# Conditioning, Correlation and Entropy Generation in Maxwell’s Demon

## Abstract

**:**

## 1. Introduction and Background

**Initial:**The cycle begins with the molecule moving freely in the unpartitioned cylinder and the memory in the “0” state.

**A.****Insertion:**The partition is inserted, trapping the molecule in either the left or right half of the cylinder. This process is considered to be both energetically and entropically benign; the partition is considered to be frictionless, and the entropy reduction that results from halving of the molecule’s accessible volume is exactly balanced by the increase that results from uncertainty about which volume (left or right) holds the molecule. (This assignment of entropy has been controversial, as will be discussed in Section 4.1.). Both the memory and environment are assumed to be unaffected.**B.****Measurement:**A reversible position measurement senses the location of the molecule and registers the outcome in the memory (“L” for left half and “R” for right half, respectively) bringing the states of the memory and molecule into correlation. The molecule and environment are assumed to be unaffected. The local entropy of the memory is increased, since it reflects the randomness of the molecule position, but the entropy of the Demon as a whole is unchanged, because of the correlation created between the molecule and memory states.**C.****Expansion:**One of the two pistons is selected for insertion based on the memory state (right piston for “L” outcome and left piston for “R” outcome). The piston is slid up to the partition, the partition is removed, and the molecule is allowed to push the piston back to the end of the cylinder and do reversible work in an isothermal expansion process. The molecule is returned to its initial state in this process, and its local entropy is unchanged (cf. Step A above), while the loss of correlation between the molecule and the memory increases the entropy of the Demon as a whole by ${k}_{B}ln\left(2\right)$. The environment entropy is, by classical thermodynamics, reduced by precisely this same amount as a result of the heat delivered to the molecule during the volume-doubling isothermal expansion, so there is no total entropy change. The memory is unaffected in this step. We should note that although Fahn took the piston insertion, partition removal, and gas expansion to three distinct steps in his work, we have combined this sequence of three processes into a single “Expansion" step for consistency with our treatment of a generalized Demon.**D.****Memory Reset:**The Demon’s memory is unconditionally reset, reducing its local entropy—and the entropy of the Demon as a whole—by ${k}_{B}ln\left(2\right)$ and returning the Demon to its initial state. Fahn attributes an equal and opposite entropy increase to the environment, citing conservation of phase-space volume and its consistency with the environmental cost of “Landauer erasure” (${k}_{B}ln\left(2\right)$ Joule per Kelvin-bit). This completes the Demon’s cycle.

**Table 1.**Entropy changes throughout the Demon’s cycle, according to Fahn’s analysis (adapted from [8]).

Step | Demon $\mathbf{\Delta}{\mathit{S}}^{\mathcal{D}}$ | Bath $\mathbf{\Delta}{\mathit{S}}^{\mathcal{B}}$ | Total $\mathbf{\Delta}{\mathit{S}}_{\mathit{tot}}$ |
---|---|---|---|

A. Insertion | 0 | 0 | 0 |

B. Measurement | 0 | 0 | 0 |

C. Expansion | +1 | −1 | 0 |

D. Memory Reset | −1 | +1 | 0 |

Cycle Total | 0 | 0 | 0 |

## 2. Referential Information and Entropy Generation in Physical Processes

#### 2.1. Referential Physical Information

**Figure 1.**Generic multicomponent physical system, built around an information-processing artifact, $\mathcal{A}$, and heat bath $\mathcal{B}$, used here to obtain bounds on the dissipative costs of processing information about $\mathcal{R}$ in $\mathcal{A}$. The composite $\mathcal{IE}$ is globally closed.

**States**: The quantum state of the global system ($\mathcal{IE}$) at any time, t, is described by a density operator, ${\widehat{\rho}}^{\mathcal{IE}}\left(t\right)\equiv \widehat{\rho}\left(t\right)$, defined on the universal Hilbert space $\mathcal{H}={\mathcal{H}}^{\mathcal{I}}\otimes {\mathcal{H}}^{\mathcal{E}}$. Local states of subsystems are described by reduced density operators obtained via the partial trace rule, e.g., so the local state of $\mathcal{RA}$ is:$${\widehat{\rho}}^{\mathcal{RA}}\left(t\right)=T{r}_{\mathcal{WE}}\left[\widehat{\rho}\left(t\right)\right]$$**Dynamics**: Consistent with Schrodinger dynamics, the global state, $\widehat{\rho}\left(t\right),$ is assumed to evolve unitarily as:$$\widehat{\rho}\left({t}_{f}\right)=\widehat{U}({t}_{i},{t}_{f})\widehat{\rho}\left({t}_{i}\right)\widehat{U}{({t}_{i},{t}_{f})}^{\u2020}$$$$\widehat{U}({t}_{i},{t}_{f})=\mathcal{T}\left(\right)open="["\; close="]">exp\left(\right)open="("\; close=")">-\frac{i}{\hslash}{\int}_{{t}_{i}}^{{t}_{f}}\widehat{H}\left({t}^{\prime}\right)d{t}^{\prime}$$

**Information**: The amount of information about $\mathcal{R}$ in $\mathcal{A}$ at time t is quantified by the quantum mutual information (or correlation entropy):$${I}^{\mathcal{RA}}\left(t\right)=S({\widehat{\rho}}^{\mathcal{R}}\left(t\right))+S({\widehat{\rho}}^{\mathcal{A}}\left(t\right))-S({\widehat{\rho}}^{\mathcal{RA}}\left(t\right))$$**Total Entropy**: The total entropy for a pair of subsystems is defined at time t as the sum of local entropies. Three such quantities will, in the context of the Demon, be of interest in this work:$${S}_{tot}^{\mathcal{RA}}\left(t\right)=S({\widehat{\rho}}^{\mathcal{R}}\left(t\right))+S({\widehat{\rho}}^{\mathcal{A}}\left(t\right))$$$${S}_{tot}^{\mathcal{AB}}\left(t\right)=S({\widehat{\rho}}^{\mathcal{A}}\left(t\right))+S({\widehat{\rho}}^{\mathcal{B}}\left(t\right))$$$${S}_{tot}\left(t\right)=S({\widehat{\rho}}^{\mathcal{RA}}\left(t\right))+S({\widehat{\rho}}^{\mathcal{E}}\left(t\right))$$

#### 2.2. Physical Processes in the Referential Approach

## 3. Maxwell’s Demon with a Generalized Szilard Engine

#### 3.1. Description of the Demon in the Referential Approach

#### 3.2. The Demon’s Cycle

**Initial:**The cycle begins at time $t=0$ with the memory and engine system in standard states, ${\widehat{\rho}}_{0}^{\mathcal{M}}$ and ${\widehat{\rho}}_{0}^{\mathcal{S}}$, respectively. The bath is in a thermal state, ${\widehat{\rho}}_{th}^{\mathcal{B}}$, and all three subsystems are uncorrelated. The initial states of $\mathcal{D}$, $\mathcal{B}$ and $\mathcal{DB}$ are thus:$${\widehat{\rho}}_{0}^{\mathcal{D}}={\widehat{\rho}}_{0}^{\mathcal{M}}\otimes {\widehat{\rho}}_{0}^{\mathcal{S}}$$$${\widehat{\rho}}_{0}^{\mathcal{B}}={\widehat{\rho}}_{th}^{\mathcal{B}}$$$${\widehat{\rho}}_{0}^{\mathcal{DB}}={\widehat{\rho}}_{0}^{\mathcal{M}}\otimes {\widehat{\rho}}_{0}^{\mathcal{S}}\otimes {\widehat{\rho}}_{th}^{\mathcal{B}}$$

**A.****Insertion:**In the “insertion” step ($0<t\le {t}_{A}$), the self-Hamiltonian of the Demon is unconditionally varied from ${\widehat{H}}_{0}^{\mathcal{D}}$ to ${\widehat{H}}_{A}^{\mathcal{D}}$. We do not make any a priori assumptions about the form of either Hamiltonian, but require that any potential barriers included in ${\widehat{H}}_{A}^{\mathcal{D}}$ are finite if arbitrarily high. (This is important for reasons to be discussed in Section 4.1.) We do not assume that this potential “insertion” is benign; we allow that the insertion potential generally changes both the energy and entropy of $\mathcal{S}$ and, because $\mathcal{S}$ and $\mathcal{B}$ remain in contact throughout the process, the energy and entropy of the environment as well.The unitary evolution operator for the control phase of the insertion step is of the form:$${\widehat{U}}_{A}^{\mathcal{IB}}\left(t\right)={\widehat{I}}^{\mathcal{M}}\otimes {\widehat{U}}_{A}^{\mathcal{SB}}\left(t\right)\otimes {\widehat{I}}^{\mathcal{W}}$$$${\tilde{\rho}}_{A}^{\mathcal{D}}={\widehat{\rho}}_{0}^{\mathcal{M}}\otimes {\tilde{\rho}}_{A}^{\mathcal{S}}$$$${\tilde{\rho}}_{A}^{\mathcal{DB}}={\widehat{\rho}}_{0}^{\mathcal{M}}\otimes {\tilde{\rho}}_{A}^{\mathcal{SB}}$$**B.****Measurement:**In the “measurement” step (${t}_{A}<t\le {t}_{B}$), $\mathcal{MS}$ unconditionally undergoes a transformation that is equivalent to a quantum measurement with discrete outcomes, with $\mathcal{S}$ and $\mathcal{M}$ playing the roles of “measured system” and “apparatus,” respectively. The i-th of these possible outcomes obtains with a probability ${p}_{i}$, which depends on the initial state, ${\widehat{\rho}}_{A}^{\mathcal{S}}$, and the nature of the measurement interaction and is registered in the memory subsystem, $\mathcal{M}$, by leaving this system in state ${\widehat{\rho}}_{i}^{\mathcal{M}}$. When the i-th outcome is realized, the subsystem, $\mathcal{S}$, is left in a corresponding post-measurement state, ${\widehat{\rho}}_{i}^{\mathcal{S}}$. We assume that the ${\widehat{\rho}}_{i}^{\mathcal{M}}$ have support on disjoint subspaces of $\mathcal{M}$’s state space, ensuring that they are mutually distinguishable and, thus, able to instantiate distinct measurement outcomes, but make no assumptions about the self entropies of the various memory states, ${\widehat{\rho}}_{i}^{\mathcal{M}}$, or their distinguishability from the standard state, ${\widehat{\rho}}_{0}^{\mathcal{M}}$. Nor do we make any assumptions about the self entropies of the various ${\widehat{\rho}}_{i}^{\mathcal{S}}$, their distinguishability from the standard state ${\widehat{\rho}}_{0}^{\mathcal{S}}$or from one another. We also allow that the environment may play a role in the measurement process, e.g., to accommodate the energy and entropy changes of $\mathcal{MS}$, environment-induced decoherence, etc., and may thus end up correlated to the measurement outcome. These liberal conditions allow for a very broad class of ideal and imperfect quantum measurements.The unitary evolution operator for the control phase of the measurement step is of the form:$${\widehat{U}}_{B}^{\mathcal{IB}}\left(t\right)={\widehat{U}}_{B}^{\mathcal{MSB}}\left(t\right)\otimes {\widehat{I}}^{\mathcal{W}}$$$${\tilde{\rho}}_{B}^{\mathcal{D}}=\sum _{i}{p}_{i}({\widehat{\rho}}_{i}^{\mathcal{M}}\otimes {\widehat{\rho}}_{i}^{\mathcal{S}})$$$${\tilde{\rho}}_{B}^{\mathcal{B}}=\sum _{i}{p}_{i}{\tilde{\rho}}_{Bi}^{\mathcal{B}}$$$${\tilde{\rho}}_{B}^{\mathcal{DB}}=\sum _{i}{p}_{i}({\widehat{\rho}}_{i}^{\mathcal{M}}\otimes {\tilde{\rho}}_{Bi}^{\mathcal{SB}})$$Note that the memory and engine are correlated at the conclusion of Step B, so there is $\mathcal{S}$-information in $\mathcal{M}$ and vice versa.**C.****Expansion:**In the crucial “expansion” step (${t}_{B}<t\le {t}_{C}$), the potential that was imposed on the engine $\mathcal{S}$ in the insertion step is evolved back to its initial configuration while delivering work to $\mathcal{W}$. Two aspects of this operation are critical. First, if the Demon is to use information to deliver work to $\mathcal{W}$ during this step—his signature maneuver—then the inserted potential must be removed in a manner that is conditioned on the state of the memory. The same is true of the $\mathcal{S}$-$\mathcal{W}$ coupling, i.e., the interactions that play the role of the linkages required in the classical case to ensure that $\mathcal{S}$ does the same work on $\mathcal{W}$ regardless of the engine state. This is to say that when the memory is in state ${\widehat{\rho}}_{i}^{\mathcal{M}}$, then the system Hamiltonian must be restored from ${\widehat{H}}_{A}^{\mathcal{S}}$ (at time ${t}_{B}$) to ${\widehat{H}}_{0}^{\mathcal{S}}$ (at time ${t}_{C}$) by a time-dependent Hamiltonian ${\widehat{H}}_{Ci}^{\mathcal{S}}\left(t\right)$ that is dictated by the measurement outcome, and an appropriate time-dependent Hamiltonian ${\widehat{H}}_{i}^{\mathcal{SW}}$ must be “activated” during this same interval. Second, unless the decision-making capability required for conditional application of the ${\widehat{H}}_{Ci}^{\mathcal{S}}\left(t\right)$ is to be “outsourced” beyond the boundaries of the composite $\mathcal{DE}$ (and any associated decision costs potentially hidden from view), then this capability must be “built in” to the global evolution operator in Step C.The unitary evolution operator:$${\widehat{U}}_{C}^{\mathcal{IB}}\left(t\right)={\widehat{U}}_{C}^{\mathcal{MSWB}}\left(t\right)=\sum _{i}\left(\right)open="("\; close=")">{\widehat{\Pi}}_{i}^{\mathcal{M}}\otimes {\widehat{U}}_{Ci}^{\mathcal{WSB}}\left(t\right)$$$${\tilde{\rho}}_{C}^{\mathcal{D}}=\left(\right)open="("\; close=")">\sum _{i}{p}_{i}{\widehat{\rho}}_{i}^{\mathcal{M}}\otimes {\widehat{\rho}}_{0}^{\mathcal{S}}$$$${\tilde{\rho}}_{C}^{\mathcal{B}}=\sum _{i}{p}_{i}{\widehat{\rho}}_{Ci}^{\mathcal{B}}$$$${\tilde{\rho}}_{C}^{\mathcal{DB}}=\sum _{i}{p}_{i}({\widehat{\rho}}_{i}^{\mathcal{M}}\otimes {\tilde{\rho}}_{Ci}^{\mathcal{SB}})$$**D.****Memory Reset:**In the “memory reset” step, the Demon’s memory is returned to its initial state. This is achieved via an unconditional local operation on $\mathcal{S}$; there are no subsystems in the accessible $\mathcal{I}$-domain that still hold $\mathcal{S}$-information and could thus be used in Step D to perform the reset conditionally.The unitary evolution operator for the control phase of the reset step is of the form:$${\widehat{U}}_{D}^{\mathcal{IB}}\left(t\right)={\widehat{I}}^{\mathcal{S}}\otimes {\widehat{U}}_{D}^{\mathcal{MB}}\left(t\right)\otimes {\widehat{I}}^{\mathcal{W}}$$$${\tilde{\rho}}_{D}^{\mathcal{D}}={\widehat{\rho}}_{0}^{\mathcal{M}}\otimes {\widehat{\rho}}_{0}^{\mathcal{S}}$$$${\tilde{\rho}}_{D}^{\mathcal{B}}=\sum _{i}{p}_{i}{\tilde{\rho}}_{Di}^{\mathcal{B}}$$$${\tilde{\rho}}_{D}^{\mathcal{DB}}={\widehat{\rho}}_{0}^{\mathcal{M}}\otimes {\tilde{\rho}}_{0}^{\mathcal{S}}\otimes {\tilde{\rho}}_{D}^{\mathcal{B}}$$

#### 3.3. Entropy and Information

Step | Demon $\mathbf{\Delta}{\tilde{\mathit{S}}}^{\mathcal{D}}$ | Bath $\mathbf{\Delta}{\tilde{\mathit{S}}}^{\mathcal{B}}$ | Total $\mathbf{\Delta}{\tilde{\mathit{S}}}_{\mathit{tot}}^{\mathcal{DB}}$ |
---|---|---|---|

A. Insertion | $\Delta {S}_{A}^{\mathcal{S}}$ | ${\tilde{I}}_{A}^{\mathcal{SB}}-\Delta {S}_{A}^{\mathcal{S}}$ | ${\tilde{I}}_{A}^{\mathcal{SB}}$ |

B. Measurement | $\Delta {S}_{B}^{\mathcal{M}}+\Delta {S}_{B}^{\mathcal{S}}$$-{I}_{B}^{\mathcal{MS}}$ | ${\tilde{I}}_{B}^{\mathcal{DB}}-\Delta {S}_{B}^{\mathcal{M}}$$-\Delta {S}_{B}^{\mathcal{S}}+{I}_{B}^{\mathcal{MS}}$ | ${\tilde{I}}_{B}^{\mathcal{DB}}$ |

C. Expansion | $-\Delta {S}_{A}^{\mathcal{S}}-\Delta {S}_{B}^{\mathcal{S}}$$+{I}_{B}^{\mathcal{MS}}$ | ${\tilde{I}}_{C}^{\mathcal{DB}}+\Delta {S}_{A}^{\mathcal{S}}$$+\Delta {S}_{B}^{\mathcal{S}}-{I}_{B}^{\mathcal{MS}}$ | ${\tilde{I}}_{C}^{\mathcal{DB}}$ |

D. Memory Reset | $-\Delta {S}_{B}^{\mathcal{M}}$ | ${\tilde{I}}_{D}^{\mathcal{MB}}+\Delta {S}_{B}^{\mathcal{M}}$ | ${\tilde{I}}_{D}^{\mathcal{MB}}$ |

Cycle Total | 0 | ${\tilde{I}}_{A}^{\mathcal{SB}}+{\tilde{I}}_{B}^{\mathcal{DB}}$$+{\tilde{I}}_{C}^{\mathcal{DB}}+{\tilde{I}}_{D}^{\mathcal{MB}}$ | ${\tilde{I}}_{A}^{\mathcal{SB}}+{\tilde{I}}_{B}^{\mathcal{DB}}$$+{\tilde{I}}_{C}^{\mathcal{DB}}+{\tilde{I}}_{D}^{\mathcal{MB}}$ |

#### 3.4. Work Extraction

#### 3.5. Analysis Summary

- A Demon who acquires information about the state of a Szilard engine’s working substance and uses it to conditionally reinitialize the engine state in an expansion process can deliver an amount of work:$$\Delta \langle {E}^{\mathcal{W}}\rangle \le -\left(\right)open="["\; close="]">\langle \Delta \langle {E}_{i}^{\mathcal{S}}\rangle \rangle -{k}_{B}Tln\left(2\right)\Delta \langle {S}_{i}^{\mathcal{S}}\rangle $$$$\langle \Delta \langle {E}_{i}^{\mathcal{B}}\rangle \rangle \ge -{k}_{B}Tln\left(2\right)\langle \Delta {S}_{i}^{\mathcal{S}}\rangle $$$$\Delta {\tilde{S}}_{tot}^{\mathcal{SD}}\ge 0$$

## 4. Discussions

#### 4.1. Generality

#### 4.2. Recovering Fahn

**A.****Insertion:**In the insertion step, the state of the engine system is transformed from the initial state, ${\widehat{\rho}}_{0}^{\mathcal{S}}$ (with entropy ${S}_{0}^{\mathcal{S}}$), to a uniform mixture:$${\tilde{\rho}}_{A}^{\mathcal{S}}=\frac{1}{2}{\tilde{\rho}}_{L}^{\mathcal{S}}+\frac{1}{2}{\tilde{\rho}}_{R}^{\mathcal{S}}$$$${\tilde{S}}_{A}^{\mathcal{S}}={H}_{2}\left(\right)open="("\; close=")">\frac{1}{2}+\frac{1}{2}{S}_{L}^{\mathcal{S}}+\frac{1}{2}{S}_{R}^{\mathcal{S}}={S}_{0}^{\mathcal{S}}$$$$S\left(\widehat{\rho}\right)=H\left(\left\{{p}_{i}\right\}\right)+\sum _{i}{p}_{i}S\left({\widehat{\rho}}_{i}\right)$$**B.****Measurement:**The measurement step corresponds to a projective measurement onto the orthogonal subspaces supporting ${\tilde{\rho}}_{L}^{\mathcal{S}}$ and ${\tilde{\rho}}_{R}^{\mathcal{S}}$, which does not affect the system state but does bring the memory into correlation with the system:$${\tilde{\rho}}_{B}^{\mathcal{MS}}=\frac{1}{2}{\tilde{\rho}}_{L}^{\mathcal{M}}\otimes {\tilde{\rho}}_{L}^{\mathcal{S}}+\frac{1}{2}{\tilde{\rho}}_{R}^{\mathcal{M}}\otimes {\tilde{\rho}}_{R}^{\mathcal{S}}$$**C.****Expansion:**$\Delta {\tilde{I}}_{C}^{\mathcal{DB}}=0$, since, for each measurement outcome from Step B, Step C requires a conditional isothermal expansion process, each of which is thermodynamically reversible with equal and opposite system-bath entropy changes ($\Delta {\tilde{I}}_{Ci}^{\mathcal{SB}}=0$).**D.****Memory Reset:**The memory reset reverses the one-bit entropy increase of $\mathcal{M}$ that occurred in Step B. ${\tilde{I}}_{D}^{\mathcal{MB}}=0$, since the reset process is assumed to be thermodynamically reversible with equal and opposite system-bath entropy changes.

#### 4.3. The No-Work Demon and the Entropy Cost of Information

**Table 3.**Entropy Changes for a Demon who acquires information, but does no work (adapted from [8]).

Step | Demon $\mathbf{\Delta}{\mathit{S}}^{\mathcal{D}}$ | Bath $\mathbf{\Delta}{\mathit{S}}^{\mathcal{B}}$ | Total $\mathbf{\Delta}{\mathit{S}}_{\mathit{tot}}$ |
---|---|---|---|

A. Insertion | 0 | 0 | 0 |

B. Measurement | 0 | 0 | 0 |

C. Expansion | +1 | 0 | +1 |

D. Memory Reset | −1 | +1 | 0 |

Cycle Total | 0 | +1 | +1 |

- A Demon who acquires information about the state of a Szilard engine’s working substance and who resets the engine state via local operations on the engine that are not conditioned on the outcomes can deliver no work:$$\Delta \langle {E}^{\mathcal{W}}\rangle =0$$$$\langle \Delta \langle {E}_{i}^{\mathcal{B}}\rangle \rangle \ge -{k}_{B}Tln\left(2\right)\langle \Delta {S}_{i}^{\mathcal{S}}\rangle +{k}_{B}Tln\left(2\right){I}_{B}^{\mathcal{MS}}$$$$\Delta {\tilde{S}}_{tot}^{\mathcal{SD}}\ge {k}_{B}Tln\left(2\right){I}_{B}^{\mathcal{MS}}$$

Comparison of our conclusions for the generalized Demon that produces work (Section 3.5) and the Demon that does not do work (this subsection) lend support to Fahn’s conjecture, and in a broader context. When the Demon does not use the memory-engine correlations to condition the expansion operations and extract work, the lower bound on the bath entropy change increases precisely by the amount of information that was embodied within these correlations prior to expansion (${I}_{B}^{\mathcal{MS}}$). This entropic cost, which is exclusively associated with the expansion step, survives in the lower bound on the increase in total Demon-bath entropy for the full cycle, while contributions from the subsystem self-entropy changes distributed throughout the Demon’s cycle ultimately sum to zero.“The entropy cost of information is the degree to which the information is not used to obtain work from the system. This cost is assessed when the correlation between information and the system is lost.”(from [8])

#### 4.4. Demon-Environment Correlations and Global Closure

## 5. Conclusions

## Acknowledgments

## Conflicts of Interest

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Anderson, N.G.
Conditioning, Correlation and Entropy Generation in Maxwell’s Demon. *Entropy* **2013**, *15*, 4243-4265.
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Conditioning, Correlation and Entropy Generation in Maxwell’s Demon. *Entropy*. 2013; 15(10):4243-4265.
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Anderson, Neal G.
2013. "Conditioning, Correlation and Entropy Generation in Maxwell’s Demon" *Entropy* 15, no. 10: 4243-4265.
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