cdots \hat{P}_{i_n}]\).

  9. The triple \((\Gamma , {{\mathcal {C}}}, \rho )\) closely resembles the construction of probability space, where \(\Gamma\) is the sample space and \(\rho\) is the probability function, except that while \({{\mathcal {C}}}\) consists of disjoint events \(P_i\), it does not satisfy the defining properties of event space because in general \(\Gamma \notin {{\mathcal {C}}}\).

  10. In addition, in literature of coarse-grained free energies [30,31,32,33,34,35,36,37,38,39,40], one wants to find a free energy functional that depends on local variables (such as energy, particle density, magnetization \(\ldots\)) and either study its dynamics, or critical behavior using methods of renormalization group. The current framework allows for rigorously defining these functionals, which seem to be equivalently described by Observational entropy with local coarse-grainings, for both classical and fully quantum systems.

  11. A local coarse-graining is equivalent to a sequence of coarse-grainings: defining trivial coarse-graining \({{\mathcal {C}}}_{\hat{I}}=\{\hat{I}\}\) which represents a situation where no measurement is performed, the definition can be written in terms of sequence of local measurements performed on different subsystems, as \(S_{{{\mathcal {C}}}_A \otimes \ldots \otimes {{\mathcal {C}}}_C}=S_{{\tilde{{{\mathcal {C}}}}}_A,\dots ,{\tilde{{{\mathcal {C}}}}}_C}\), where \({\tilde{{{\mathcal {C}}}}}_A={{\mathcal {C}}}_A\otimes {{\mathcal {C}}}_{\hat{I}} \otimes \cdots \otimes {{\mathcal {C}}}_{\hat{I}}\), \(\dots\), \({\tilde{{{\mathcal {C}}}}}_C={{\mathcal {C}}}_{\hat{I}}\otimes {{\mathcal {C}}}_{\hat{I}} \otimes \cdots \otimes {{\mathcal {C}}}_C\).

  12. Observational entropy is therefore quite unlike the von Neumann entropy, which remains constant in an isolated system. See Ref. [5] for a detailed study of fluctuations in one type of observational entropy.

  13. Also known as the surface entropy, or the Boltzmann entropy, although in our framework, since we consider general coarse-grainings, this would be called the Boltzmann entropy with energy coarse-graining. See  for an alternative definition of microcanonical entropy—the volume entropy—and references therein.

  14. The particle coarse-graining is defined as \({{\mathcal {C}}}_{\hat{N}}=\{\hat{P}_n\}\), where \(\hat{P}_n\) is a projector onto subspace of n particles, and energy coarse-graining as \({{\mathcal {C}}}_E =\{\hat{P}_E\}\), where \(\hat{P}_E=\sum _{E\le \tilde{E} <E+{\Delta \! E}}|{\tilde{E}}\rangle \langle {\tilde{E}}|\) is a projector onto subspace of wave functions within an energy shell \([E,E+{\Delta \! E})\).

  15. As well as when inserted with a microcanonical state \({\hat{\rho }}=\frac{1}{Z}\sum _{E\le {{\tilde{E}}}<E+{\Delta \! E}}|{\tilde{E}}\rangle \langle \tilde{E}|\). The volume microcanonical entropy  is obtained by inserting \({\hat{\rho }}=\frac{1}{Z}\sum _{0\le {{\tilde{E}}}<E}|{\tilde{E}}\rangle \langle \tilde{E}|\), the canonical by \({\hat{\rho }}=\frac{1}{Z}e^{-\beta \hat{H}}\), and grandcanonical by \({\hat{\rho }}=\frac{1}{Z}e^{-\beta (\hat{H}-\mu \hat{N})}\).

  16. The spatial volume \({\mathcal {V}}\) is assumed to be fixed implicitly here, but in general it might not be, for example when considering a work-extraction protocol using a piston.

  17. Equation (17) can be generalized to any number of conserved observables, see Eq. (23).

  18. Here we assume multipartite system \({\mathcal {H}}={\mathcal {H}}_1\otimes \cdots {\mathcal {H}}_m\). Each \({\mathcal {H}}_i\) is a space of all quantum states that can occur within a spatial region of volume \({\mathcal {V}}_i\), and \({{\mathcal {C}}}_{\hat{N}_i}\) and \({{\mathcal {C}}}_{E_i}\) correspond to a particle and energy measurement of this spatial region.

  19. Proven to be an approximate upper bound in weakly interacting systems [2].

  20. Additional exact bounds are given by Eqs. (5), (6). When local Hamiltonians and local particle operators commute, \([\hat{H}_i,\hat{N}_i]=0\), Eq. (6) provides two bounds: one connected to observational entropy with just local particle numbers, and one with local energies.

  21. For example the first half of ice and the first half of water would together form the first subsystem, while the rest would form the second subsystem.

  22. Although simulations [2] show that integrable case also converges, just not that well.

  23. Denoted \(S_x\) in [2] where its time evolution is illustrated in Fig. 2.

  24. The case of \({\Delta \! E}=0\) has been studied in detail in [2] under the name of “Factorized Observational entropy” or FOE for short, and denoted \(S_F\).

  25. This entropy has been studied closely in the classical case [3] where it has been denoted \(S_F\), and where also the quantum equivalent is mentioned for the first time. Since in quantum case, this definition behaves the same (in its time evolution in particular) as (2b) for \({\Delta \! E}=0\), apart from giving non-zero value for local energy eigenstates, we refer reader to [2] for its detailed properties.

  26. This entropy has been studied in detail both in classical [3] and quantum case [2] where it was denoted \(S_{xE}\).

  27. Time evolution of this entropy has been studied in the quantum case in Appendix H of [2], where it was denoted \(S_{Ex}\), but it was realized only later in [3] that it has a meaningful interpretation. Classically, \(S_{xE}\) and \(S_{Ex}\) are identical.

  28. This entropy has been studied in detail by Strasberg and Winter in [4, 6].

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Acknowledgements

This research was supported by the Foundational Questions Institute (FQXi.org), of which AA is Associate Director, and by the Faggin Presidential Chair Fund. D\v S acknowledges additional funding by the Institute for Basic Science in Korea (IBS-R024-Y2 and IBS-R024-D1).

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Appendix: Physically Relevant Coarse-Grainings

Appendix: Physically Relevant Coarse-Grainings

In this appendix we collect and/or introduce, and discuss, a number of physically relevant coarse-grainings, generally with some relevance to thermodynamics, some of which have been studied in detail before [1,2,3,4, 6, 43].

In terms of generic properties, any observational entropy generated by an observable (which is typically the case of (1) below) that is conserved in the system (i.e., commutes with the Hamiltonian) will be constant during the time evolution [2]. Moreover, any entropy that consists solely of local coarse-grainings (which is the case of (2) below) will be additive on independent systems as per Eq. (13), and bounded as per (15). All convergence in the long-time limit discussed below in points (2) and (3) assumes particle conserving non-integrableFootnote 22 Hamiltonian with short range interactions, so that particles tend to thermalize well and the interaction energy between the subsystems is negligible. The convergence holds up to some corrections due to finite particle number and finite size-effect effects. These limits are approximate upper bounds for the non-equilibrium entropies, which is why we say “grows to.” The exact upper bounds follow from Eq. (6). For simplicity we also consider non-degenerate Hamiltonian (both globally and locally), which means that each energy has a unique associated particle number, so instead of common eigenstate \(|n,E\rangle\) of the particle operator and the Hamiltonian we can write simply \(|E\rangle\). The two observational entropies discussed in the main body of this paper are (1c) and (2c).

1.1 1a: Global Particle Number Coarse-Graining

Defining \({{\mathcal {C}}}_{\hat{N}}=\{\hat{P}_n\}\), where \(\hat{P}_n\) is a projector onto subspace of n particles,

$$\begin{aligned} S_{{{\mathcal {C}}}_{\hat{N}}}=-\sum _np_n\ln \frac{p_{n}}{V_{n}} \end{aligned}$$
(A1)

measures the uncertainty about the particle number in the system.

1.2 1b: Global Energy Coarse-Graining

Defining \({{\mathcal {C}}}_E\equiv C_{\hat{H}^{({\Delta \! E})}}=\{\hat{P}_E\}\), where \(\hat{P}_E=\sum _{E\le {\tilde{E}} <E+{\Delta \! E}}|{\tilde{E}}\rangle \langle {\tilde{E}}|\) is a projector onto subspace of wave functions within an energy shell \([E,E+{\Delta \! E})\) (and \(\hat{H}^{({\Delta \! E})}=\sum _E E \hat{P}_E\) is the coarse-grained Hamiltonian),

$$\begin{aligned} S_{{{\mathcal {C}}}_E}=-\sum _E p_E \ln \frac{p_E}{V_E} \end{aligned}$$
(A2)

measures the equilibrium thermodynamic entropy of a system with a fixed number number of particles.

Details \({\Delta \! E}\) is the resolution in energy of the measuring apparatus. If restricted to a Hilbert space with a fixed number of particles, for small but non-zero \({\Delta \! E}\) this entropy gives microcanonical entropy for both energy eigenstates and a microcanonical state, and it (approximately) gives Gibbs entropy \(\ln Z-\beta \langle E\rangle\) for the Gibbs state \(\frac{1}{Z}e^{-\beta \hat{H}}\). The case of \({\Delta \! E}>0\) cannot be applied to Hilbert space which includes variable number of particles, because the energy subspace would include wave-functions with any particle numbers, and would be typically infinite in size. For \({\Delta \! E}=0\) it does not have this problem (since energy eigenstate uniquely determines the particle number in common particle-conserving Hamiltonians), but it gives zero for energy eigenstates.

1.3 1c: Global Particle Number with Global Energy Coarse-Graining

$$\begin{aligned} S_{{\mathrm {th}}}\equiv S_{{{\mathcal {C}}}_{\hat{N}},{{\mathcal {C}}}_E}=-\sum _{n,E} p_{nE} \ln \frac{p_{nE}}{V_{nE}} \end{aligned}$$
(A3)

measures the equilibrium thermodynamic entropy.

Details Macrostates now distinguish both energy and the number of particles, which means it can be used for systems with a variable number of particles. For common particle-conserving Hamiltonians, the case of \({\Delta \! E}=0\) reduces to \(S_{{{\mathcal {C}}}_E}\). It gives \(S_{{{\mathcal {C}}}_{\hat{N}},{{\mathcal {C}}}_E}(|E\rangle )=\ln V_{nE}=S_{\mathrm {micro}}(E,{\mathcal {V}},n)\) microcanonical entropy, for a global energy eigenstate. \({\mathcal {V}}\) is the spatial volume of the system.

1.4 2a: Local Particle Number Coarse-GrainingFootnote 23

$$\begin{aligned} S_{{{\mathcal {C}}}_{\hat{N}_1}\otimes \cdots \otimes {{\mathcal {C}}}_{\hat{N}_m}}=-\sum _{{\varvec{n}}}p_{{\varvec{n}}}\ln \frac{p_{{{\varvec{n}}}}}{V_{{{\varvec{n}}}}} \end{aligned}$$
(A4)

where \({{\varvec{n}}}=(n_1,\dots ,n_m)\) are energies of the subsystems, and measures how uniformly are particles distributed over the subsystems.

Details It grows to (1a) in the long-time limit: when particles spread uniformly throughout the system, they fill uniformly every particle shell.

1.5 2b: Local Energy Coarse-GrainingFootnote 24

$$\begin{aligned} S_{{{\mathcal {C}}}_{E_1}\otimes \cdots \otimes {{\mathcal {C}}}_{E_m}}=-\sum _{{\varvec{E}}}p_{{\varvec{E}}}\ln \frac{p_{{{\varvec{E}}}}}{V_{{{\varvec{E}}}}} \end{aligned}$$
(A5)

where \({{\varvec{E}}}=(E_1,\dots ,E_m)\) are energies in the subsystems, how uniformly is energy distributed over the subsystems.

Details For local energy eigenstates \(S_{{{\mathcal {C}}}_{E_1}\otimes \cdots \otimes {{\mathcal {C}}}_{E_m}}(|\tilde{E}_1\rangle \cdots |{\tilde{E}}_m\rangle )=\sum _{i=1}^m\ln V_{{\tilde{E}}_i}\). Despite from what it may seem from (1b), for \({\Delta \! E}>0\), \(\ln V_{{\tilde{E}}_i}\) does not describe thermodynamic entropy in each subsystem, because in a non-equilibrium system, number of particles in each subsystem typically varies, even though the total number of particles may be conserved. Macrostate \({\mathcal {H}}_{E_i}\) contains all states with energy \(E_i\), even though these states might have different particle numbers. In case of Hamiltonians which conserve particles locally, each local eigenstate uniquely determines its particle number, which implies that the case of \({\Delta \! E}=0\) is identical to (2c), having all of its dynamical properties. However, for \({\Delta \! E}=0\), \(S_{{{\mathcal {C}}}_{E_1}\otimes \cdots \otimes {{\mathcal {C}}}_{E_m}}(|\tilde{E}_1\rangle \cdots |{\tilde{E}}_m\rangle )=0\), which is undesirable for a physically meaningful thermodynamic entropy. It grows to (1b) in the long-time limit.

1.6 2c: Local Particle Number with Local Energy Coarse-GrainingFootnote 25

$$\begin{aligned} S_{{\mathrm {th}}}^{\mathrm {non-eq.}}\equiv S_{{{\mathcal {C}}}_{\hat{N}_1}\otimes \cdots \otimes {{\mathcal {C}}}_{\hat{N}_m},{{\mathcal {C}}}_{E_1}\otimes \cdots \otimes {{\mathcal {C}}}_{E_m}}=-\sum _{{{\varvec{n}}},{{\varvec{E}}}} p_{{{\varvec{n}}}{{\varvec{E}}}}\ln \frac{p_{{{\varvec{n}}}{{\varvec{E}}}}}{V_{{{\varvec{n}}}{{\varvec{E}}}}} \end{aligned}$$
(A6)

measures non-equilibrium thermodynamic entropy of the system.

Details At some intermediate time t (when the system has only partially equilibrated) its value can be interpreted as the equilibrium thermodynamic entropy the system would attain in the long-time limit if (hypothetically) starting from time t the subsystems were not allowed to exchange either energy or particles [3]. For \({\Delta \! E}>0\), it gives \(S_{{{\mathcal {C}}}_{\hat{N}_1}\otimes \cdots \otimes {{\mathcal {C}}}_{\hat{N}_m},{{\mathcal {C}}}_{E_1}\otimes \cdots \otimes {{\mathcal {C}}}_{E_m}}(|E_1\rangle \cdots |E_m\rangle )=\sum _{i=1}^m\ln V_{nE}=\sum _{i=1}^mS_{\mathrm {micro}}(E_i,{\mathcal {V}}_i,n_i)\), the sum of local microcanonical entropies, for a local energy eigenstate. \({\mathcal {V}}_i\) denote the local spatial volumes. It grows to (1c) in the long-time limit.

1.7 3a: Local Particle Number then Global Energy Coarse-GrainingFootnote 26

$$\begin{aligned} S_{{{\mathcal {C}}}_{\hat{N}_1}\otimes \cdots \otimes {{\mathcal {C}}}_{\hat{N}_m},{{\mathcal {C}}}_{E}}=-\sum _{{{\varvec{n}}},E} p_{{{\varvec{n}}}E}\ln \frac{p_{{{\varvec{n}}}E}}{V_{{{\varvec{n}}}E}} \end{aligned}$$
(A7)

is a different type of non-equilibrium thermodynamic entropy of the system.

Details Is not additive. At some intermediate time t, its value can be interpreted as the equilibrium thermodynamic entropy the system would attain in the long-time limit if (hypothetically) starting from time t the subsystems were allowed to exchange energy but not particles [3]. It is upper bounded by (2a), and it grows to (1c).

1.8 3b: Global Energy Then Local Particle Number Coarse-GrainingFootnote 27

$$\begin{aligned} S_{{{\mathcal {C}}}_{E},{{\mathcal {C}}}_{\hat{N}_1}\otimes \cdots \otimes {{\mathcal {C}}}_{\hat{N}_m}}=-\sum _{E,{{\varvec{n}}}} p_{E{{\varvec{n}}}}\ln \frac{p_{ E{{\varvec{n}}}}}{V_{E{{\varvec{n}}}}} \end{aligned}$$
(A8)

is similar in behavior to (3a), but differs when quantum effects become significant, such as at low energies and when subsystems are small so that effects of non-commutation between \(\hat{N_i}\) and \({\hat{H}}\) intensify.

Details It is upper bounded by (1b), and it grows to (1c). For \({\Delta \! E}=0\), it is identical to (1b) and (1c).

1.9 4: Combination of Arbitrary Local and Local Energy Coarse-GrainingFootnote 28

$$\begin{aligned} S_{{{\mathcal {C}}}\otimes {{\mathcal {C}}}_{\hat{E}_1}\otimes \cdots \otimes {{\mathcal {C}}}_{E_m}}=-\sum _{i, {{\varvec{E}}}} p_{i {{\varvec{E}}}}\ln \frac{p_{i {{\varvec{E}}}}}{V_{i {{\varvec{E}}}}} \end{aligned}$$
(A9)

is the total entropy of a small well-controlled subsystem plus large bath(s), with applications in open system non-equilibrium thermodynamics.

Details Change in this entropy defines entropy production which, unlike the formulation based on von Neumann entropy, does not depend explicitly on the temperature(s) of the bath(s). Moreover, with this it is possible to define work as the part of the “useful” internal energy that can be recovered from the system, while heat is the part of internal energy that is irreversibly lost.

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Šafránek, D., Aguirre, A., Schindler, J. et al. A Brief Introduction to Observational Entropy. Found Phys 51, 101 (2021). https://doi.org/10.1007/s10701-021-00498-x

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