# Enhanced Energy Distribution for Quantum Information Heat Engines

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## Abstract

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## 1. Introduction

## 2. Short Description of a Quantum Information Heat Engine

## 3. Scenario for Energy Distribution through Quantum Information in Multipartite Systems

- (i)
- any user ${A}_{i}$ can independently make that the other ones (${A}_{j\ne i}$) access the energy of one bit in their part of ${\rho}_{{A}_{1},\dots ,{A}_{n}}$. This possibility can be implemented by using generalized Greenberger–Horne–Zeilinger (GHZ) states, also known as cat states. An n-partite GHZ state is a pure state:$$\left|{\mathrm{GHZ}}_{n}\right.\u232a:={2}^{-1/2}\left({\left|0\right.\u232a}^{\otimes n}+{\left|1\right.\u232a}^{\otimes n}\right),$$
- (ii)
- only if all users agree, one of them can convert the energy. For this purpose, we may use a classically correlated mixed state:$${\rho}_{EP}:=\frac{1}{{2}^{n}}\phantom{\rule{0.166667em}{0ex}}\sum _{{i}_{1},\dots ,{i}_{n}}(1+{(-1)}^{{i}_{1}+\dots +{i}_{n}})\phantom{\rule{0.166667em}{0ex}}{\sigma}_{{i}_{1},\dots ,{i}_{n}}\phantom{\rule{0.166667em}{0ex}},$$$${\sigma}_{{i}_{1},\dots ,{i}_{n}}:=\left|{i}_{1},\dots ,{i}_{n}\right.\u232a\left.\u2329{i}_{1},\dots ,{i}_{n}\right|\phantom{\rule{0.166667em}{0ex}}.$$According to Equation (8), ${\rho}_{EP}$ is a mixture of all even parity pure states. It can only furnish one bit of information upon knowing all but one bits measured on the standard basis. We will now prove this. For that purpose, we will show that any operation on $n-2$ qubits, say $3,\dots ,n$, gives no information about qubit 1. Let ${K}_{j}$ be the Krauss operators; then, ignoring a possible normalization factor, the state of the system after it is:$${\rho}^{\prime}=\sum _{j}{K}_{j}{\rho}_{EP}{K}_{j}^{\u2020}\phantom{\rule{0.166667em}{0ex}},$$$${\rho}^{\prime}=\sum _{j}\sum _{{i}_{1},\dots ,{i}_{n}\in E}{k}_{j}{\sigma}_{{i}_{3},\dots ,{i}_{n}}{k}_{j}^{\u2020}\otimes {\sigma}_{{i}_{1},{i}_{2}}\phantom{\rule{0.166667em}{0ex}}.$$Next, we separate the ${i}_{3},\dots ,{i}_{n}\in E$ (set of even parity $n-2$-tuples of bits) and ${i}_{3},\dots ,{i}_{n}\in O$ (set of odd parity $n-2$-tuples of bits) contributions so that$$\begin{array}{ccc}\hfill \phantom{\rule{-14.22636pt}{0ex}}{\rho}^{\prime}=& & \phantom{\rule{-17.07164pt}{0ex}}\sum _{{i}_{3},\dots ,{i}_{n}\in E,j}\phantom{\rule{-14.22636pt}{0ex}}{k}_{j}{\sigma}_{{i}_{3},\dots ,{i}_{n}}{k}_{j}^{\u2020}\otimes ({\sigma}_{0,0}+{\sigma}_{1,1})\hfill \\ & +& \phantom{\rule{-17.07164pt}{0ex}}\sum _{{i}_{1},\dots ,{i}_{n}\in O,j}\phantom{\rule{-14.22636pt}{0ex}}{k}_{j}{\sigma}_{{i}_{3},\dots ,{i}_{n}}{k}_{j}^{\u2020}\otimes ({\sigma}_{0,1}+{\sigma}_{1,0}).\hfill \end{array}$$Subsequently, we trace over all but the first and second qubits to obtain ${\rho}_{12}^{\prime}$; let the constants ${C}_{E},{C}_{O}$ be defined by:$${C}_{E/O}:=\sum _{\begin{array}{c}{i}_{3},\dots ,{i}_{n}\in E/O\\ {\ell}_{3},\dots ,{\ell}_{n}\end{array},j}\phantom{\rule{-14.22636pt}{0ex}}{\left|\left.\u2329{\ell}_{3},\dots ,{\ell}_{n}\right|{k}_{j}\left|{i}_{3},\dots ,{i}_{n}\right.\u232a\right|}^{2},$$$${\rho}_{12}^{\prime}={C}_{E}({\sigma}_{0,0}+{\sigma}_{1,1})+{C}_{O}({\sigma}_{0,1}+{\sigma}_{1,0}),$$$${\rho}_{1}^{\prime}={C}_{E}({\sigma}_{0}+{\sigma}_{1})+{C}_{O}({\sigma}_{1}+{\sigma}_{0})=({C}_{E}+{C}_{O})\phantom{\rule{0.166667em}{0ex}}{\mathbb{1}}_{1},$$

## 4. Mutual Limitation between Communication and Energy Extraction

## 5. Simultaneous Supply of Information and Work Extraction Capability

- there is a typical subspace ${\Lambda}_{L,\u03f5,\delta}\phantom{\rule{0.166667em}{0ex}}\subset \phantom{\rule{0.166667em}{0ex}}{\mathcal{H}}^{\otimes \phantom{\rule{0.166667em}{0ex}}L}$, whose orthogonal projector is ${\Pi}_{L,\u03f5,\delta}$ that verifies$$\mathrm{T}\mathrm{r}\phantom{\rule{0.166667em}{0ex}}\left\{{\Pi}_{L,\u03f5,\delta}\phantom{\rule{0.166667em}{0ex}}{\rho}_{L}\right\}>1-\u03f5,$$$${\rho}_{B}:=\sum _{a=1}^{\mathcal{N}}{p}_{a}{\rho}_{a};$$
- the dimension ${d}_{\Lambda}$ of Λ is bounded by$${d}_{\Lambda}:=\phantom{\rule{0.166667em}{0ex}}dim\phantom{\rule{0.166667em}{0ex}}(\Lambda )\phantom{\rule{0.166667em}{0ex}}\le \phantom{\rule{0.166667em}{0ex}}{2}^{L(\delta +S({\rho}_{B}))};$$
- there is a communication protocol between Alice and Bob, whose information is coded (in [59], the codes are constructed by choosing a number of codewords independently, according to the a priori string probability for each codeword. The choice is supposed to be random and the results concerning the probability of error are averaged over the different possibilities) in the order in which Alice arranges the letters ${\rho}_{a}$ before sending them to Bob, where the frequency of letter ${\rho}_{a}$ is ${p}_{a}$, that achieves $\chi -5\delta $ bits per letter with a probability of error ${P}_{E}\le 10\u03f5$. When Bob receives the codewords sent by Alice, he performs a positive-operator valued measure (POVM) (positive-operator measure (POM) in the reference article [59]), defined by a collection of effects ${E}_{j}=\left|{\mu}_{j}\right.\u232a\phantom{\rule{0.166667em}{0ex}}\left.\u2329{\mu}_{j}\right|$, where all of the kets $\left|{\mu}_{j}\right.\u232a$ belong to the typical subspace Λ. All the ensuing process of decoding relays only on the result of this POVM.

## 6. Results and Conclusions

- Simple transmission of messenger systems whose state is not completely depolarized for the receiver. He can extract a work equal to ${k}_{B}\phantom{\rule{0.166667em}{0ex}}T\phantom{\rule{0.166667em}{0ex}}(ln2)\phantom{\rule{0.166667em}{0ex}}({S}_{max}-S(\rho ))$.
- Encrypted transmission of messenger systems though the use of previously entangled bipartite systems. This technique makes the transmitted system useless for illegitimate users that might intercept them. Quantum systems prove to be able to supply twice as much work as classical ones because of the same physics that is behind the feature of superdense coding in quantum communication protocols.
- In a multi-user scenario, where users initially share generalized GHZ states, any users can enable all the other ones to extract work.
- Also in a multi-user environment, if some correlated classical states are shared among all users, all but one can enable the other to extract work.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**One particle Szilard cylinder engine; a measurement is carried out to determine whether the particle is in the $\left|L\right.\u232a$ or $\left|R\right.\u232a$ state and the result is stored in the $\left|\ell \right.\u232a$ or $\left|r\right.\u232a$ state of an indicator which, in turn, determines whether the mechanical plunder is attached to the left or right end of the cable depicted in the figure. The work stored in the potential energy of the plunder after full expansion of the barrier in the cylinder is ${k}_{B}\phantom{\rule{0.166667em}{0ex}}(ln2)\phantom{\rule{0.166667em}{0ex}}T$, and is exactly equal to the Landauer’s work that would be needed to reset the indicator before the beginning of the next cycle. If the system works as a QIHE, instead of this energy, a fresh entropy-free indicator bit is supplied.

**Figure 2.**Magnetic Quantum Information Heat Engine; a spin-$\frac{1}{2}$ particle S lies in the magnetic field generated by an electrical current $I(t)$ circulating through a coil C. An ancilla A is used to measure the z component of the magnetic moment of S; the result determines the value of the magnetic field generated by $I(t)$. Then, the field is gradually turned off, while S keeps in thermal equilibrium with a reservoir $R(T)$ at temperature T. At the end of the process, an energy W is stored at the electrical battery B; its value is $W={k}_{B}\phantom{\rule{0.166667em}{0ex}}(ln2)\phantom{\rule{0.166667em}{0ex}}T$, on account of the increase of entropy of S, which enters the isothermal process with zero entropy and exits completely depolarized. Again, this energy is equal to the Landauer’s energy needed to reset the ancilla for the next cycle. If the system works as a QIHE, instead of this energy, a fresh entropy-free ancilla bit is supplied.

**Figure 3.**Part (

**a**) represents a way of supplying energy to a remote station at temperature ${T}_{2}$, consisting of sending physical systems in a quantum state ${\rho}_{i}$ whose entropy increases after fueling a QIHE and return as completely unpolarized states. As an example, photons can be sent to the remote station under a particular polarization and return completely unpolarized. The extracted work is ${W}_{2}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{k}_{B}\phantom{\rule{0.166667em}{0ex}}(ln2)\phantom{\rule{0.166667em}{0ex}}{T}_{2}$; In (

**b**), a remote Carnot cycle is depicted, where entropy is interchanged between the primary station at temperature ${T}_{1}$, where entropy is pumped out of the traveling physical system at a work cost ${W}_{1}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{k}_{B}\phantom{\rule{0.166667em}{0ex}}(ln2)\phantom{\rule{0.166667em}{0ex}}{T}_{1}$ , and the remote station. In this scenario, the net work gain per transmitted qubit is $W={W}_{2}-{W}_{1}=\phantom{\rule{0.166667em}{0ex}}{k}_{B}\phantom{\rule{0.166667em}{0ex}}(ln2)\phantom{\rule{0.166667em}{0ex}}(\phantom{\rule{0.166667em}{0ex}}{T}_{2}-{T}_{1})$ and, taking into account that the heat taken from ${T}_{2}$ is ${Q}_{2}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{k}_{B}\phantom{\rule{0.166667em}{0ex}}(ln2)\phantom{\rule{0.166667em}{0ex}}{T}_{2}$, it follows that $W=\phantom{\rule{0.166667em}{0ex}}{Q}_{2}\phantom{\rule{0.166667em}{0ex}}(\phantom{\rule{0.166667em}{0ex}}1-\frac{{T}_{1}}{{T}_{2}}),$ which is the standard result for Carnot cycles.

**Figure 4.**The source P sends ${\rho}_{{A}_{1}{A}_{2}{A}_{3}}$ states to remote systems ${A}_{1},{A}_{2},{A}_{3}$.

**Figure 5.**When some source of quantum states is used to convey classical information, it may also serve as an ancilla to extract work from a thermal source. Equation (20) sets a mutual limitation for both functions. This figure represents a distribution of the functionality of a source $\{{\rho}_{a},{p}_{a}\}$ per transmitted letter between energy conversion $\mathcal{E}$ (hatched), communication $\mathcal{C}$ (white) and a useless part $\u2329{S}_{a}\u232a$ (gray), showing that they add up to $\mathcal{M}$.

**Figure 6.**Pictorial view of the intuitive idea behind the refactorization process described in the main text. A codeword is a sequence of L letters and, accordingly, is an element of a ${2}^{L\phantom{\rule{0.166667em}{0ex}}\mathcal{M}}$-dimensional Hilbert space ${\mathcal{H}}_{L}$. Codewords are not evenly distributed in ${\mathcal{H}}_{L}$, but are concentrated in a typical subspace Λ of dimension ${d}_{\Lambda}$. After enlarging ${\mathcal{H}}_{L}$ by a tensor product of the codewords with a pure state ${\left|0\right.\u232a}_{D}$ of an ancillary ${d}_{\Lambda}$-dimensional system D, we have a bigger system whose states are still concentrated in a ${d}_{\Lambda}$-dimensional subspace. A unitary transformation can always be found to map this subspace into the subspace defined by the tensor product of a pure state ${\left|0\right.\u232a}_{L}\in {\mathcal{H}}_{L}$ and ${\mathcal{H}}_{D}$. This space is factorizable; the first part is fed into a QIHE and the second part undergoes the decoding process to extract the information sent from the emitter.

**Figure 7.**Schematic view of the process described in Section 5 for the simultaneous extraction of work and transmission of messages. There is a source of copies of a physical system M available to user A (Alice). Copies are in one of a predefined set of states ${\rho}_{a}$ (letters), each with a probability ${p}_{a}$. Alice can arrange blocks of L copies in predefined codewords, which have been previously agreed with user B (Bob). The relative frequencies expected for each letter must follow the probability distribution. When the codeword reaches Bob, it undergoes a tensor product with a pure state ${\left|0\right.\u232a}_{D}$ of an ancillary system D whose dimension is equal to that of the typical subspace Λ. The enlarged codeword then undergoes a unitary transformation U which, with at least $1-\u03f5$ probability, factorizes it into two parts; one of them is a pure state ${\left|0\right.\u232a}_{L}$ in a ${2}^{L\mathcal{M}}$-dimensional space. The other factor is equivalent to the projection of the codeword on the typical space Λ, which can be further decoded to recover the original message. The state ${\left|0\right.\u232a}_{L}$ can be fed to a QIHE (QIHE${}_{1}$) to extract a work ${W}_{1}={k}_{B}\phantom{\rule{0.166667em}{0ex}}T\phantom{\rule{0.166667em}{0ex}}(ln2)\phantom{\rule{0.166667em}{0ex}}\mathcal{M}\phantom{\rule{0.166667em}{0ex}}L$. Part of it, ${W}_{D}={k}_{B}\phantom{\rule{0.166667em}{0ex}}T\phantom{\rule{0.166667em}{0ex}}(ln2)\phantom{\rule{0.166667em}{0ex}}{log}_{2}{d}_{\Lambda}$, has to be used to operate another QIHE (QIHE${}_{2}$) acting reversely to generate a pure state ${\left|0\right.\u232a}_{D}$ out of completely depolarized systems.

**Figure 8.**Graphical representation of the mutual limitation between $\mathcal{E}$ (extractable work per letter) and $\mathcal{C}$ (communication bits per letter) in the asymptotic limit. The hatched area contains the achievable values for $\mathcal{E},\phantom{\rule{0.166667em}{0ex}}\mathcal{C}$. The maximum value of $\mathcal{C}$ is Holevo’s bound $\chi :=S({\rho}_{B})-\u2329\phantom{\rule{0.166667em}{0ex}}{S}_{a}\phantom{\rule{0.166667em}{0ex}}\u232a$. If Alice and Bob agree on reducing $\mathcal{C}$ below this value, a higher value for $\mathcal{E}$ can be achieved, with a maximum of ${\mathcal{E}}_{M}:={k}_{B}\phantom{\rule{0.166667em}{0ex}}T\phantom{\rule{0.166667em}{0ex}}(ln2)\phantom{\rule{0.166667em}{0ex}}\left(\mathcal{M}-\u2329\phantom{\rule{0.166667em}{0ex}}{S}_{a}\phantom{\rule{0.166667em}{0ex}}\u232a\right)$. The blue line represents the case when the letters are orthogonal pure states with equal probabilities. This is the situation which offers maximum possibilities to the users.

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Diaz de la Cruz, J.M.; Martin-Delgado, M.A. Enhanced Energy Distribution for Quantum Information Heat Engines. *Entropy* **2016**, *18*, 335.
https://doi.org/10.3390/e18090335

**AMA Style**

Diaz de la Cruz JM, Martin-Delgado MA. Enhanced Energy Distribution for Quantum Information Heat Engines. *Entropy*. 2016; 18(9):335.
https://doi.org/10.3390/e18090335

**Chicago/Turabian Style**

Diaz de la Cruz, Jose M., and Miguel Angel Martin-Delgado. 2016. "Enhanced Energy Distribution for Quantum Information Heat Engines" *Entropy* 18, no. 9: 335.
https://doi.org/10.3390/e18090335