# Quantum Information Remote Carnot Engines and Voltage Transformers

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## Abstract

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## 1. Introduction

## 2. Antecedents

## 3. Elements of a Quieg

- A fermionic system F with only one available state. It can be either empty or occupied by a single electron.
- A thermo-chemical reservoir $\mathcal{R}$, at temperature T and electron chemical potential $\mu $.
- Two terminals ${\mathcal{T}}^{+},{\mathcal{T}}^{-}$ of a battery at potentials ${\mathcal{V}}^{+},{\mathcal{V}}^{-}$, respectively, used to power an electrical load L (direct mode) or to draw energy from an external source (reverse mode). They are equivalent to thermo-chemical reservoirs for electrons at temperature T and chemical potentials ${\mathcal{V}}^{+},{\mathcal{V}}^{-}$, respectively.
- Three gates ${G}_{1},{G}_{2},{G}_{3}$ used to commute between the following four possible situations for F:
- i.
- it is isolated
- ii.
- it is in equilibrium with the terminal ${\mathcal{T}}^{+}$
- iii.
- it is in equilibrium with the terminal ${\mathcal{T}}^{-}$
- iv.
- it is in equilibrium with the reservoir $\mathcal{R}$.

- A tunable energy $\mathcal{E}$ for F.

## 4. Stages of a Quieg Working in the Direct Mode

- Measurement. The qubit $\mathcal{M}$ from $\mathcal{C}$ acts as a target qubit of a CNOT gate controlled by F. The bipartite system $\mathcal{M}\phantom{\rule{0.166667em}{0ex}}F$ results in the joint state$${\rho}_{CNOT}^{\left(\mathcal{M}\phantom{\rule{0.166667em}{0ex}}F\right)}=\frac{1}{2}\left(\right)open="("\; close=")">m\left|00\right.\u232a\left.\u232900\right|+m\left|11\right.\u232a\left.\u232911\right|+(1-m)\left|01\right.\u232a\left.\u232901\right|+(1-m)\left|10\right.\u232a\left.\u232910\right|$$After measuring, $\mathcal{M}$ will be either in $\left|0\right.\u232a$ or $\left|1\right.\u232a$ state with equal probabilities. The states of F, conditioned to the outcomes, are$$\left(\right)$$
- Branching. According to the value of $\mathcal{M}$, the process may take two possible ways. First, we assume that F is in state ${\rho}_{1}^{F}$, where it is more likely to be occupied (case a) and then we describe the steps taken if F is in state ${\rho}_{0}^{F}$ (case b), when it is more likely to be empty.

- a.3.
- Fitting. The energy level $\mathcal{E}$ is tuned to ${\mathcal{E}}_{1}$, while F is isolated. The value of ${\mathcal{E}}_{1}$ is the one for which the equilibrium state, when F is in contact with ${\mathcal{T}}^{+}$, is ${\rho}_{1}^{F}$. It is given by$$m=\frac{1}{1+{e}^{\beta ({\mathcal{E}}_{1}-{\mathcal{V}}^{+})}}.$$
- a.4.
- Plunging. F is put in contact with ${\mathcal{T}}^{+}$ and, keeping the equilibrium, $\mathcal{E}$ is raised from ${\mathcal{E}}_{1}$ to ${\mathcal{E}}_{a}$, where there is a probability of occupation ${p}_{a}$ to be determined later. Accordingly, ${\mathcal{E}}_{a}$ results from$${p}_{a}=\frac{1}{1+{e}^{\beta ({\mathcal{E}}_{a}-{\mathcal{V}}^{+})}}\phantom{\rule{0.166667em}{0ex}}.$$
- a.5.
- Leveling. F is isolated and $\mathcal{E}$ is taken from ${\mathcal{E}}_{a}$ to ${\mathcal{E}}_{2}$ chosen so that ${p}_{a}$ is the occupation probability for F at equilibrium with $\mathcal{R}$. ${\mathcal{E}}_{2}$ satisfies the equation$${p}_{a}=\frac{1}{1+{e}^{\beta ({\mathcal{E}}_{2}-\mu )}}\phantom{\rule{0.166667em}{0ex}}.$$
- a.6.
- Relaxation. While in contact with $\mathcal{R}$, $\mathcal{E}$ is taken back from ${\mathcal{E}}_{2}$ to ${\mathcal{E}}_{0}$ as it drives F back to its initial state ${\rho}_{\mathrm{init}}^{F}$.

- b.3.
- Fitting. The energy level $\mathcal{E}$ is tuned to ${\mathcal{E}}_{1}^{\prime}$, while F is isolated. The value of ${\mathcal{E}}_{1}^{\prime}$ is the one for which the equilibrium state when F is in contact with ${\mathcal{T}}^{-}$ is ${\rho}_{0}^{F}$. It is given by$$1-m=\frac{1}{1+{e}^{\beta ({\mathcal{E}}_{1}^{\prime}-{\mathcal{V}}^{-})}}\phantom{\rule{0.166667em}{0ex}}.$$
- b.4.
- Plunging. F is put in contact with ${\mathcal{T}}^{-}$ and $\mathcal{E}$ is lowered from ${\mathcal{E}}_{1}^{\prime}$ to ${\mathcal{E}}_{b}$, where there is a probability of occupation ${p}_{b}$ to be determined later. Accordingly, ${\mathcal{E}}_{b}$ results from$${p}_{b}=\frac{1}{1+{e}^{\beta ({\mathcal{E}}_{b}-{\mathcal{V}}^{-})}}\phantom{\rule{0.166667em}{0ex}}.$$
- b.5.
- Leveling. F is isolated and $\mathcal{E}$ is taken from ${\mathcal{E}}_{b}$ to ${\mathcal{E}}_{2}^{\prime}$ chosen so that ${p}_{b}$ is the occupation probability for F at equilibrium with $\mathcal{R}$. It satisfies the equation$${p}_{b}=\frac{1}{1+{e}^{\beta ({\mathcal{E}}_{2}^{\prime}-\mu )}}\phantom{\rule{0.166667em}{0ex}}.$$
- b.6.
- Relaxation. While in contact with $\mathcal{R}$, $\mathcal{E}$ is taken back from ${\mathcal{E}}_{2}^{\prime}$ to ${\mathcal{E}}_{0}$ as it drives F back to its initial state ${\rho}_{\mathrm{init}}^{F}$.

## 5. Quieg Working in the Reverse Mode

- Branching. The qubit $\mathcal{M}$ is projected into its computational basis and one of two paths is selected. The value 1 (case c) branches into a sequence of steps which correspond to the upper part of Figure 6, whereas the value 0 (case d) selects the lower part of the figure.

- c.2.
- Polarization. It is the inverse of stage a.6. of Section 4. Initially, the system F is in equilibrium with reservoir $\mathcal{R}$. The energy $\mathcal{E}$ is taken from ${\mathcal{E}}_{0}$ to ${\mathcal{E}}_{2}$, as defined in Equation (7) and the state of F is driven from ${\rho}_{\mathrm{init}}^{F}$ to ${\rho}_{a}^{F}$, given by$${\rho}_{a}^{F}={p}_{a}\left|0\right.\u232a\left.\u23290\right|+(1-{p}_{a})\left|1\right.\u232a\left.\u23291\right|\phantom{\rule{0.166667em}{0ex}},$$
- c.3.
- Adjustment. In this stage, $\mathcal{E}$ is tuned from ${\mathcal{E}}_{2}$ to ${\mathcal{E}}_{a}$ as given by Equation (6) and is the inverse of a.5. F stays isolated and its state remains ${\rho}_{1}^{F}$.
- c.4.
- Extraction. While F is in equilibrium with ${\mathcal{T}}^{+}$, $\mathcal{E}$ is lowered from ${\mathcal{E}}_{a}$ to ${\mathcal{E}}_{1}$, so that the probability of occupation for F increases. Its state is$${\rho}_{m}^{F}=m\left|0\right.\u232a\left.\u23290\right|+(1-m)\left|1\right.\u232a\left.\u23291\right|\phantom{\rule{0.166667em}{0ex}}.$$This stage is the inverse of a.4.
- c.5.
- Zeroing. This stage adiabatically takes the energy $\mathcal{E}$ back to ${\mathcal{E}}_{0}$.

- d.2.
- Polarization. This corresponds to the inverse of stage b.6. of Section 4. The system F is in equilibrium with reservoir $\mathcal{R}$. The energy $\mathcal{E}$ is taken from ${\mathcal{E}}_{0}$ to ${\mathcal{E}}_{2}^{\prime}$, as defined in Equation (10) and the state of F is driven from ${\rho}_{\mathrm{init}}^{F}$ to$${\rho}_{b}^{F}={p}_{b}\left|0\right.\u232a\left.\u23290\right|+(1-{p}_{b})\left|1\right.\u232a\left.\u23291\right|\phantom{\rule{0.166667em}{0ex}}.$$
- d.3.
- Adjustment. In this stage, $\mathcal{E}$ is tuned from ${\mathcal{E}}_{2}^{\prime}$ to ${\mathcal{E}}_{b}$ as given by Equation (9) and is the inverse of b.5. F stays isolated.
- d.4.
- Extraction. While F is in equilibrium with ${\mathcal{T}}^{-}$, $\mathcal{E}$ is lowered from ${\mathcal{E}}_{b}$ to ${\mathcal{E}}_{1}^{\prime}$, so that the probability of occupation for F decreases. This stage is the inverse of b.4.
- d.5.
- Zeroing. This stage adiabatically takes the energy $\mathcal{E}$ back to ${\mathcal{E}}_{0}$.

- 6.
- Reset. The final step of the cycle is a CNOT gate controlled by F targeted at $\mathcal{M}$. The state of the $\mathcal{M}-F$ system before applying the gate is$${\rho}_{5}^{(\mathcal{M}-F)}=\frac{1}{2}m\left(\right)open="("\; close=")">\left|00\right.\u232a\left.\u232900\right|+\left|11\right.\u232a\left.\u232911\right|\phantom{\rule{0.166667em}{0ex}}.$$In addition, after the CNOT, is$${\rho}_{6}^{(\mathcal{M}-F)}=\frac{1}{2}m\left(\right)open="("\; close=")">\left|00\right.\u232a\left.\u232900\right|+\left|01\right.\u232a\left.\u232901\right|\phantom{\rule{0.166667em}{0ex}},$$$${\rho}_{6}^{(\mathcal{M}-F)}=\left(\right)open="["\; close="]">m\left|0\right.\u232a\left.\u23290\right|+(1-m)\left|1\right.\u232a\left.\u23291\right|\phantom{\rule{0.166667em}{0ex}}.$$Therefore, it leaves F in a maximally mixed state and $\mathcal{M}$ in state ${\rho}_{\mathrm{init}}^{\mathcal{M}}$ given by Equation (1).Figure 6 represents the whole schedule. Figure 7 contains the evolution of the occupation probability p of F and the tunable energy $\mathcal{E}$.The energy balance follows that of Section 4. The only difference is that the processes are reversed. Accordingly, the number of $\mathcal{M}$ qubits generated per electron transfer from ${\mathcal{T}}^{+}$ to ${\mathcal{T}}^{-}$ is$$\phantom{\rule{0.166667em}{0ex}}\left(\right)open="\langle "\; close="\rangle ">\phantom{\rule{0.166667em}{0ex}}{N}_{{\mathcal{T}}^{+}\phantom{\rule{0.166667em}{0ex}}\to \phantom{\rule{0.166667em}{0ex}}{\mathcal{T}}^{-}}\phantom{\rule{0.166667em}{0ex}}$$

## 6. Informational Voltage Transformer

## 7. Remote Informational Carnot Engine

## 8. Network

## 9. Security

- First, we consider a generalization of the technique explained in our previous paper [71]. It is depicted in Figure 10. A and B can share a set of entangled qubit pairs, all of them in the same state, upon which they have previously agreed. Their Quiegs are set to work with a set of qubits in the state ${\rho}_{\mathrm{init}}^{\mathcal{M}}$ given by Equation (1). Accordingly, A uses qubits in a maximally mixed state and generates qubits in a ${\rho}_{\mathrm{init}}^{\mathcal{M}}$ state, according to the reverse mode of the Quieg.Next, A applies an entangling unitary operation to a pair of qubits ${\mathcal{M}}_{a},{\mathcal{M}}_{b}$. First, a Hadamard gate is applied to ${\mathcal{M}}_{b}$, so that it transforms into the state$$\begin{array}{cc}\hfill {\rho}_{2}^{{\mathcal{M}}_{b}}& =\frac{m}{2}\phantom{\rule{0.166667em}{0ex}}\left(\right)open="("\; close=")">\left|0\right.\u232a+\left|1\right.\u232a\phantom{\rule{0.166667em}{0ex}}\left(\right)open="("\; close=")">\left.\u23290\right|+\left.\u23291\right|\hfill & +\frac{1-m}{2}\phantom{\rule{0.166667em}{0ex}}\left(\right)open="("\; close=")">\left|0\right.\u232a-\left|1\right.\u232a\\ \phantom{\rule{0.166667em}{0ex}}\left(\right)open="("\; close=")">\left.\u23290\right|-\left.\u23291\right|\end{array}$$Then, a CNOT controlled by ${\mathcal{M}}_{b}$ takes the system from the joint state$$\begin{array}{cc}\hfill {\rho}_{2}^{{\mathcal{M}}_{a}{\mathcal{M}}_{b}}& =\left(\right)open="("\; close=")">m\phantom{\rule{0.166667em}{0ex}}\left|0\right.\u232a\left.\u23290\right|+(1-m)\phantom{\rule{0.166667em}{0ex}}\left|1\right.\u232a\left.\u23291\right|\otimes \frac{1}{2}\left(\right)open="("\; close=")">\left|0\right.\u232a\left.\u23290\right|+\left|1\right.\u232a\left.\u23291\right|+(2m-1)\left(\right)open="["\; close="]">\left|0\right.\u232a\left.\u23291\right|+\left|1\right.\u232a\left.\u23290\right|\hfill \end{array}$$$$\begin{array}{cc}\hfill {\rho}_{3}^{{\mathcal{M}}_{a}{\mathcal{M}}_{b}}=& \frac{1}{2}\left(\right)open="("\; close=")">m\left(\right)open="["\; close="]">\left|00\right.\u232a\left.\u232900\right|+\left|11\right.\u232a\left.\u232911\right|+m(2m-1)\left(\right)open="["\; close="]">\left|00\right.\u232a\left.\u232911\right|+\left|11\right.\u232a\left.\u232900\right|\hfill \\ +\end{array}$$Next, B takes the ${\mathcal{M}}_{b}$ part of the pair. The reduced states for ${\mathcal{M}}_{a}$ and ${\mathcal{M}}_{b}$ are equal. They are the partial traces of the matrix representing the state given by Equation (42) with respect to ${\mathcal{M}}_{b}$ and ${\mathcal{M}}_{a}$, respectively. They are given by$${\rho}_{3}^{{\mathcal{M}}_{a}}=\frac{1}{2}\left(\right)open="("\; close=")">\left|0\right.\u232a\left.\u23290\right|+\left|1\right.\u232a\left.\u23291\right|$$Then, when B needs to fuel its Quieg working in the direct mode, takes ${\mathcal{M}}_{a}$, so that it has the system ${\mathcal{M}}_{a},{\mathcal{M}}_{b}$ in the state ${\rho}_{3}^{{\mathcal{M}}_{a}{\mathcal{M}}_{b}}$ given by Equation (42). Subsequently, B applies a CNOT gate controlled by ${\mathcal{M}}_{b}$ and targeted at ${\mathcal{M}}_{a}$ followed by a Hadamard on ${\mathcal{M}}_{b}$. This takes the pair to the state given by$${\rho}_{4}^{{\mathcal{M}}_{a}{\mathcal{M}}_{a}}:=\phantom{\rule{0.166667em}{0ex}}\left(\right)open="("\; close=")">m\left|0\right.\u232a\left.\u23290\right|+(1-m)\left|1\right.\u232a\left.\u23291\right|\phantom{\rule{0.166667em}{0ex}}.$$This represents two $\mathcal{M}$ qubits in the ${\rho}_{\mathrm{init}}^{\mathcal{M}}$ state, with which the Quieg at B can execute two cycles. So that the net effect is equivalent to the remote informational voltage transformer of Section 6 or a Carnot engine of Section 7. The only qubit traveling through the line is ${\mathcal{M}}_{a}$ whose reduced state is maximally mixed and, therefore, useless for any unauthorized user E.
- If A and B have no previous entangled states or secret key, then they must obtain a private key communicating through a public line. The classical RSA protocol can be used, but quantum computers could crack the key. On the other hand, QKD protocols could be used that offer quantum privacy. However, QKD consumes resources, and this implies that there is an additional cost for security. There have been several QKD protocols described. The best-known one is the BB84 [72], after Bennet and Brassard, who proposed it in 1984. These protocols consume resources because at some point they generate entropy. To reset the system for another cycle, the entropy must be evacuated elsewhere and this needs qubits in a not maximally mixed state (for example, the BB84 generates entropy when Bob measures the qubits sent by Alice in the wrong basis).

## 10. Discussion

## 11. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Simplified schematic representation of the function of a Quantum Information Electrical Generator (Quieg) $\mathcal{Q}$. It features a communication line $\mathcal{C}$ through which qubits flow (dotted line). They enter $\mathcal{Q}$ in some state and exit at another one. A thermo-chemical reservoir $\mathcal{R}$ contains electrons at temperature T and chemical potential $\mu $. They can be moved to or from two terminals ${\mathcal{T}}^{+},{\mathcal{T}}^{-}$ that also work as thermo-chemical reservoirs at temperature T and chemical potentials ${\mathcal{V}}^{+},{\mathcal{V}}^{-}$ (dashed line). $\mathcal{Q}$ may work in the direct (

**a**) or reverse (

**b**) mode. In the first case, the entropy of the qubits in $\mathcal{C}$ increases, whereas in the second one it decreases.

**Figure 2.**Elements of a Quieg. In the direct mode, qubits $\mathcal{M}$ travel through the communication line $\mathcal{C}$. They are used as targets in a CNOT gate which is controlled by a fermionic system F. F features a tunable energy level $\mathcal{E}$ and can be either empty or filled with one single electron. After measuring $\mathcal{M}$ in the computational basis, F undergoes a suitable process whose outcome is either (1) extracting an electron from the thermo-chemical reservoir $\mathcal{R}$ at electro-chemical potential $\mu $ and deliver it to the negative terminal ${\mathcal{T}}^{-}$ at voltage ${\mathcal{V}}^{-}$ or (2) extracting an electron from the positive terminal ${\mathcal{T}}^{+}$ at voltage ${\mathcal{V}}^{+}$ and deliver it to the thermo-chemical reservoir $\mathcal{R}$. Gates ${G}_{1},{G}_{2},{G}_{3}$ represent the possibilities of putting F in thermo-chemical equilibrium with ${\mathcal{T}}^{+},{\mathcal{T}}^{-},R$, respectively. L is an electrical load that draws electrons from ${\mathcal{T}}^{-}$ at voltage ${\mathcal{V}}^{-}$ and delivers them to ${\mathcal{T}}^{+}$ at voltage ${\mathcal{V}}^{+}$.

**Figure 3.**Nodes in a five-station ring-network. Each one should feature the elements represented in Figure 2. The communication line $\mathcal{C}$ contains qubit strings with an identification of source and destination ports. Each station can either order or disorder the qubits so that another station should reverse the operation. The net effect is that the nodes whose Quiegs work in the direct mode can deliver power to a passive load, whereas the nodes with Quiegs running in the reverse mode will act as power stations.

**Figure 4.**Symbolic diagram of the stages of the system. The clock-like elements represent tuning knobs for setting the adequate values of $\mathcal{E}$. The shaded ovals indicate equilibrium with a thermo-chemical reservoir. First, F is measured by means of a CNOT gate and the result is placed at $\mathcal{M}$. According to its value, the upper or the lower part of the graph is used. If $\mathcal{M}$ is 1, the energy $\mathcal{E}$ is lowered so that F can be reversibly put in contact with ${\mathcal{T}}^{+}$. Then it is raised keeping F in equilibrium with ${\mathcal{T}}^{+}$, until the probability of occupation reaches the value ${p}_{a}$. At this point, F is put in isolation and $\mathcal{E}$ is lowered to put F in equilibrium with reservoir $\mathcal{R}$. Then, in contact with $\mathcal{R}$, $\mathcal{E}$ is further raised until the initial state is reached. If $\mathcal{M}$ is 0, the energy $\mathcal{E}$ is raised so that F can be reversibly put in contact with ${\mathcal{T}}^{-}$. Then it is lowered keeping F in equilibrium with ${\mathcal{T}}^{-}$, until the probability of occupation reaches the value ${p}_{b}$. At this point, F is put in isolation and $\mathcal{E}$ is raised to put F in equilibrium with reservoir $\mathcal{R}$. Then, in contact with $\mathcal{R}$, $\mathcal{E}$ is further lowered until the initial state is reached.

**Figure 5.**Evolution of the occupation probability p of F (

**a**) and the tunable energy $\mathcal{E}$; (

**b**) for a Quieg working in the direct mode.

**Figure 6.**Stages of the Quieg working in the reverse mode. First, $\mathcal{M}$ is measured. According to the result, the system follows the upper branch or the lower one. If the measurement determines that $\mathcal{M}$ is in state $\left|1\right.\u232a$, then steps a.6, a.5, a.4, a.3 are reversed and F is left in state $m\phantom{\rule{0.166667em}{0ex}}\left|0\right.\u232a\left.\u23290\right|\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}(1-m)\phantom{\rule{0.166667em}{0ex}}\left|1\right.\u232a\left.\u23291\right|$. If the measurement determines that $\mathcal{M}$ is in state $\left|0\right.\u232a$, then steps b.6, b.5, b.4, b.3 are reversed and F is left in state $(1-m)\phantom{\rule{0.166667em}{0ex}}\left|0\right.\u232a\left.\u23290\right|\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}m\phantom{\rule{0.166667em}{0ex}}\left|1\right.\u232a\left.\u23291\right|$. A CNOT controlled by F and targeted at $\mathcal{M}$ factorizes the system in a maximally mixed state for F and a $m\phantom{\rule{0.166667em}{0ex}}\left|0\right.\u232a\left.\u23290\right|\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}(1-m)\phantom{\rule{0.166667em}{0ex}}\left|1\right.\u232a\left.\u23291\right|$ state for $\mathcal{M}$.

**Figure 7.**Evolution of the occupation probability p of F (

**a**) and the tunable energy $\mathcal{E}$; (

**b**) for a Quieg working in the reverse mode.

**Figure 8.**The

**left**part of the figure represents a traditional voltage transformer. Magnetic coupling through a ferromagnetic core makes two electrical circuits exchange energy. The relation between the voltages is given by the ratio of the windings. In the

**right**system, the coupling is mediated by a communication line $\mathcal{C}$. Two nodes, each one with a Quieg share the qubits that travel through $\mathcal{C}$. By setting the ${p}_{a}$ parameter, the voltage difference at each node is selected.

**Figure 9.**When the temperatures at the two nodes of an informational voltage transformer are different, the electrical energy supplied at one end may be bigger than the one consumed at the other. In fact, energy is drawn from the temperature difference. This follows immediately if we consider the temperature as the work cost of resetting a maximally mixed qubit which is the same as the work obtained by completely mixing a pure state qubit (within some multiplicative factors).

**Figure 10.**Outline of the security based on shared entangled qubit pairs. The Quieg ${\mathcal{Q}}_{A}$ works in the reverse mode, outputting qubits in a ${\rho}_{\mathrm{init}}^{\mathcal{M}}$ state. ${\mathcal{M}}_{a},{\mathcal{M}}_{b}$ are two of them. After they are generated, ${\mathcal{M}}_{b}$ undergoes a Hadamard gate and controls a CNOT targeted at ${\mathcal{M}}_{a}.$. Then, ${\mathcal{M}}_{b}$ is taken to node B. This transfer is made before the system begins to work and may be done for a big number of ${\mathcal{M}}_{a},{\mathcal{M}}_{b}$ pairs. For example, it may take place before B signs-in into the network or when it renovates its subscription. It does not occur over the communication line $\mathcal{C}$. Then, ${\mathcal{M}}_{a}$ qubits circulate over $\mathcal{C}$. When B needs to use them at ${\mathcal{Q}}_{B}$, undoes the transformations performed by the CNOT and Hadamard gates. Then, B obtains two qubits in a ${\rho}_{\mathrm{init}}^{\mathcal{M}}$ ready to be used in the ${\mathcal{Q}}_{b}$ Quieg. Please note that if user E want to use ${\mathcal{M}}_{a}$, it will be useless for a Quieg working in the direct mode, because it is in a maximally mixed state.

© 2019 by the authors. 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.; Martin-Delgado, M.A.
Quantum Information Remote Carnot Engines and Voltage Transformers. *Entropy* **2019**, *21*, 127.
https://doi.org/10.3390/e21020127

**AMA Style**

Diazdelacruz J, Martin-Delgado MA.
Quantum Information Remote Carnot Engines and Voltage Transformers. *Entropy*. 2019; 21(2):127.
https://doi.org/10.3390/e21020127

**Chicago/Turabian Style**

Diazdelacruz, Jose, and Miguel Angel Martin-Delgado.
2019. "Quantum Information Remote Carnot Engines and Voltage Transformers" *Entropy* 21, no. 2: 127.
https://doi.org/10.3390/e21020127