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Experimental activation of strong local passive states with quantum information

Strong local passivity is a property of multipartite quantum systems from which it is impossible to locally extract energy. A popular example is the vacuum state of a quantum field theory from which it is impossible to extract zero-point energy. However, adding classical communication between different partitions of the system through so-called 'quantum energy teleportation' protocols makes it possible to activate local energy of strong local passive states, including the ground state of quantum systems if this state displays entanglement. Here we report the first experimental realization of both the activation of a strong local passive state and the demonstration of a quantum energy teleportation protocol by using nuclear magnetic resonance on a bipartite quantum system.

Introduction.-Methods to extract and transfer energy from physical systems at the quantum scale have been developed recently using tools from quantum information processing and quantum thermodynamics . But can these tools allow us to activate energy extraction from quantum systems in which the presence of entanglement locks outgoing energy flows through general local interactions [24,25]? Furthermore, can these quantum information tools activate energy locally hidden in regions of an entangled ground state? One may be tempted to answer 'no', since this sounds akin to the activation of zero-point energy, which is widely believed to be impossible. However, we will discuss that the answer to these questions is nuanced and that zero-point energy density can be activated using quantum informational tools.
The quantum states from which any local action that attempts to extract energy through a single subsystem would fail, merely boosting the total energy across the system, receive the name of strong local passive (SLP) states [24,25], see Fig. 1, also known as CP-local passive states. This distinctive property of strong local passivity in a quantum system is associated with the presence of entanglement in the low-energy eigenstates, as well as the limitations to creating correlations within the system via local operations alone. Indeed, a sufficient condition for a quantum system to exhibit this phenomenon is to have a Hamiltonian with a non-degenerate fully entangled ground state, for which there will be a threshold population probability (and a critical temperature for the case of Gibbs states) to be in an SLP state [24]. The necessary and sufficient conditions to have this property were presented as a theorem in [25].
Strong local passivity provides new insights into understanding the emergent thermodynamic behavior arising from the interplay between entanglement and localization, such as understanding the allowed flows of energy and information within an entangled quantum system. Along these lines, a fundamental question is how/when strong local passivity can be broken. Finding methods to activate the non directly available energy in SLP states can bring fascinating physical scenarios. One of them is the case of activating entangled ground states. An entangled ground state of a multipartite quantum system can possess regions with positive (and negative) energy density due to its entanglement. However, any locally positive energy density is not directly available since any action attempting to extract it directly will only give energy to the whole system. Would it be possible to activate this energy by driving the system out of the ground state in a different way?
This question was answered by Masahiro Hotta, who showed that, under some conditions, it is possible to activate strong local passive states by allowing local operations and classical communication (LOCC) to exploit correlations between distant points of the system [24,[26][27][28][29][30][31][32][33]. He introduced the family of protocols known under the general name of quantum energy teleportation (QET). Specifically, in a QET protocol, first, a local re-arXiv:2203.16269v1 [quant-ph] 30 Mar 2022 mote measurement is made on a subsystem (A), far from the subsystem where the energy is blocked (B), and then the outcome is communicated to B. Because of the correlations, the information gained allows B, to some extent, to predict and design an informed local operation to activate the previously inaccessible energy, see Fig. 1. It is important to note that the energy injected during A's measurement does not disappear from A's surroundings, nor appears around B. The protocol guarantees this by using measurement operators that commute with both the local Hamiltonian of B and the interacting Hamiltonian term for A and B. This means that if the system was originally in an entangled ground state, the QET protocol could excite and drive the system's state in such a way that activation of regions with initially zero-point energy can be possible, as was shown for the activation of the vacuum state of a field theory [30]. Beyond the importance of QET in the activation of passive states, QET has also been suggested to be relevant to understanding a broad range of situations -from the black-hole information loss problem [26,[34][35][36] and violations of energy conditions in quantum field theory in curved spacetimes [37], to technological applications to local cooling of many-body systems with restricted measurements [12].
However, QET had not yet been realized in an experiment. Experimental proposals did exist (for instance, using a semiconductor exhibiting the quantum Hall effect [38]), but no actual experiments. This work presents the first experimental realization of quantum energy teleportation, demonstrating both energy extraction from a strong local passive state and the activation of local zeropoint energy density. The experiment was carried out using nuclear magnetic resonance on a system of three qubits in the ground state of an interacting simulated Hamiltonian. The experimental results show local energy extraction from a subsystem with respect to which a quantum state is SLP, beyond local and environmental noise, without energy transfer through the system. Our experiment demonstrates the feasibility of the control required for a QET protocol and the evidence of activation of local zero-point energy density in an entangled ground state under experimental conditions. Furthermore, we present an optimized fully-unitary QET model and an analytical solution for the maximum extractable energy via QET protocols, providing a physical understanding of upper bounds derived in previous work [32].
We start with a summary of the minimal QET model [32] to explain how it is possible to break strong local passivity, particularly to activate regions of an entangled ground state. Then, we present our optimized fully unitary version of QET implemented in the experiment. Finally, we show the experimental results of energy activated in strong local passive states and conclusions.
Minimal QET model.-Consider two interacting qubits, A and B, with a Hamiltonian that has a nondegenerate fully entangled ground state. An example of such a Hamiltonian is with where h a , h b and κ are positive constants and the function f (h a , h b , κ) is chosen such that the ground state of the full Hamiltonian, |g , has vanishing expectation values for each term of the Hamiltonian (i.e. g| H A |g = g| H B |g = g| V |g = 0), for convenience and without loss of generality. In particular for this case, The system has a non-degenerate fully entangled ground state |g , where , satisfying the sufficient condition to have SLP states (see ref. [24] for its critical temperature and threshold probability). Note that, while the total Hamiltonian is a nonnegative operator (H ≥ 0, since its lowest eigenvalue is zero) H b and H b + V have negative eigenvalues which could yield negative energy density at B. The QET protocol can locally activate that energy in B surroundings as described below. Step The minimal QET protocol consists of the following steps implemented by Alice and Bob (Fig. 2): Step 1. Alice measures subsystem A using a POVM with measurement operators that commute with H b and V to ensure that the energy injected during the measurement on A does not raise the energy on B.
Step 2. Alice communicates the result of the measurement, µ, to Bob in a time t µ shorter than the coupling time scale t c ∼ 1/κ; to avoid that the energy infused on A in the first step propagates to B during that time.
Step 3. Based on the outcome µ, Bob implements an optimized local unitary on B, U B (µ), to extract energy locally through B.
Note that the extracted energy corresponds only to previously locked energy in B's surroundings since the steps are designed so that no energy is propagated from A to B during the protocol.
The total effect of repeatedly and identically applying the QET protocol's steps on a strong local passive state, ρ, is described by the following density matrix [32]: where M a (µ) is the measurement operator of a POVM on A, that commutes with H b and V, µ is the corresponding outcome, and U b (µ) is an informed unitary that maximizes the energy extraction depending on the outcome. Then, the amount of energy extracted locally from B on average is given by since [M a , V ] = [M a , H b ] = 0 and given that the expectation value of each term of the Hamiltonian was set to have a vanishing value (see appendix C3 in [39] and [32] for detailed calculation). Note that when the measurement outcome is not communicated to B (U b independent of µ), it is impossible to activate and extract energy on average from B. From eq.(5), the amount of activable energy is bounded as 0 ≤ −∆E b ≤ −λ min , where λ min is the most negative eigenvalue of H b + V . This upper bond is tight when the POVMs are proportional to projective operators. In particular, by implementing the QET protocol on the ground state of the Hamiltonian in eq.(1), the maximum amount of activable energy in B is given by This particular activation can be achieved by using projection operators of observable σ a x ; in this case, the amount of energy injected into A on average will be E a = h a / 1 + 4κ 2 (ha+hb) 2 . The energy injected stays in A and is not extracted during the protocol, with E a ≥ (−∆E b ) max . Since no energy was propagated from A to B, the activated energy corresponds to previously locked energy within local zero-point energy density.
Experimental implementation.-For the experimental implementation, we designed an optimized fully unitary QET protocol by introducing an auxiliary system that mediates the measurement on A and transmits information to B. Here, the role of the measurement device is played by an auxiliary qubit (An); and the information encoded in the post-measurement state of An is used to activate energy in B.
The fully unitary QET protocol consists of the following steps implemented by Alice and Bob: Step 1. Alice applies a joint unitary U AnA on qubits A and An that commutes with V and H b . Through this interaction, An gains information about Alice's qubit A without raising the energy on B.
Step 2. Then, instead of classical communication, the auxiliary qubit An itself is sent to Bob.
Step 3. Finally, Bob implements a joint unitary U BAn on B and An to extract energy from the system by acting locally on B.
We optimized the three steps to achieve the maximum amount of activated energy (−∆E b ) max , given by eq.(6), as explained below.

SLP state preparation Maximizing mutual information between A & An
Optimized energy extraction System setup. The experiments were performed on a Bruker Avance III 700 MHz nuclear magnetic resonance (NMR) spectrometer. We used 13 C-labelled transcrotonic acid (shown in Fig. 4a) dissolved in acetone-d6 as our sample. Carbons labeled as C 1 , C 2 and C 3 were used as B, An and A, respectively. The whole experiment took place at an ambient temperature of 298 K. The overview of the experimental scheme is shown in Fig. 3.
Step 0. As a preliminary step, to prepare the system's initial state, we start with the NMR pseudo-pure state |000 followed by a global unitary U prep . The unitary U prep required for the state in eq.(3), in a product state with |0 An , consists of first a rotation Y (θ) = e −iσyθ/2 on qubit B, followed by a CNOT on An and B (with B as the control), and finally a SWAP between A and An (see Fig.3). The explicit form of Y (θ) is where F ± = 1 ± ha+hb √ 4κ 2 +(ha+hb) 2 .
Step 1. To make An gain information about Alice's qubit A without increasing the energy in B, we optimized the unitary U AnA to create the maximum amount of mutual information I (A;An) between A and An subject to the condition that [U AnA , V ] = 0. For our aforementioned prepared state, an optimized unitary is given by which achieves the maximum possible I (A:An) . This unitary was performed through the circuit in Fig. (3).
Steps 2 and 3. We use An to transmit information to B by applying a unitary U BAn on both B and An. We optimized the energy activation in B under all possible 2qubit unitaries U BAn . For our system setup, an optimized U BAn that achieves the bound (−∆E b ) max is The quantum gates implemented in the experiment, shown in Fig. 3, were realized using GRAPE pulses [40], with a slight modification utilizing the technique described in ref. [41], which resulted in very smooth radio frequency (RF) pulses. Moreover, all the designed pulses had theoretical fidelity of over 0.998 and were robust against small imperfections in RF power.
The expectation values of the interaction term operator − X A X B and B's local hamiltonian operator Z B were measured directly in the experiments at the end of the quantum circuit. The experimental results are shown in Fig. 4  The results are shown in Fig. 4. The curves for the ideal scenario represent the outcome when the circuit is implemented perfectly without any decoherence. The curves for the simulation were made using optimized GRAPE pulses that take decoherence into account. For the decoherence simulation, we assumed that the environment is Markovian, the qubits relax independently, and the dissipator commutes with the total Hamiltonian for the discretization time of GRAPE pulses (∆t = 2µs). These assumptions simplify the implementation of master equations for each ∆t, the evolution under the propagator of the GRAPE pulses, and the implementation of the dissipator. All of the experimental results give energy extraction. The discrepancy between the simulation and the experiment is due to the decoherence assumptions and the transfer function of the spectrometer, which implements the GRAPE pulses slightly differently than assumed. The error bars shown in the results refer only to the statistical error of the experiment.
Numerical tests show that the protocol is stable under some uncertainty in the local Hamiltonians. To study this sensitivity quantitatively, we added perturbations in the local parts of the Hamiltonian while performing the QET protocol that is optimized for the non-perturbed case. In particular, we considered perturbed local Hamiltonians of the form h ν ∝ (1 + ) h ν σ ν z + h ν f (h a , h b , κ)1 1. We found that if the parameter (quantifying the relative difference between the Hamiltonian assumed to optimize the protocol and the actual Hamiltonian of the system) is small, ≤ 0.3, then the relative impact of the error in the implementation of the protocol is neglectable.
Conclusions.-We presented the first experimental activation of strong local passive states and the first experimental demonstration of a quantum energy teleportation (QET) protocol proposed by Hotta in [29,30]. Furthermore, our experiment confirms for the first time that the presence of entanglement in a ground state allows for the activation of local zero-point energy density without energy transfer through the system to the activated point.
Specifically, we show the experimental energy extraction from a bipartite system, initially in a strong local passive state but activated through a quantum energy teleportation protocol consisting of local operations and communication without energy transfer between the parties. For this, we designed a fully unitary quantum energy teleportation protocol optimized to maximize the amount of activated energy under the constraints of our experimental setup. The experiment was carried out using nuclear magnetic resonance, demonstrating that the control necessary for a quantum energy teleportation protocol can be achieved in realistic experimental scenarios.
Furthermore, the optimization of the fully unitary QET protocol demonstrates that the maximum possible amount of activated energy can only be achieved after the measurement device and the measured subsystem gain full mutual information between them and the measurement outcome is transmitted to the target subsystem.
The QET protocol has the potential to be a valuable tool both for our fundamental understanding of quantum thermodynamics and for quantum technologies. On the fundamental side, quantum energy teleportation helps understand quantum fluctuations and their role in fundamental scenarios, from quantum field theory in curved spacetimes to quantum thermodynamics: from black hole physics [26,[34][35][36] to violations of energy conditions in quantum field theory in curved spacetimes [37]. On the technological side, it has been theoretically discussed that QET can be used to improve purity locally by exploiting interaction-induced correlations in algorithmic cooling protocols [12]. Another application appears in scenarios where strong local passivity can potentially impose restrictions on thermodynamic tasks and the regime in which some quantum machines can perform, especially those that rely on energy exchange through local quenching and/or pulses that are fast compared to the dynamics of the system, for example in refs. [42][43][44][45]. This exper-iment demonstrating QET for the first time paves the way for the experimental exploration of these protocols in controlled quantum systems.