Quantum energy teleportation in phase space quantum mechanics

The phase space representation of quantum mechanics, also referred as deformation quantization by many authors, provides a general method to investigate quantum systems implementing notions from classical physics [1],[2]. The distinctive characteristic of this quantization approach lies in the role assumed by the algebra of quantum observables, which is not represented by a set of operators defined on a Hilbert space [3]. Instead the observables are associated with smooth complex-valued functions on the classical phase space, where the usual commutative point-wise product is replaced by a non-commutative product termed as the star-product [4]. A central feature within this formulation is determined by the Wigner distribution function. This function provides a phase space representation of the density operator and captures all auto-correlation properties and transition amplitudes of a quantum system. The phase space formalism has undoubtedly made significant contributions not only to pure mathematics [5], [6], but it has also proved to be a reliable technique in the understanding of many physical complex systems [7],[8], including more recently, finite dimensional multi-qubit systems [9],[10], distinct aspects of the loop representation of quantum cosmology and quantum gravity [11],[12],[13],[14] and novel connections with the path integral formalism [15], [16], to cite a few examples. However, despite the broad relevance of the phase space formalism, a limited understanding remains regarding its applicability to recent discovered phenomena involving the interplay between quantum information theory and quantum field theory, such as Quantum Energy Teleportation (QET) [17],[18], black hole loss information problem [20] and entanglement harvesting [19], among others.

Following some ideas formulated in [9],[21], in this paper we propose to analyse the protocol of Quantum Energy Teleportation within the phase space formalism of quantum mechanics. This protocol, originally developed within the framework of quantum field theory and many body systems, were first propose by Hotta [17] as a method to transfer energy, through local operations and classical communication (LOCC) with an entangled partner, by locally generating regions with negative energy density. This technique, based on the entanglement of the ground state, has recently found many applications in cold trapped ions [22], relativistic quantum information [23] and black hole evaporation [24]. With the aim to describe the QET protocol, we compute the variation of its associated Wigner quasi-probability distribution under the sequence of local unitarities applied on the initial ground state. As we will demonstrate below, the Wigner and the Husimi phase space functions enable us to quantify the amount of coherence and entanglement consumed throughout the protocol, with the aim to extract energy without requiring any physical transport between the source and the receiver. Finally, a correlation between the entanglement negativity and the Wehrl entropy for the system is discussed.

The paper is organized as follows, in section 2 we briefly introduce the fundamental concepts of phase space quantization, Wigner functions and star products in multi-qubit systmes. Then, in section 3 we analyse the QET protocol within the phase space formalism, focusing our attention on the evolution of the entanglement. Finally, we introduce some concluding remarks in section 4.

The phase space formulation of quantum mechanics possesses an extensive history. In 1932 Wigner introduced his renowned function, which has been widely applied in numerous fields of physics, mathematics, electronics and geophysics [1],[2]. In 1949, Moyal found that the inverse of the Weyl correspondence rule can be obtained by using the Wigner transform, enabling the association of an operator defined on a Hilbert space with a function on the classical phase space [3]. Consequently, the expectation value of an operator can be expressed as the statistical average of the corresponding phase-space function, where the probability density is determined by the Wigner function related to the density matrix of the quantum state. In this manner, quantum mechanics can be formally framed as a statistical-like theory on classical phase space. This formalism, known as phase space quantum mechanics proves to be equivalent to the standard Schrödinger and Heisenberg representations. Lately, in the seminal papers [4], it was shown that the phase space formulation of quantum mechanics inherently possesses a noncommutative deformation of the algebraic structure of classical observables, where the usual pointwise product of functions should be replaced by the so called star product (also known as the Moyal product) in order to recover the operator algebra structure of quantum mechanics. This approach currently referred to as deformation quantization consists in a entirely new formulation of quantum mechanics in terms of phase space algebras instead of employing operators on Hilbert spaces. Most recently, M. Kontsevich proved that there is a one-to-one correspondence between equivalent classes of star products and Poisson structures, showing that every Poisson manifold, and as a consequence every classical system, admits a canonical quantization, which is in fact unique up to formal equivalence [5].

Originally, the conventional expression for the Wigner function was restricted to continuous non-relativistic spinless systems, such as quantum particles on flat phase spaces. It is customarily introduced via the Weyl-Wigner transform, which maps an operator $\hat{A}$ defined on a Hilbert space to a classical phase space function $W_{\hat{A}}(x,p)$ [2], as

For simplicity, we will focus on systems with a single degree of freedom, but the generalization to more dimensions follows straightforwardly. Now, let $\hat{\rho}$ be a normalized density operator associated with the quantum state $\psi(x)$, that is, a self-adjoint positive semi-definite operator with trace one. By making use of the Weyl-Wigner transform (1), one may see that its corresponding phase space function up to multiplication by a constant reads

This special representation of the density operator embodies the celebrated Wigner function [1], which allows us to characterize both pure and mixed quantum states by using probability distributions in phase space. However, in order to reflect the probabilistic nature of quantum theory, the Wigner distribution may take negative values in certain regions of phase space. As a result, it cannot be interpreted as a true probability density and is often referred to as a quasi-probability distribution in the literature [7]. Furthermore, the Wigner function plays an essential role in determining the expectation values of operators. This is achieved by integrating the associated functions over the phase space as follows

Moreover, the Wigner function obeys the normalization condition

These expressions underscore the utility of the Wigner function in connecting quantum mechanical operators with their classical counterparts, enabling a bridge between both classical and quantum descriptions. This formalism can also be written in terms of the displacement and parity operators given by the relation

where the operator $\hat{D}(x,p)$ denotes the unitary displacement operator and $\hat{\Pi}$ represents the parity operator which acts according to

where $\ket{x}$ and $\ket{p}$ comprise a complete set of eigenstates of the position and momentum operators respectively [25]. It has been recently understood that a generalization of the phase space quantization approach for quantum systems with intrinsic symmetry groups, such as spin systems, can be achieved by using the Stratonovich-Weyl correspondence [21]. The premise behind the Stratonovich-Weyl correspondence implies that a linear bijective mapping between operators acting on a Hilbert space and functions defined on phase space, also called symbols, can be realized via an appropriate kernel $\hat{\Delta}(\Omega)$, where $\Omega$ is any parametrization of the phase space, which satisfies the following physical motivated postulates [26]:

1. 1.

   The mapping $W_{\hat{A}}(\Omega)=\tr\left[\hat{A}\hat{\Delta}(\Omega)\right]$ is a one-to-one linear map.

2. 2.

   $W_{\hat{A}}(\Omega)$ is a real valued function, which implies that the operator kernel $\hat{\Delta}(\Omega)$ must be hermitian.

3. 3.

   $W_{\hat{A}}(\Omega)$ satisfies standarization, which means that the integral over all phase space $\int_{\Omega}W_{\hat{A}}(\Omega)d\Omega=\tr\hat{A}$ exists and $\int_{\Omega}\hat{\Delta}(\Omega)d\Omega=\hat{1}$.

4. 4.

   Traciality, that is $\int_{\Omega}W_{\hat{A}}(\Omega)W_{\hat{B}}(\Omega)d\Omega=\tr\left[\hat{A}\hat{B}\right]$.

5. 5.

   Covariance, which indicates that if an operator $\hat{A}$ is invariant under global unitary operators, then $W_{\hat{A}}(\Omega)$ also remains invariant.

In the case of systems with continuous variables, the kernel operator $\hat{\Delta}(\Omega)$ proves to be proportional to $\hat{D}(x,p)\hat{\Pi}\hat{D}^{\dagger}(x,p)$, where the parametrization is given by $\Omega=\left\{x,p\right\}$. For other systems, it is crucial to select both the kernel operator and the parametrization phase space coordinates in a manner that the symmetries of the physical system under study are precisely captured [9].

In this section, we analyse the protocol called Quantum Energy Teleportation (abbreviated as QET) within the phase space formalism of quantum mechanics. The QET model, were first propose by Hotta [17], as a method to transfer energy through local operations and classical communication (LOCC) with an entangled partner, but without requiring any physical transport between the source and the receiver. This approach maintains local energy conservation and preserves causality [18],[32],[33],. The protocol of QET was originally developed within the framework of quantum field theory and many body systems as a technique to generate locally regions with negative energy density [33], although it has also found applications in spin chains systems [17], entangled qubits [18], on maximizing negative energy distributions for stress-energy tensor [23], algorithmic cooling protocols [34] and extracting energy from black holes [24], to name a few examples. Recently it has been experimentally confirmed using magnetic nuclear resonance on a bipartite quantum system [35], and in a superconducting quantum hardware [36].

where $h$ and $k$ are positive constants with dimensions of energy and the operators $\sigma^{A}_{z}$, $\sigma^{B}_{z}$, $\sigma^{A}_{x}$ and $\sigma^{B}_{x}$ denote the $z$ and $x$ components of the Pauli operators for Alice and Bob respectively. Since the Hamiltonian operator is a positive-semidefinite operator, i.e. $\hat{H}\geq 0$, this implies that all its eigenvalues are non-negative [37]. In particular, the ground state is given by

where $\ket{0}_{A}$, $\ket{0}_{B}$ denote the eigenstates of $\sigma^{A}_{z}$, $\sigma^{B}_{z}$ with eigenvalue $+1$ and $\ket{1}_{A}$, $\ket{1}_{B}$ stand for the eigenstates with eigenvalue $-1$ respectively. By using expression (16) the Wigner function associated to the ground state reads

where $\hat{\rho}_{g}=\ket{g}\bra{g}$ is the density operator associated to the ground state (22). From the Wigner function $W_{\hat{\rho}_{g}}(\vec{\theta},\vec{\phi})$ obtained above, we can observe that it cannot be factored into two separate functions, each depending on $(\theta_{1},\phi_{1})$ or $(\theta_{2},\phi_{2})$ only. This property shows that the non-factorizability of the ground state is associated with the non-factorizability of the corresponding Wigner function. Moreover, the Wigner function depends on the variables $\phi_{1}$ an $\phi_{2}$ only through the non-classical interference term $\cos(\phi_{1}+\phi_{2})$, characteristic feature of entanglement within the phase space description [38], which are related to the non-trivial topological properties of the phase space $\mathbb{S}^{2}\times\mathbb{S}^{2}$. Moreover, the Wigner function (23) satisfies the normalization condition

where $\Omega=\mathbb{S}^{2}\times\mathbb{S}^{2}$, with $0\leq\theta_{1},\theta_{2}\leq\pi$ and $0\leq\phi_{1},\phi_{2}\leq 2\pi$, and the integral Haar measure is written as $d\Omega=(1/2\pi)\sin\theta\,d\theta\,d\phi$. The expectation value of the Hamiltonian on the ground state can be obtained as

where $W_{\hat{H}}(\vec{\theta},\vec{\phi})$ corresponds to the phase space symbol associated to the Hamiltonian operator $\hat{H}$, and has de form

From this expression we can observe that the first term encompasses the information of the free Hamiltonians $\hat{H}_{A}$ and $\hat{H}_{B}$, the second term incorporates the interaction $\hat{V}$, and the constant last term is added to reduce to zero the expectation value of the Hamiltonian operator for the ground state. Furthermore, the Wigner function of the ground state satisfies the star-genvalue equation

where the star product for a two qubit system, in the integral representation reads

where the region $\Omega=(\mathbb{S}^{2}\times\mathbb{S}^{2})^{2}$ and $d\Omega$ represents the normalized Haar measure of a qubit phase space as state previously.

In this paper we have analysed the QET protocol within the phase space formulation of quantum mechanics. Following some ideas introduced by the Weyl-Wigner quantization of finite dimensional spin systems, we have obtained the Wigner function and the star-product of the minimal QET model, which corresponds to a pair of two entangled qubits in the presence of a transverse magnetic field. Then, by using the integral properties of the Winger quasi-probability distribution and the symbols corresponding to a sequence of unitarities, we showed that the teleported energy is proportional to the amount of entanglement present in the initial ground state, which is consistent with previous results. Finally, with the aid of the Husimi $Q$-function, we analysed the Wehrl entropy of the protocol. As we have seen, the Wehrl entropy enables us to quantify the consumption of coherence and entanglement of the initial ground state in order to extract energy, resulting in a final state with purely classical correlations. We expect the results established here may clarify several aspects of the energy teleportation protocols via local operations and classical communication, found in the literature. As future work, we pretend to implement our quantization approach to the case of quantum fields, in order to explore how quasi-probability distributions describe the violation of classical energy conditions. This will enable us to investigate the QET protocol in more general scenarios. This will be done elsewhere.

The authors would like to acknowledge support from CONAHCYT-Mexico. JBM thanks to the Dipartimento di Fisica “Ettore Pancini” for the kind invitation and its generous hospitality. Computations have been made using wxMaxima.