Minimal excess thermodynamic cost of quantum operations: tightness, attainability, and the exceptional spectral-symmetry cases

Bedingham and Maroney (New J. Phys. 18, 113050, 2016) proved that implementing a quantum operation — a completely positive trace-preserving map that must act correctly on every input state of an ensemble, not merely on average — carries an operation-specific excess heat $\epsilon_Q$ beyond the Landauer term $kT\ln 2[S(\rho_S)-S(\rho'_S)]$, whenever the output states do not share a common diagonal basis whose eigenvalues are related to the input diagonal statistics by a left stochastic map, barring exceptional spectral symmetries. Their argument yields nonzero lower bounds on $\epsilon_Q$ via trace-norm and spectral-ratio inequalities, but no matching construction, and they explicitly state that demonstrating a protocol which reaches $\epsilon_Q$ is an open problem. The task is to determine the exact characterization of the minimal excess heat: whether the derived bounds are tight, which physical protocols attain the minimum for general quantum operations, and which operations possessing eigenvalue-ratio coincidences $\lambda_i/\lambda_j=\lambda'_k/\lambda'_l$ admit dissipation-free ($\epsilon_Q=0$) implementations. The problem remains open: the most direct continuation (Dunlop, Cerisola, Tabanera-Bravo and Anders, Phys. Rev. Res. 6, 033005, 2024) constructs optimisable protocols only for two-input single-qubit operations, states that before their work no prescription for optimal implementation and no upper bound on the energy cost existed, and notes that their own analytic bound is not tight; the general tightness, attainability, and exceptional-case questions remain unresolved.

Background

Landauer's principle states that erasing one bit of information while in contact with an environment at temperature $T$ dissipates at least $kT\ln 2$ of heat, where $k$ is Boltzmann's constant; reaching this bound is equivalent to performing the operation in a thermodynamically reversible way. In quantum theory the entropy entering the bound is the von Neumann entropy $S(\rho)=-\mathrm{Tr}[\rho\log_2\rho]$ of the density matrix $\rho$, so an operation taking an average input state $\rho_S$ to an average output state $\rho'_S$ should satisfy $\Delta E_Q \ge kT\ln 2[S(\rho_S)-S(\rho'_S)]$, where $\Delta E_Q$ is the heat dumped into the bath. A quantum operation - formally a completely positive trace-preserving map on density matrices, the most general evolution of an open system's state, realizable as unitary interaction with an environment whose part is then discarded - must map each possible input state $\rho^n_S$ of a known ensemble to its designated output $\rho'^n_S$, not just the average state, and this per-input requirement is what generates the excess cost studied here. Bedingham and Maroney (2016) analyzed an implementation model in which the system $S$ is coupled to an auxiliary system $A$ and a heat bath $B$, the auxiliary starts in a standard catalyst state $\rho_A$ that the operation must restore on average and the heat bath starts in a canonical thermal state $\rho_B$ at temperature $T$, the joint evolution is a unitary, and $\Delta E_Q$ is the resulting energy change of the bath. They proved that for every quantum operation $Q$ there is a non-negative, implementation-independent quantity $\epsilon_Q$ with $\Delta E_Q \ge kT\ln 2[S(\rho_S)-S(\rho'_S)]+\epsilon_Q$. Writing the spectral decompositions of the average input and output states as $\rho_S=\sum_i\lambda_i|\phi_i\rangle\langle\phi_i|$ and $\rho'_S=\sum_k\lambda'_k|\phi'_k\rangle\langle\phi'_k|$, and the individual inputs and outputs in these bases as $\rho^n_S=\sum_{ij}\mu^n_{ij}|\phi_i\rangle\langle\phi_j|$ and $\rho'^n_S=\sum_{kl}\mu'^n_{kl}|\phi'_k\rangle\langle\phi'_l|$, they showed that $\epsilon_Q=0$ is possible whenever all output states share a common diagonal basis and the output eigenvalues are related to the input diagonal elements by a common left stochastic map $P_Q(k|i)$ (non-negative coefficients with $\sum_k P_Q(k|i)=1$ and $\mu'^n_{kl}=\delta_{kl}\sum_i P_Q(k|i)\mu^n_{ii}$); classical information processing, erasure to a standard state, pure unitary operations, and single-input operations fall in this class, and explicit reversible constructions were given. Otherwise, barring exceptional spectral symmetries — coincidences of eigenvalue ratios $\lambda_i/\lambda_j=\lambda'_k/\lambda'_l$ for some $i\neq j$ or $k\neq l$ — every physical implementation of the operation has $\epsilon_Q>0$. The proof rewrites any implementation through complex coefficients $q(kl|ij)$ (with $\mu'^n_{kl}=\sum_{ij}q(kl|ij)\mu^n_{ij}$) and yields nonzero lower bounds on $\epsilon_Q$ from a trace-norm inequality and from spectral-ratio inequalities of the form $\|\rho'-\rho^\star\|_1 \ge |\lambda_j\lambda'_k-\lambda_i\lambda'_l|/(\lambda'_k+\lambda'_l)\cdot|q(kl|ij)|$, where $\rho'$ is the joint final state of system, auxiliary, and bath actually produced by the implementation unitary, $\rho^\star=\rho'_S\otimes\rho_A\otimes\rho_B$ is the target decoupled final state (system in the correct average output, auxiliary and bath uncorrelated and back in their initial states $\rho_A$ and $\rho_B$), and $\|\cdot\|_1$ is the trace norm (sum of singular values of its argument). The authors explicitly caution that these bounds may not be tight — their results only place a nonzero lower bound on the excess energy cost — and that the mere existence of spectral-ratio coincidences is not sufficient for $\epsilon_Q=0$, since an implementation must in addition have $q(kl|ij)=0$ for every index quadruple with $\lambda_i/\lambda_j\neq\lambda'_k/\lambda'_l$. Later work has circumscribed but not resolved the question: Dunlop, Cerisola, Tabanera-Bravo and Anders (2024) developed explicit optimisable protocols, feasibility conditions, and a complementary (explicitly non-tight) bound on dissipated work for the two-input single-qubit case, noting that prior to their work no prescription for thermodynamically optimal multipurpose operations and no upper bound on their energy cost existed; Chiribella, Meng, Renner and Yung (2022) bounded the nonequilibrium resources needed for approximately accurate information processing in a resource-theoretic framework, a complementary setting that does not determine $\epsilon_Q$ for exact operations; experimental energy accounting for a driven single-qubit gate (Stevens et al., 2022) and lower bounds on the field energy required by Hamiltonian gates (Stevens and Deffner, 2025) use different resource accounting and leave $\epsilon_Q$ untouched.

Problem Statement

In the Bedingham–Maroney implementation model — a system $S$ coupled to an auxiliary system $A$ and a heat bath $B$, with $A$ initially in a standard catalyst state $\rho_A$ restored on average and $B$ initially in a canonical thermal state at temperature $T$, joint unitary evolution, and $\Delta E_Q$ the resulting energy change of the bath — determine the minimal excess thermodynamic cost of an arbitrary quantum operation $Q$ acting on an ensemble of input states $\{\rho^n_S\}$ with probabilities $p_n$ and designated outputs $\rho'^n_S = Q(\rho^n_S)$. Specifically: (1) For operations failing the reversible condition of Theorem 1 (the outputs do not share a common diagonal basis related to the input diagonal statistics by a common left stochastic map $P_Q(k|i)$), determine the exact minimum of the excess heat $\epsilon = \Delta E_Q - kT\ln 2[S(\rho_S)-S(\rho'_S)]$ over all physical implementations of $Q$ (the minimum across implementations being the implementation-independent operation-level quantity $\epsilon_Q$ of the Background); decide whether the lower bounds derived from the trace-norm inequality $\epsilon_Q \ge \frac{1}{2}kT\|\rho'-\rho^\star\|_1^2$ (with $\rho'$ the actual joint final state of system, auxiliary, and bath produced by the implementation and $\rho^{\star}=\rho'_S\otimes\rho_A\otimes\rho_B$ the decoupled target state) and the spectral-ratio inequalities $\|\rho'-\rho^{\star}\|_1 \ge |\lambda_j\lambda'_k-\lambda_i\lambda'_l|/(\lambda'_k+\lambda'_l)\cdot|q(kl|ij)|$ are tight; and determine whether the minimal excess heat is attained by a physical protocol, constructing attaining protocols wherever they exist. (2) Classify the exceptional spectral-symmetry cases: characterize, in terms of the spectral data $\{\lambda_i\}$, $\{\lambda'_k\}$, $\{\mu^n_{ij}\}$, $\{\mu'^n_{kl}\}$ of the operation, which operations possessing eigenvalue-ratio coincidences $\lambda_i/\lambda_j=\lambda'_k/\lambda'_l$ (with $i\neq j$ or $k\neq l$) admit implementations with $\epsilon_Q=0$, and construct such dissipation-free implementations, noting that the existence of the coincidences alone is not sufficient. The two parts are jointly required and together characterize the minimal excess heat $\epsilon_Q$ of an arbitrary quantum operation of arbitrary system dimension acting on arbitrary input ensembles (not only qubits or two-input cases), and the exact boundary between thermodynamically reversible and irreversible quantum operations.

References

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