# Entropy Distribution in a Quantum Informational Circuit of Tunable Szilard Engines

## Abstract

**:**

## 1. Introduction

## 2. Antecedents

## 3. Reference Quantum Szilard Engine

- (I)
- Insertion of the wall in the middle of the box.
- (II)
- Measurement of which side of the wall the particle is on.
- (III)
- Expansion of the wall until $\ell =L$ if the particle is on the left side, or until $\ell =0$, otherwise.
- (IV)
- Removal of the wall.

## 4. Protected Information Transmission with GHZ States

## 5. Partial Information Cycles for Motors and Generators

- (I)
- Insertion. Instead of reinserting the wall in the middle of the box, we consider the case where it is reintroduced at an arbitrary position, determined by the value ${\ell}_{1}$ of ℓ. It is assumed that ${\ell}_{1}\ge L/2$. Considering the left-right symmetry of the cylinder, this restriction does not imply a loss of generality. At the end of this stage there is no tunneling between the left and the right compartments.
- (II)
- Measurement. We assume that the fresh qubit M which is used to host the result of the measurement enters a partially mixed state. The which-side qubit I, defined in Appendix B and determined by the left ($\left|0\right.\u232a$) or right ($\left|1\right.\u232a$) situation of the particle, enters the stage in a state$${\rho}_{0}^{\left(I\right)}=\alpha \left|0\right.\u232a\left.\u23290\right|+(1-\alpha )\left|1\right.\u232a\left.\u23291\right|,$$$$z\left(\ell \right):=\frac{{Z}_{0}\left(\ell \right)}{{Z}_{0}\left(\ell \right)+{Z}_{0}(L-\ell )}.$$In the original RefQSZ, the measurement was perfect, which is equivalent to having a pure ${\chi}_{0}=1$ initial state for M; it also assumed that ${\ell}_{1}=L/2$, which implied $\alpha =1/2$.
- (III)
- Expansion. Considering that the left/right measurement is not completely certain, the expansion will stop at a suitable value ${\ell}_{2}^{(M=0,1)}$ of ℓ, controlled by the result $M=0,1$ of the measurement.
- (IV)
- Removal. This stage differs from the RQuSZE only in that the wall is pulled-out at $\ell ={\ell}_{2}^{(M=0,1)}$, instead of at $\ell =L$ or $\ell =0$. The positions ${\ell}_{2}^{(M=0,1)}$ are chosen so that the extraction is reversible. Therefore, they are determined by the equations:$$\left\{\begin{array}{ccc}\hfill z\left({\ell}_{2}^{(M=0)}\right)& =& {r}_{00}\hfill \\ \hfill z\left({\ell}_{2}^{(M=1)}\right)& =& {r}_{01}\hfill \end{array}\right.,$$

- (I’)
- Insertion. The position of the insertion ${\ell}_{1}^{(M=0,{1}^{\prime})}$ is controlled by M. This creates a correlation between M and the left/right position of the particle.
- (II’)
- Compression. The wall is displaced from its initial position to a final value of ${\ell}_{2}^{\prime}$.
- (III’)
- Erasure. A CNOT gate, controlled by the position of the particle, is applied to M. As a consequence, some of the entropy of M is transferred to I and their joint state factorizes.
- (IV’)
- Removal. The wall is pulled-out at $\ell ={\ell}_{2}^{\prime}$.

## 6. Qubit Processing by A Cylinder

- (a)
- Motor. In the first stage, the introduction of the wall at $\ell ={\ell}_{1}$ sets the state of I to be partially mixed and is given by Equation (6).Next, in stage (II), the M qubit is used as target in a CNOT gate controlled by the which-side qubit I. After the CNOT gate, the joint state of qubits I,M reads$$\begin{array}{cc}\hfill {\rho}_{1}^{\left(IM\right)}=& \alpha \frac{1+{\chi}_{0}}{2}\left|00\right.\u232a\left.\u232900\right|+(1-\alpha )\frac{1-{\chi}_{0}}{2}\left|10\right.\u232a\left.\u232910\right|+\hfill \\ & \alpha \frac{1-{\chi}_{0}}{2}\left|01\right.\u232a\left.\u232901\right|+(1-\alpha )\frac{1+{\chi}_{0}}{2}\left|11\right.\u232a\left.\u232911\right|.\hfill \end{array}$$The M qubit then controls the final position in the expansion stage. Its two states $\{\left|0\right.\u232a,\left|1\right.\u232a\}$ have probabilities ${p}_{0},{p}_{1}$, that are given by$$\left\{\begin{array}{ccc}\hfill {p}_{0}& =& {\displaystyle \frac{1-{\chi}_{0}+2\alpha {\chi}_{0}}{2}}\hfill \\ \hfill {p}_{1}& =& {\displaystyle \frac{1-{\chi}_{0}+2(1-\alpha ){\chi}_{0}}{2}}\hfill \end{array}\right..$$The states of I, conditioned on M are:$$\left\{\begin{array}{ccc}\hfill {\rho}_{M=0}^{I}& =& {r}_{00}\left|0\right.\u232a\left.\u23290\right|+{r}_{10}\left|1\right.\u232a\left.\u23291\right|\hfill \\ \hfill {\rho}_{M=1}^{I}& =& {r}_{01}\left|0\right.\u232a\left.\u23290\right|+{r}_{11}\left|1\right.\u232a\left.\u23291\right|\hfill \end{array}\right.,$$$$\left\{\begin{array}{ccc}\hfill {r}_{00}& =& {\displaystyle \frac{\alpha (1+{\chi}_{0})}{1-{\chi}_{0}+2\alpha {\chi}_{0}}}\hfill \\ \hfill {r}_{10}& =& {\displaystyle \frac{(1-\alpha )(1-{\chi}_{0})}{1-{\chi}_{0}+2\alpha {\chi}_{0}}}\hfill \\ \hfill {r}_{01}& =& {\displaystyle \frac{\alpha (1-{\chi}_{0})}{1-{\chi}_{0}+2(1-\alpha ){\chi}_{0}}}\hfill \\ \hfill {r}_{11}& =& {\displaystyle \frac{(1-\alpha )(1-{\chi}_{0})}{1-{\chi}_{0}+2(1-\alpha ){\chi}_{0}}}\hfill \end{array}\right.,$$The M qubit leaves the cycle in the state$${\rho}_{1}^{\left(M\right)}={p}_{0}\left|0\right.\u232a\left.\u23290\right|+{p}_{1}\left|1\right.\u232a\left.\u23291\right|,$$$${\chi}_{1}:=2{p}_{0}-1,$$$${\rho}_{1}^{\left(M\right)}={\chi}_{1}\left|0\right.\u232a\left.\u23290\right|+\frac{1-{\chi}_{1}}{2}\phantom{\rule{0.166667em}{0ex}}U,$$$${\chi}_{1}={\chi}_{0}\phantom{\rule{0.166667em}{0ex}}(2\alpha -1).$$
- (b)
- Generator. This mode is essentially the inverse of the motor mode. However, for the sake of disambiguation, we outline it in detail. The objective of the process is to output the M qubit in state$${\rho}_{1}^{\left(M\right)}={\chi}_{1}^{\prime}\phantom{\rule{0.166667em}{0ex}}\left|0\right.\u232a\left.\u23290\right|+\frac{1-{\chi}_{1}^{\prime}}{2}\phantom{\rule{0.166667em}{0ex}}{U}_{M},$$$${\rho}_{0}^{\left(M\right)}={\chi}_{0}^{\prime}\left|0\right.\u232a\left.\u23290\right|+\frac{1-{\chi}_{0}^{\prime}}{2}\phantom{\rule{0.166667em}{0ex}}{U}_{M},$$$${\alpha}^{\prime}=\frac{1}{2}+\frac{{\chi}_{0}^{\prime}}{2{\chi}_{1}^{\prime}},$$$${p}_{0}^{\prime}=\frac{1+{\chi}_{0}^{\prime}}{2}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{0.166667em}{0ex}}{p}_{1}^{\prime}=1-{p}_{0}^{\prime},$$$$\left\{\begin{array}{ccc}\hfill {\rho}_{M=0}^{{I}^{\prime}}& =& {\displaystyle \frac{1}{2{p}_{0}^{\prime}}\left({\alpha}^{\prime}(1+{\chi}_{1}^{\prime})\left|0\right.\u232a\left.\u23290\right|+(1-{\alpha}^{\prime})(1-{\chi}_{1}^{\prime})\left|1\right.\u232a\left.\u23291\right|\right)}\hfill \\ \hfill {\rho}_{M=1}^{{I}^{\prime}}& =& {\displaystyle \frac{1}{2{p}_{1}^{\prime}}\left(\alpha (1-{\chi}_{1}^{\prime})\left|0\right.\u232a\left.\u23290\right|+(1-\alpha )(1+{\chi}_{1}^{\prime})\left|1\right.\u232a\left.\u23291\right|\right)}\hfill \end{array}\right..$$States ${\rho}_{M=0}^{I}{\phantom{\rule{0.166667em}{0ex}}}^{\prime},{\rho}_{M=1}^{I}{\phantom{\rule{0.166667em}{0ex}}}^{\prime}$ determine ${\ell}_{2}^{(M=0)}{\phantom{\rule{0.166667em}{0ex}}}^{\prime},{\ell}_{1}^{(M=1)}{\phantom{\rule{0.166667em}{0ex}}}^{\prime}$ as the positions for a reversible reintroduction of the wall, so that it is controlled in Stage (I’) by qubit M. Accordingly, with probabilities ${p}_{0}^{\prime}/{p}_{1}^{\prime}$, the I qubit exits (I’) in states ${\rho}_{M=0}^{{I}^{\prime}}/{\rho}_{M=1}^{{I}^{\prime}}$, respectively.The next stage is still controlled by M and leads to a position of the wall given by ${\ell}_{2}^{\prime}$. It is determined by the condition that if the wall was reinserted at ${\ell}_{2}^{\prime}$, the reduced state of I would be$${\rho}_{1}^{\left({I}^{\prime}\right)}=\phantom{\rule{0.166667em}{0ex}}{\alpha}^{\prime}\phantom{\rule{0.166667em}{0ex}}\left|0\right.\u232a\left.\u23290\right|\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}(1-{\alpha}^{\prime})\phantom{\rule{0.166667em}{0ex}}\left|1\right.\u232a\left.\u23291\right|,$$Then, after the CNOT on M controlled by I, the joint state factorizes into$${\rho}_{1}^{\left(I{M}^{\prime}\right)}={\rho}_{1}^{\left({I}^{\prime}\right)}\otimes \left[\frac{1+{\chi}_{1}^{\prime}}{2}\left|0\right.\u232a\left.\u23290\right|+\frac{1-{\chi}_{1}^{\prime}}{2}\left|1\right.\u232a\left.\u23291\right|\right],$$

## 7. Work Evaluation

- (a)
- Motor. After addition of the (I), (III) and (IV) contributions, we arrive at$$<{W}^{\left(I\right)}+{W}^{\left(III\right)}+{W}^{\left(IV\right)}>={F}^{\left(I{I}_{f}\right)}-{F}^{\left(I{I}_{i}\right)}.$$Following the measurement, there are two possible states for the particle, ${\rho}_{M=0}^{\left(P\right)},{\rho}_{M=1}^{\left(P\right)}$, with probabilities ${p}_{0},{p}_{1}$, respectively. The average work expected when $M=0$ is$$<W{>}_{M=0}=\mathrm{Tr}\phantom{\rule{0.166667em}{0ex}}\left\{({\rho}_{M=0}^{\left(P\right)}-{\rho}_{0}^{\left(P\right)})H\right\}+{k}_{B}\left(\mathrm{ln}2\right)T({S}_{0}^{\left(P\right)}-{S}_{M=0}^{\left(P\right)}),$$$$<W{>}_{M=1}=\mathrm{Tr}\phantom{\rule{0.166667em}{0ex}}\left\{({\rho}_{M=1}^{\left(P\right)}-{\rho}_{0}^{\left(P\right)})H\right\}+{k}_{B}\left(\mathrm{ln}2\right)T({S}_{0}^{\left(P\right)}-{S}_{M=1}^{\left(P\right)}),$$Consequently, the average value of the work per cycle yields$$\begin{array}{cc}\hfill <W>=& {p}_{0}<W{>}_{0}+{p}_{1}<W{>}_{1}\hfill \\ \hfill =& \mathrm{Tr}\phantom{\rule{0.166667em}{0ex}}\left\{({p}_{0}\phantom{\rule{0.166667em}{0ex}}{\rho}_{M=0}^{\left(P\right)}+{p}_{1}\phantom{\rule{0.166667em}{0ex}}{\rho}_{M=1}^{\left(P\right)}-{\rho}_{0}^{\left(P\right)})H\right\}+{k}_{B}\left(\mathrm{ln}2\right)T\left({S}_{0}^{\left(P\right)}-{p}_{0}{S}_{M=0}^{\left(P\right)}-{p}_{1}{S}_{M=1}^{\left(P\right)}\right),\hfill \end{array}$$$$<W>={k}_{B}\left(\mathrm{ln}2\right)T\left({S}_{0}^{\left(P\right)}-{p}_{0}{S}_{M=0}^{\left(P\right)}-{p}_{1}{S}_{M=1}^{\left(P\right)}\right).$$We can derive another equation by considering that the CNOT gate preserves the entropy of the M-P system. Before the CNOT, M,P are independent. Therefore,$${S}_{0}^{\left(MP\right)}={S}_{0}^{\left(P\right)}+{S}_{0}^{\left(M\right)}.$$On the other hand, after the CNOT, the two possible states of P: ${\rho}_{M=0}^{\left(P\right)},{\rho}_{M=0}^{\left(P\right)}$ occur with probabilities ${p}_{0},{p}_{1}$. Then,$${S}_{1}^{\left(MP\right)}=g\left({p}_{0}\right)+{p}_{0}{S}_{M=0}^{\left(P\right)}+{p}_{1}{S}_{M=1}^{\left(P\right)},$$$${S}_{0}^{\left(P\right)}+{S}_{0}^{\left(M\right)}={S}_{1}^{\left(M\right)}+{p}_{0}{S}_{M=0}^{\left(P\right)}+{p}_{1}{S}_{M=1}^{\left(P\right)},$$$$<{W}_{c}>={k}_{B}\left(\mathrm{ln}2\right)T\phantom{\rule{0.166667em}{0ex}}\left[{S}_{1}^{\left(M\right)}-{S}_{0}^{\left(M\right)}\right]={k}_{B}\left(\mathrm{ln}2\right)T\phantom{\rule{0.166667em}{0ex}}\Delta {S}^{\left(M\right)},$$As has been shown, the CNOT gate plays a central role in the obtention of the average work per cycle. The following remarks may shed some additional intuition on the previous derivation. The only contribution to the energy of the system, before and after the CNOT, comes from the particle, because the M qubit is assumed to have a completely degenerate Hamiltonian. Accordingly, only the reduced state of the particle is relevant. Note that it is not changed by the CNOT gate. However, this does not imply that the energies of the conditional states ${\rho}_{M=0,1}^{\left(P\right)}$ are equal. Considering the entropy, the distinctive character of the CNOT stems from the fact that it is the only interaction that exchanges information between the particle and the M qubits. Despite the overall entropy being conserved by the gate, the conditional states ${\rho}_{M=0,1}^{\left(P\right)}$ and ${\rho}_{0}^{\left(P\right)}$ show different values and the entropy of M is raised. This increment is the prize paid for the obtained work.
- (b)
- Generator. Considering that the generator and motor modes are the reverse of one another, Equation (34) applies. The only obvious difference is that in the generator mode, $<{W}_{c}>,\Delta {S}^{\left(M\right)}$ are both negative.

## 8. Force Matching Tweak

- A new stage (IIa) known as Precompression is defined between (II) and (III). The wall is moved from ${\ell}_{1}$ to a suitable position ${\ell}_{3}^{(M=0,1)}$, controlled by M and keeping the load disengaged.
- Stage (III) now proceeds from ${\ell}_{3}^{(M=0,1)}$ to ${\ell}_{4}^{(M=0,1)}$. The load is engaged only in this stage, and the values of ${\ell}_{4}^{(M=0,1)}$ are also controlled by M.
- A second new stage (IIIa), called
**Relaxation**, follows (III). The wall is moved to ${\ell}_{2}^{(M=0,1)}$ defined in Section 5, keeping the load disengaged.

- A new stage (I’a) that we also call Precompression is defined between (I’) and (II’). The wall is moved to a suitable position ${\ell}_{3}^{(M=0,{1}^{\prime})}$, keeping the load (in the generator mode the load acts supplying energy, as a pump in a hydraulic system) disengaged. The value of ${\ell}_{3}^{(M=0,{1}^{\prime})}$ is controlled by M.
- Stage (II’) now proceeds from ${\ell}_{3}^{(M=0,{1}^{\prime})}$ to ${\ell}_{4}^{(M=0,{1}^{\prime})}$. The load is engaged only in this stage and the value of ${\ell}_{4}^{(M=0,{1}^{\prime})}$ is also controlled by M.
- A second new stage (II’a), also called Relaxation, follows (II’). Te wall is moved to ${\ell}_{2}^{\prime}$ keeping the load disengaged.

## 9. Carnot Cycles with Szilard Cylinders

## 10. Discussion

## 11. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Energies of a Particle in A Cylindrical Compartment

## Appendix B. Location Qubit

## Appendix C. Employment of A CNOT Gate to Locate the Particle

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**Figure 1.**Subfigure (

**a**) shows a basic RefQSZ. The length of the cylinder is L and the position of the wall is specified by ℓ. Subfigure (

**b**) represents the function $z\left(\ell \right)$, defined in Equation (7), that denotes the probability for the particle to end in the left compartment when the wall is removed and slowly reintroduced at position ℓ. The saturation elbows are quantum effects that are explained physically in References [66,67].

**Figure 2.**The drawings depict stages (

**I**) to (

**IV**) of the Quantum Szilard Engine described in Reference [1] for the particular case of only one particle in the cylinder.

**Figure 3.**Ring of n Szilard cylinders ${Z}_{1},{Z}_{2},\dots ,{Z}_{n}$ sharing a quantum communication line C. The M qubits transit the line from a cylinder to the next in a maximally mixed reduced state. Each engine may work either in a motor or a generator mode. In the former, some work is done on the load, whereas in the latter some energy is borrowed.

**Figure 4.**Representation of the M qubit processing by consecutive cylinders. First, a CNOT is applied, controlled by the local security qubit ${G}_{k}$. It unlocks the M qubit, so that it can undergo a second CNOT, controlled by the internal qubit I. After ${Z}_{k}$ has finished its cycle, a new CNOT, again controlled by ${G}_{k}$ prepares M for the transmission to ${Z}_{k+1}$.

**Figure 5.**On the left, subfigure (

**a**) depicts the stages of the motor cycle. The vertical axis represents the conditional probability of finding the particle on the left side of the wall. The right subfigure (

**b**) corresponds to the generator mode. Function $z\left(\ell \right)$ is defined in Equation (7).

**Figure 6.**On the left, subfigure (

**a**) depicts the stages of the motor cycle with the additional (IIa), (IIIa) stages. Subfigure (

**b**) corresponds to the generator mode with the additional (I’a), (II’a) stages. The vertical axis represents the conditional probability of finding the particle on the left side of the wall for the two alternatives $M=0,1$. Function $z\left(\ell \right)$ is defined in Equation (7).

**Figure 7.**Graphical representations of partition functions (

**a**), free energies (

**b**) and forces (

**c**) as a function of the wall position ℓ. Graphs (

**b**) and (

**c**) represent the case of the particle on the left side (dashed line), on the right side (dotted line) and a linear combination with weights ${r}_{00},{r}_{10}$ of both.

© 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Diazdelacruz, J. Entropy Distribution in a Quantum Informational Circuit of Tunable Szilard Engines. *Entropy* **2019**, *21*, 980.
https://doi.org/10.3390/e21100980

**AMA Style**

Diazdelacruz J. Entropy Distribution in a Quantum Informational Circuit of Tunable Szilard Engines. *Entropy*. 2019; 21(10):980.
https://doi.org/10.3390/e21100980

**Chicago/Turabian Style**

Diazdelacruz, Jose. 2019. "Entropy Distribution in a Quantum Informational Circuit of Tunable Szilard Engines" *Entropy* 21, no. 10: 980.
https://doi.org/10.3390/e21100980