Maxwell’s Demon in Photonic Systems

Abstract

Photons are massless, noninteracting particles, and thermodynamics seems to be completely inappropriate in their description. Here we present two examples of the opposite: connecting thermodynamics with information through Maxwell’s Demon provides interesting insight on properties of light fields. This does not amount to directly applying thermodynamics to photons, but rather helps to obtain tools and concepts from thermodynamics to manipulate and evaluate the information content of light. The examples presented here pinpoint some of the challenges that arise when putting a thought experiment into practice and provide new insights into the relation between thermodynamic work and information.

Appendix

In this appendix we revise the demonstrations of the results (39.6), (39.8), and (39.12) which have been stated in this chapter.

Derivation of the work-information equality The work-information theorem by Sagawa and Ueda (see Chap. 10), [14] assumes interaction with a single thermal bath, but this is not the case for the photonic Maxwell’s demon experiment. However, in [8], a relation similar to that of Sagawa and Ueda is derived to describe this scenario. This work-information relation connects the work extraction with and without an action by the demon. The cases are denoted by the subscripts “0” for the unperturbed state, and “f” when including measurement and feedback.

The language adopted to describe the model is that of quantum mechanics for the sake of generality, but it also applies to states and operations with a valid classical counterpart. Three parts can be distinguished: (i) the system $\mathcal {S}$, starting in thermal equilibrium with inverse temperature $\beta$, representing the working medium (i.e. the two thermal light modes). Its initial and final energy eigenstates are $\left| s \right\rangle$ and $\left| S \right\rangle$ respectively, and its Hamiltonian is $\hat{H}_{\mathcal {S}}$. This is the only part on which measurement and feedback operate. The measurement outcomes are indicated by m, associated to the measurement operator $\hat{M}_j^{(m)}$, with an extra index j due to the possibility of several measurement operators associated with a single measurement outcome (e.g. APD click). The measurement operators must sum to the identity operator: $\sum _{m,j} \hat{M}_j^{(m)\dagger }\hat{M}_{j}^{(m)}=\hat{1}$. The feed-forward operation is described by $\hat{U}_m$. (ii) The work reservoir, which in the experiment is a battery $\mathcal {B}$ which provides an initial voltage drop $U_0$ across the capacitor. Its initial and final energy levels are $\left| b \right\rangle$ and $\left| B \right\rangle$ with energies $E_b$ and $E_B$, respectively. Its Hamiltonian is $\hat{H}_{\mathcal {B}}$. The extracted work is defined as the energy increase of this system: $W=E_B-E_b$. (iii) All the remaining ancillary systems as well as the environment: these are collectively denoted as the rest, $\mathcal {R}$. The Hamiltonian is $\hat{H}_{\mathcal {R}}$, giving initial and final energy levels r and R respectively. The final joint energy of $\mathcal {S}$ and $\mathcal {R}$ is $E_{SR}\equiv E_S+E_R$. The set of random variables of interest is given by $v\equiv \{s,b,r,m, SR\}$.

Owing to decoherence, the total initial state can be assumed to be diagonal in the product basis of the free Hamiltonians: the initial energies of the three systems then represent classical variables. The work reservoir undergoes decoherence in its energy eigenbasis at the end, hence its final energy is well defined. A priori, the sum of the free-Hamiltonian energies of the systems is not necessarily conserved: however, by modelling light-matter interaction with the Jaynes–Cummings interaction Hamiltonian plus decoherence in the energy eigenbasis, energy conservation is ensured. Furthermore, connecting or disconnecting the capacitor from the battery requires in principle no energy. These conditions lead to $E_s+E_b+E_r=E_S+E_B+E_R$, which, in turn, implies that $W=E_s+E_r-E_{SR}$.

The left hand side of (39.6) is given by

$$\begin{aligned} \langle e^{\beta W-I}\rangle _{\text {f}}=\sum _v\, p(s,b,r,m, SR)\,e^{\beta W-I} = \sum _v\, p(m, SR \vert s,b,r) p(s,b,r)\,e^{\beta W-I}, \end{aligned}$$

(39.13)

where Bayes’ theorem has been used in the second step. The probabilities for the initial system is given by the thermal distribution $p(s,b,r)=\frac{1}{Z}e^{-\beta E_s}p(b,r)$, where Z is the partition function. If the expressions for W and $I=\log \left[ p(m\vert s)\right] -\log \left[ m\right]$ are introduced one gets

$$\begin{aligned} \begin{aligned} \langle e^{\beta W-I}\rangle _{\text {m+f}}&=\sum _v\, p(m,SR \vert s,b,r)\frac{1}{Z}e^{-\beta E_s}p(b,r)\,e^{\beta \left( E_s+E_r-E_{SR}\right) }\frac{p(m)}{p(m\vert s)}\\ {}&=\sum _v\, p(m,SR \vert s,b,r)\frac{p(m)}{p(m\vert s)}\frac{1}{Z}p(b,r)\,e^{\beta \left( E_r-E_{SR}\right) }. \end{aligned} \end{aligned}$$

(39.14)

If the sum over the initial states s is isolated, one gets

$$\begin{aligned} \begin{aligned}&\sum _s\, p(m,SR \vert s,b,r)/p(m\vert s)=\\&{{\,\mathrm{Tr}\,}}\left[ \left| SR \right\rangle \left\langle SR \right| \hat{V}\left( \hat{U}_m\left( \sum _s\frac{\sum _{j}\hat{M}_j^{(m)}\left| s \right\rangle \left\langle s \right| \hat{M}_j^{(m)\dagger }}{p(m\vert s)}\right) \hat{U}_m^\dagger \otimes \left| b\, r \right\rangle \left\langle b\, r \right| \right) \hat{V}^\dagger \right] , \end{aligned} \end{aligned}$$

(39.15)

where the fact that system is closed, hence its evolution is described by the unitary $\hat{V}$, has been made explicit. The sum over s contains all normalized states produced by the demon’s measurement, for a sharp energy input. A non-disturbing measurement would leaves the state $\left| s \right\rangle \left\langle s \right|$ unchanged: in this limit the sum would reduce to the identity. The measurement of the demon does not meet this requirement, however, it can be shown to satisfy the relation [8]:

$$\begin{aligned} \sum _s\!\sum _{j}\!\frac{\hat{M}_j^{(m)}\!\left| s \right\rangle \left\langle s \right| \hat{M}_j^{(m)\dagger }}{p(m\vert s)}=\hat{1}, \end{aligned}$$

(39.16)

which can be taken as a generalization of the notion of non-disturbance. Invoking this and $\hat{U}_m \hat{U}_m^\dagger =\hat{1}$, one gets

$$\begin{aligned} \sum _s\, p(m,SR \vert s,b,r)/p(m\vert s)={{\,\mathrm{Tr}\,}}\!\left[ \left| SR \right\rangle \left\langle SR \right| \,\hat{V}\!\!\left( \hat{I}\!\otimes \!\left| b\, r \right\rangle \left\langle b\, r \right| \right) \!\hat{V}^\dagger \right] =\sum _s p_0(SR\vert s,b,r) \end{aligned}$$

(39.17)

where $p_0$ is the probability distribution for the free evolution of the systems, i.e. without implementing measurement and conditional operations. The final result is obtained:

$$\begin{aligned} \langle e^{\beta W-I}\rangle _{\text {f}}=\sum _v\, p_0(SR\vert s,b,r)p(s,b,r)\,e^{\beta W}=\langle e^{\beta W}\rangle _0. \end{aligned}$$

(39.18)

Informational bound on work extraction The bound (39.6) is hard to relate directly to measurable quantities. For this purpose, one can apply the concave $\log$ function to both sides of the equation, and use Jensen’s inequality:

$$\begin{aligned} \beta \left\langle W \right\rangle _{\text {f}}-\langle I \rangle < \log {\left\langle e^{\beta W}\right\rangle _0}. \end{aligned}$$

(39.19)

It is convenient to act with the substitution $\left\langle e^{\beta W}\right\rangle _0=e^{\beta \left\langle W\right\rangle _0}\left\langle e^{\beta \left( W-\left\langle W\right\rangle _0\right) }\right\rangle _0$ to manipulate the expression above:

$$\begin{aligned} \left\langle W \right\rangle _{\text {f}}-\left\langle W \right\rangle _{\text {0}} < \beta ^{-1} \langle I \rangle +\beta ^{-1}\,\log {\left\langle e^{\beta \left( W-\left\langle W\right\rangle _0\right) }\right\rangle _0}. \end{aligned}$$

(39.20)

In order to develop an intuition on these quantities, one can perform an expansion up to second order in $\beta$:

$$\begin{aligned} \log {\left\langle e^{\beta \left( W-\left\langle W\right\rangle _0\right) }\right\rangle _0}=\beta ^2/2\,\mathrm {Var}(W)_0+\mathcal {O}(\beta ^3), \end{aligned}$$

(39.21)

which is exact for Gaussian work distributions. This leads to the definition:

$$\begin{aligned} \sigma (\beta ,W)\equiv \sqrt{2\,\beta ^{-2}\log {\left\langle e^{\beta \left( W-\left\langle W\right\rangle _0\right) }\right\rangle _0}}. \end{aligned}$$

(39.22)

This coincides with the standard deviation of W for a Gaussian distribution, and it is an approximation up to first order in $\beta$ for the general case. Equation (39.20) can then be rewritten as:

$$\begin{aligned} \left\langle W \right\rangle _{\text {f}}-\left\langle W \right\rangle _{\text {0}} < \beta ^{-1} \langle I \rangle +\beta \,[\sigma (\beta ,W)]^2/2. \end{aligned}$$

(39.23)

Statistical considerations are drawn by taking into account N runs of the experiment. Let $U_k$ denote the voltage created across the capacitor C in the $k^\text {th}$ iteration: work extraction relies on storing the energy accumulated in the capacitor in a work reservoir. The capacitor is connected to a battery with voltage $U_0$: the charge $C(U_k-U_0)$ is transferred to the battery, increasing its energy by $C U_0 (U_k-U_0)$. The voltage $U_0$ can be held as constant, since $U_k$ can be made arbitrarily small. The capacitor then relaxes to 0 voltage. Over the N cycles the work extracted is $W^{(N)}=C U_0\sum _k^N (U_k-U_0)$, while the mutual information gain is $I^{(N)}\equiv \sum _k^N I_k$. Consequently, the sum of voltages across the capacitor over N runs is $U^{(N)}\equiv \sum _k^N U_k$. The energy scale with respect to the temperature is indicated by the parameter $\xi \equiv \beta C U_0$. If these expressions are inserted in (39.6), Jensen’s inequality is applied as above, one obtains:

$$\begin{aligned} \left\langle U^{(N)} \right\rangle _{\text {f}}-\left\langle U^{(N)} \right\rangle _{\text {0}} < \xi ^{-1}\langle I^{(N)} \rangle +\xi \,[\sigma (\xi ,U^{(N)})]^2/2. \end{aligned}$$

(39.24)

The voltages $U_k$ are independently and identically distributed, implying:

$$\begin{aligned} \left\langle e^{x\,U^{(N)}}\right\rangle _0=\left\langle e^{x\,U}\right\rangle _0^N, \end{aligned}$$

(39.25)

a general property of the moment generating function. Equation (39.22) results in the fact that $[\sigma (\xi ,U^{(N)})]^2=N [\sigma (\xi ,U)]^2$. Furthermore, the properties hold: $\left\langle U^{(N)}\right\rangle =N \left\langle U\right\rangle$ and $\left\langle I^{(N)}\right\rangle =N \left\langle I\right\rangle$. Equation (39.24) then becomes

$$\begin{aligned} \left\langle U \right\rangle _{\text {f}}-\left\langle U \right\rangle _{\text {0}} < \frac{1}{\xi } \langle I \rangle +\xi \,[\sigma (\xi ,U)]^2/2. \end{aligned}$$

(39.26)

Since this holds for arbitrary $U_0$, minimization of the right-hand term of the inequality can be performed with respect to $\xi$, delivering the tightest bound:

$$\begin{aligned} \left\langle U \right\rangle _{\text {f}}-\left\langle U \right\rangle _{\text {0}} < \mathrm {min}_\xi \left\{ \frac{1}{\xi } \langle I \rangle +\xi \,[\sigma (\xi ,U)]^2/2\right\} . \end{aligned}$$

(39.27)

For a Gaussian distribution of U, one has that $[\sigma (\xi ,U)]=\sqrt{\mathrm {Var}(U)_0}$, thus $\mathrm {min}_\xi \left\{ \frac{1}{\xi } \langle I \rangle +\xi \,[\sigma (\xi ,U)]^2/2\right\} =\sqrt{2 I}\sqrt{\mathrm {Var}(U)_0}$. This yields the bound

$$\begin{aligned} \frac{\vert \left\langle U \right\rangle _{\text {f}}-\left\langle U \right\rangle _{\text {0}}\vert }{\sqrt{\mathrm {Var}(U)_0}} < \sqrt{2 \langle I\rangle } \end{aligned}$$

(39.28)

and, using the definition of work in the setup

$$\begin{aligned} \frac{\vert \left\langle W \right\rangle _{\text {f}}-\left\langle W \right\rangle _{\text {0}}\vert }{\sqrt{\mathrm {Var}(W)_0}} < \sqrt{2 \langle I\rangle }. \end{aligned}$$

(39.29)

The bound given by (39.28) is a stronger version of in (39.27), which is valid for small values of $\langle I\rangle$.

Bounding the extractable work by separable quantum state We start by considering a generic separable quantum state in the form $\rho = \sum _i p_i \rho _i^A\otimes \rho _i^B$, where $\rho _i^A$ and $\rho _i^B$ are a set of local state at Alice’s and Bob’s locations, jointly showing with probability $p_i$. When measuring the observables $\hat{A}$ and $\hat{B}$, one can introduce two operators $\hat{A}^1$ and $\hat{A}^0$ such that the probabilities for Alice to get the outcome j are given by $p^A_j=\sum _i p_i {{\,\mathrm{Tr}\,}}[\hat{A}_j \rho ^A_i]$, and similarly for Bob’s side. This allows to express the conditional entropy in terms of the conditional probabilities $p(B^0|A^j)$ as $H(B|A)=\sum _j p_j^A H(p(B^0|A^j))$. Bayes’ theorem can be invoked to find the conditional probability as:

$$\begin{aligned} \begin{aligned} p(B^0|A^j)=&\frac{p(A^j,B^0)}{p^A_j}=\frac{\sum _i p_i {{\,\mathrm{Tr}\,}}[A^j\rho ^A_i] {{\,\mathrm{Tr}\,}}[B^0\rho ^B_i]}{p^A_j}\\ =&\frac{\sum _i p_i \,p^A_{j,i}\, p^B_{0,i}}{p^A_j}, \end{aligned} \end{aligned}$$

(39.30)

where the shorthand notation has been introduced $p^A_{j,i}={{\,\mathrm{Tr}\,}}[A^j\rho ^A_i]$ for Alice, and similarly for Bob. This leads to

$$\begin{aligned} \begin{aligned} H(B|A) =&\sum _j p_j^A H\left( \frac{\sum _i p_i \,p^A_{j,i}\, p^B_{0,i}}{p^A_j}\right) \\&\ge \sum _j \sum _i p_i \,p^A_{j,i} H(p_{0,i}^B) = \sum _i p_i H(p_{0,i}^B), \end{aligned} \end{aligned}$$

(39.31)

due to the concavity of Shannon entropy. The inequality is saturated when a single component is present: $\rho = \rho ^A_0\otimes \rho ^B_0$, and $H(p_{0,0}^B)$ is minimal for a pure state. QED.