Quantum energy teleportation

There are two main factors involved in how QET works: how energy is transferred from Alice to Bob, and how Bob can extract energy from his spin. Spin chains QET is studied through analyzing spin chain models. A spin chain is a type of model where a one dimensional chain of sites are assigned certain spin value at each site, typically +1/2 or -1/2 when considering spin-1/2. The spin of one individual site can interact with the spin of its adjacent neighbours, causing the entire system to be coupled together. Spin chains are useful for QET due to the fact that they can be entangled even in the ground state. This means that even without external energy being added to the system, the ground state exhibits quantum correlations across the chain. Alice and Bob are both in possession of an entangled state from a spin chain system. This can provide a rudimentary explanation of how energy can be transferred from Alice's spin to Bob's spin, since any action on Alice's spin can have an effect on Bob's spin. Vacuum fluctuations The other key component to understanding the QET mechanism is vacuum fluctuations and the presence of negative energy density regions within the energy distribution of a quantum mechanical system. Vacuum fluctuations are a consequence of the Heisenberg uncertainty principle, specifically the uncertainty between the field amplitude and its conjugate momentum, which is analogous to the position-momentum uncertainty principle. The commutation relation, [\varphi(x,y,z),\Pi(x',y',z')]=i\hbar\delta (x-x')\delta(y-y')\delta(z-z') , gives rise to uncertainty in energy densities at different spatial points. Consequently, the energy fluctuates around the zero-point energy density of the state The vacuum fluctuations in certain regions can have lower amplitude fluctuations due to the effect of local operations. These regions possess a negative energy density since the vacuum fluctuations already represent the zero-energy state. Therefore, fluctuations of lower amplitude relative to the vacuum fluctuations represent a negative energy density region. Since the entire vacuum state still has zero-energy, there exist other regions with higher vacuum fluctuations with a positive energy density. Negative energy density in the vacuum fluctuations plays an important role in QET since it allows for the extraction of energy from the vacuum state. Positive energy can be extracted from regions of positive energy density which can be created by regions of negative density region elsewhere in the vacuum state. == QET in a spin chain system ==

Framework of the quantum energy teleportation protocol The QET process is considered over short time scales, such that the Hamiltonian of the spin chain system is approximately invariant with time. It is also assumed that local operations and classical communications (LOCC) for the spins can be repeated several times within a short time span. Alice and Bob share entangled spin states in the ground state |g\rangle with correlation length \ell. Alice is located at site n_A of the spin chain system and Bob is located at site n_B of the spin chain system such that Alice and Bob are far away from each other, |n_A - n_B| \gg 1. The QET protocol Conceptually, the QET protocol can be described by three steps: • Alice performs a local measurement on her spin at site n_A, measuring eigenvalue \mu . When Alice acts on her spin with the local operator, energy E_A is inputted into the state. • Alice then communicates to Bob over a classical channel what her measurement result \mu was. It is assumed that over the time the classical message is travelling that Alice and Bob's state does not evolve with time. • Based on the measurement Alice got on her spin \mu , Bob applies a specific local operator to his spin located at site n_B. After the application of the local operator, the expectation value of the Hamiltonian at this site \hat{H}_{n_B} is negative. Since the expectation of \hat{H}_{n_B} is zero before Bob's operation, the negative expectation value of \hat{H}_{n_B} after the local operation implies energy was extracted at site n_B while the operation was being applied. Intuitively, one would not expect to be able to extract energy from the ground state in such a manner. However, this protocol allows energy to be teleported from Alice to Bob, despite Alice and Bob sharing entangled spin states in the ground state. == Mathematical description ==

Local measurement by Alice The QET protocol can be worked out mathematically. The derivation in this section follows the work done by Masahiro Hotta in "Quantum Energy Teleportation in Spin Chain Systems". Therefore, it is impossible for Alice to extract the energy E_A while only using local operators. == Quantum energy distribution ==

Quantum energy distribution (QED) is a protocol proposed by Masahiro Hotta in "A Protocol for Quantum Energy Distribution" which proposes an extension of QET with quantum key distribution (QKD). This protocol allows an energy supplier S to distribute energy to M consumers denoted by C_m. Quantum energy distribution protocol The supplier S and any consumer C_m share common short keys k which are used for identification. Using the short keys k, S and C_m can perform secure QKD which allows S to send classical information to the consumers. It is assumed that S and C_m share a set of many spin states in the ground state |g \rangle. The protocol follows six steps: • S performs a local measurement of the observable \hat{U}_S = \sum_{\mu=0,1} (-1)^\mu \hat{P}_S(\mu) on the ground state |g \rangle and measures \mu. S must input energy E_S = \sum_{\mu=0,1} \langle g | \hat{P}_S(\mu) \hat{H} \hat{P}_S(\mu) | g \rangle into the spin chain. • S confirms the identity of C_m through use of the shared secret short keys k. • S and C_m share pseudo-random secret keys K by use of a QKD protocol. • S encodes the measurement result \mu using secret key K and sends it to C_m. • C_m decodes the measurement result \mu using secret key K. • C_m performs the local unitary operation \hat{V}_m(\mu) to their spin. C_m receives energy E_m = \frac{1}{2} \left[ \sqrt{\xi_m^2 + \eta_m^2} - \xi_m \right] where \xi_m = \langle g| \hat{U}^\dagger_m \hat{H} \hat{U}_m |g \rangle, \eta_m = \langle g| \hat{U}_s \dot{\hat{U}}_m |g \rangle, \hat{U}_m = \vec{n}_m \cdot \vec{\sigma}_{n_{C_{m}}}, \dot{\hat{U}}_m = i [\hat{H}_{C_{m}}, \hat{U}_m], \vec{n}_m is a unit vector, and \vec{\sigma}_{n_{C_{m}}} is the Pauli spin matrix vector at site n_{C_{m}}. Robustness against thieves This process is robust against an unidentified consumer, a thief D, at site n_D attempting to steal energy from the spin chain. After step 6, the post-measurement state is given by \hat{\rho} = \sum_{\mu=0,1} \left( \prod_m \hat{U}_m(\mu) \right) \hat{P}_S(\mu) |g \rangle \langle g| \hat{P}_S(\mu) \left( \prod_m \hat{U}^\dagger_m(\mu) \right).Since D has no information on \mu and therefore randomly acts with either \hat{U}_D(0) or \hat{U}_D(1) where \hat{U}_D(\mu) = \hat{I} \text{cos} \theta + i (-1)^\mu \vec{n}_D \cdot \vec{\sigma}_{n_{D}}\text{sin} \theta. The post-measurement state becomes a sum over the possible guesses D makes of \mu, 0 or 1. Taking the expectation value of the localized energy operator \hat{H}_D yields: \text{Tr}[\hat{\rho}_D \hat{H}_D] = \frac{1}{2} \sum_{\mu=0,1} \langle g|\hat{P}_S(\mu) \left( \prod_m \hat{U}^\dagger_m(\mu) \right) \hat{U}^\dagger_D(\mu) \hat{H}_D \hat{U}_D(\mu) \left( \prod_m \hat{U}_m(\mu) \right) \hat{P}_S(\mu) |g \rangle.\hat{H}_D is positive semi-definite by definition. This means that all expectation values of \hat{H}_D, even the ones altered by \hat{U}_D(\mu), are greater than or equal to zero. At least one of the values in the sum of the trace will be positive, the one where D guesses the wrong value of \mu. This is because the operation \hat{U}_D(\mu)|g\rangle will add energy to the system when \mu does not match the value measured by Alice. Therefore, \text{Tr}[\hat{\rho}_D \hat{H}_D] > 0 which implies that on average D will have to input energy to the spin chains without gain. This protocol is not perfect as theoretically D could guess \mu on their first attempt, which would be a 50% chance to guess \mu correctly, and would immediately profit energy. However, the idea is that over multiple attempts D will lose energy since the energy output from a correct guess is lower than that of the energy input required when making an incorrect guess. == Experimental implementation ==

QET was experimentally demonstrated in 2022 by IQC group in the publication "Experimental Activation of Strong Local Passive States with Quantum Information", and in 2023 by Kazuki Ikeda in the publication "Demonstration of Quantum Energy Teleportation on Superconducting Quantum Hardware". The basic QET protocol discussed early was verified using several IBM superconducting quantum computers. Some of the quantum computers that were used include ibmq_lima, and ibm_cairo, and ibmq_jakarta which provided the most accurate results for the experiment. These quantum computers provide two connected qubits with high precision for controlled gate operation. The Hamiltonian used accounted for interactions between the two qubits using the \hat{X} and \hat{Z} Pauli operators. Protocol The entangled ground state was first prepared using the \widehat{\text{CNOT}} and \hat{R}_Y quantum gates. Alice then measured her state using the Pauli operator \hat{X}, injecting energy E_0 into the system. Alice then told Bob her measurement result over a classical channel. The classical communication of measurement results was on the order of 10 nanoseconds and was much faster than the energy propagation timescale of the system. Bob then applied a conditional rotational operation on his qubit dependent on Alice's measurement. Bob then performed a local measurement on his state to extract energy from the system E_1. Results The observed experimental values are dimensionless and the energy values correspond to the eigenvalues of the Hamiltonian. For quantum computers, energy scales tend to be limited by the qubit transition frequency which is often on the order of GHz. Therefore, the typical energy scale is on the order of 10^{-24} Joules. Ikeda experimented with varying the parameters in the Hamiltonian, specifically the local energy h and interaction strength k, to see if the QET protocol improved under certain conditions. For differing experimental parameters, the experimental values for Alice's input energy E_0 was around 1 and matched the experimental results very closely when error mitigation was applied. Bob's extracted energy E_1, for certain experimental parameters, was observed to be negative when error mitigation was applied. This indicates that the QET protocol was successful for certain experimental parameters. Depending on the experimental parameters, Bob would receive around 1-5% of Alice's inputted energy. Quantum error correction Quantum computers are currently the most viable platform for experimentally realizing QET. This is mainly due to their ability to implement quantum error correction. Quantum error correction is important specifically for implementing QET protocols experimentally due to the high precision needed to calculate the negative energy Bob receives in the QET protocol. Error correction in this experiment greatly improved the amount of energy Bob could extract from the system. In some cases without error correction, Bob's extracted energy would be positive, indicating the QET protocol did not work. However, after error correction, these values could be brought closer to the experimental values and in some cases even become negative, causing the QET protocol to function. The quantum error correction employed in this experiment allowed Ikeda to observe negative expectation values of the extracted energy E_1, which had not been experimentally observed before. High precision is also required for experimental implementation of QET due to the subtle effects of negative energy density. Since negative energy densities are a consequence of vacuum fluctuations, they can easily be overshadowed by measurement noise in the instrumentation. So, higher precision can lead to better distinguishability between negative energy signals and noise. == See also ==