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/).

A new scenario for energy distribution, security and shareability is presented that assumes the availability of quantum information heat engines and a thermal bath. It is based on the convertibility between entropy and work in the presence of a thermal reservoir. Our approach to the informational content of physical systems that are distributed between users is complementary to the conventional perspective of quantum communication. The latter places the value on the unpredictable content of the transmitted quantum states, while our interest focuses on their certainty. Some well-known results in quantum communication are reused in this context. Particularly, we describe a way to securely distribute quantum states to be used for unlocking energy from thermal sources. We also consider some multi-partite entangled and classically correlated states for a collaborative multi-user sharing of work extraction possibilities. In addition, the relation between the communication and work extraction capabilities is analyzed and written as an equation.

The study of the relation between computational irreversibility and energy can be traced back as far as 1961 when Landauer [

Another fruitful relation between energy and information began in the

Szilard engines and Maxwell demons analysis are often prone to misunderstandings. In our view, they could be avoided with careful accounting for information and energy trade-off in the presence of feed-back systems [

It is now well understood that the availability of a thermal bath allows for a trade-off between information and work. The devices that carry out this conversion are known as

Other magnetic machines (classical and quantum) have been proposed for other different thermodynamic cycles, especially cooling systems [

The role of standard entropy in Information Heat Engines is relevant for average values of extracted work. Generalizations to the so called

In this contribution, a new scenario for communication and energy distribution is defined that allows the consideration of novel possibilities concerning the security and conditions of use for both the information and energy convertibility stored in a shared quantum state.

Quantum information resources for communication have already been adapted to energy distribution [

In

Information Heat Engines are devices that cyclically convert thermal energy into useful work at the expense of degrading information. They are periodically fueled with auxiliary systems, whose entropy is increased, although their energy is conserved. The power delivered by the engine is drawn from a single thermal reservoir. This is in contrast with other cyclic engines that decrease the internal energy of the systems that are supplied as fuel. It should also be remarked that they comply with all three principles of thermodynamics. In the following, we assume that the engine is a physical system with a local Hamiltonian and no energy is exchanged with the fueling system. If the evolution is divided into intervals with either Hamiltonian evolution or reversible thermal equilibrium with a bath at temperature

In a single particle cylinder Szilard engine, as shown in

In a magnetic quantum information heat engine [

The resource theory behind energy extraction [

In the previous sections, it has been established that a physical system whose quantum state is not completely depolarized can be transformed into another state with greater entropy and obtain work in the process, provided the availability of a thermal bath and an Information Heat Engine, like a Szilard or a magnetic information engine as described in

Now, we set up a new scenario for energy distribution: a power information plant

The next important issues about this scenario are the security and shareability. As a first scenario, one could send some messenger qubits from the power plant

As an example,

It is important to note that interception of just the qubit being sent from

A classical version of this protocol can be devised where a set of completely correlated random bits would be used instead of the Bell states. However, completely correlated pairs of random bits carry one bit of entropy, and, consequently, the extractable work

There are further possibilities: one can use multipartite states

any user

only if all users agree, one of them can convert the energy. For this purpose, we may use a classically correlated mixed state:

According to Equation (

Next, we separate the

Subsequently, we trace over all but the first and second qubits to obtain

According to the previous paragraphs, posting messenger quantum states from

Holevo [

It is understood that the communication protocol assigns a sequence of states in the alphabet, or a

We are now going to use this theorem [

Let us now describe a bipartite scenario, with

According to Equation (

The derivation of Equation (

Given any positive real numbers

there is a typical subspace

the dimension

there is a communication protocol between Alice and Bob, whose information is coded (in [

Our next purpose is to use this result to closely factorize the state

In order to guarantee the possibility of factorization in the general case, we enlarge the Hilbert space

The enlarged codewords live now in the

Next, we define the enlarged typical subspace

As the next step, we define a unitary transformation

We also know, from the theory of QIHE, and ultimately from the Second Principle of Thermodynamics, that

Given the set of states

A way to do it would be a previous agreement on always sending blocks of

In the limit,

This value can also be reached if Alice and Bob arrange a particular ordering of the states, declining to use them for communication.

New domains for the application of quantum information theory, especially cryptography and entanglement have been presented. It is assumed that users have access to suitable QIHEs. In particular, this paper describes protocols for work extraction from a single thermal bath through the distribution of messenger qubits, with increasingly complex features. Procedures for requiring collaboration from other users to unlock work extraction are also presented using both strongly entangled and classically correlated multipartite quantum ancillas. Specifically, the following possibilities have been presented:

Simple transmission of messenger systems whose state is not completely depolarized for the receiver. He can extract a work equal to

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.

In order to find a relation between the two possible uses of messenger systems, a mutual limitation between communication and energy has been derived in

We thank the Spanish MINECO (Ministerio de Economia y Competitividad) grant FIS2012-33152, FIS2015-67411, the CAM (Comunidad Autonoma de Madrid) research consortium QUITEMAD+ S2013/ICE-2801, and the U.S. Army Research Office through grant W911NF-14-1-0103 for partial financial support.

Jose M. Diaz de la Cruz and Miguel Angel Martin-Delgado conceived and discussed the contents of the article; Jose M. Diaz de la Cruz wrote the paper; Miguel Angel Martin-Delgado supervised the work. Both authors have read and approved the final manuscript.

The authors declare no conflict of interest.

One particle Szilard cylinder engine; a measurement is carried out to determine whether the particle is in the

Magnetic Quantum Information Heat Engine; a spin-

Part (

The source

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 (

Pictorial view of the intuitive idea behind the refactorization process described in the main text. A codeword is a sequence of

Schematic view of the process described in

Graphical representation of the mutual limitation between