Quantum dichotomies and coherent thermodynamics beyond first-order asymptotics

TL;DR

This work develops a unified second-order framework for comparing quantum statistical models via transformations of quantum dichotomies, covering small, moderate, large, and zero-error regimes. By introducing the sesquinormal distribution and pinched hypothesis testing, it derives explicit transformation rates for arbitrary inputs and commuting outputs, and shows that thermal operations attain these rates for Gibbs-state targets in coherent thermodynamics. The results reveal resonance phenomena that can dramatically reduce finite-size dissipation, and extend to entanglement transformations under LOCC, providing quantum-thermodynamic laws beyond the thermodynamic limit. The approach connects quantum hypothesis testing with resource theories, offering practical insights for finite-size quantum thermodynamics, coherence-enabled protocols, and entanglement manipulation.

Abstract

We address the problem of exact and approximate transformation of quantum dichotomies in the asymptotic regime, i.e., the existence of a quantum channel $\mathcal E$ mapping $ρ_1^{\otimes n}$ into $ρ_2^{\otimes R_nn}$ with an error $ε_n$ (measured by trace distance) and $σ_1^{\otimes n}$ into $σ_2^{\otimes R_n n}$ exactly, for a large number $n$. We derive second-order asymptotic expressions for the optimal transformation rate $R_n$ in the small, moderate, and large deviation error regimes, as well as the zero-error regime, for an arbitrary pair $(ρ_1,σ_1)$ of initial states and a commuting pair $(ρ_2,σ_2)$ of final states. We also prove that for $σ_1$ and $σ_2$ given by thermal Gibbs states, the derived optimal transformation rates in the first three regimes can be attained by thermal operations. This allows us, for the first time, to study the second-order asymptotics of thermodynamic state interconversion with fully general initial states that may have coherence between different energy eigenspaces. Thus, we discuss the optimal performance of thermodynamic protocols with coherent inputs and describe three novel resonance phenomena allowing one to significantly reduce transformation errors induced by finite-size effects. What is more, our result on quantum dichotomies can also be used to obtain, up to second-order asymptotic terms, optimal conversion rates between pure bipartite entangled states under local operations and classical communication.

Figures (6)

• Figure 1: Summary of our main results. Asymptotics of transformation rates between quantum dichotomies $(\rho_1,\sigma_1)\to(\rho_2,\sigma_2)$ with an error of at most $\epsilon_n$ allowed on the first state. The table summarises the different error regimes, i.e., the different manners in which the error $\epsilon_n$ and rate $R_n$ can scale. In the above, the first-order rate is $C:=D\!\left( \rho_{1} \middle\| \sigma_{1} \right)/D\!\left( \rho_{2} \middle\| \sigma_{2} \right)$ and zero-error rate is $Z$. For each result we just have upper bounds for general target dichotomies, but for commuting targets, $[\rho_2,\sigma_2]=0$, we have upper and lower bounds. The final column denotes whether these bounds coincide, which they do in all-but-one regime.
• Figure 2: Coherent resonance in thermodynamic transformations of two-level systems. (a) The ratio $V(\rho\|\gamma)/D(\rho\|\gamma)$ (encoding the resonance condition) for qubit states lying in the $xz$ plane of the Bloch sphere for a thermal state $\gamma=\mathrm{diag}(0.95,0.05)$ (indicated by a white triangle). The white disk corresponds to the final state $\rho_2=\mathrm{diag}(0.75,0.25)$, while the dashed white line indicates a family of initial states $\rho_1(x)$ with diagonal $(0.85,0.15)$ and off-diagonal elements equal to $\sqrt{0.85\cdot 0.15}\cdot x$ for $x\in[0,1]$. (b) Threshold transformation error $\epsilon$ required to achieve the asymptotic transformation rate $D(\rho_1(x)\|\gamma)/D(\rho_2\|\gamma)$ for finite number $n$ of transformed systems (i.e., $\epsilon$ such that the second-order correction term in Eq. \ref{['eq:small_deviation']} disappears). Resonance is obtained when the relative free energy fluctuations $V/D$ are the same for the initial state $\rho_1(x)$ and the final state $\rho_2$, i.e., when $\xi=1$.
• Figure 3: Weak and strong resonance phenomena. Left: Weak resonance, in which the small and moderate regimes at rates $R<C$ collapse, but the large and extreme regimes persist, i.e. $Z<C$. Right: Strong resonance, in which all error regimes at rates below $C$ collapse, i.e. $Z=C$. See \ref{['fig:sigmoid_rate']} for an explanation of the various error regimes indicated, as well as the definitions of $Z$ and $C$.
• Figure 4: Trade-off between the optimal type-I and -II errors. An illustrative sketch of the trade-off between the optimal type-I and -II errors of the hypothesis test between two states $\rho^{\otimes n}$ and $\sigma^{\otimes n}$ as $n$ grows. Here $\alpha_n$ is the optimal type-I error, $\beta_n$ the optimal type-II error, and $-\frac{1}{n} \log \beta_n$ the type-II error exponent. Each of the grey curves correspond to a trade-off $(\alpha_n,\beta_n)$, for a given $n$, with darker curves correspond to growing $n$. The fact these curves approach a step at the relative entropy is equivalent to Stein's Lemma, \ref{['lem:ht_stein']}. Each of the coloured regions corresponds to a deviation regimes in which we will consider refinements to Stein's lemma in this subsection. In the table we present the scaling in each regime. For the details and explicit expressions for all of the scaling constants see the corresponding lemmas, \ref{['lem:ht_smalldev', 'lem:ht_moddev', 'lem:ht_largedev', 'lem:ht_extremedev']}. The final column denotes whether the asymptotics of the pinched and non-pinched variants of hypothesis testing are identical, which they are in all regimes by where both errors are exponentially decreasing.
• Figure 5: Trade-off using log odds. The trade-off between the type-I and -II error log odds per copy---$\lim\limits_{n\to\infty}\frac{1}{n}L[\alpha_n]$ and $\lim\limits_{n\to\infty}\frac{1}{n}L[\beta_n]$ respectively---for the hypothesis test between $\rho^{\otimes n}$ and $\sigma^{\otimes n}$, in the limit of growing $n$. The bottom-left quadrant corresponds to the regime in which both errors are decaying exponentially, with exponents bound by the relative entropies $D(\sigma\|\rho)$ and $D\!\left( \rho_{} \middle\| \sigma_{} \right)$ respectively. The top-left regime corresponds to a type-I error which is decaying even more rapidly, causing the type-II error to instead increase towards 1, and the bottom-right the converse of this. This curve is generated by plotting $\Gamma_\lambda\!\left( \rho_{} \middle\| \sigma_{} \right)$ from \ref{['lem:ht_largedev']} for two randomly generated $d=5$ qudit states.

Theorems & Definitions (74)

• Lemma 1: Sesquinormal distribution
• proof
• Theorem 1: First-order rate
• Theorem 1: Small deviation rate
• Theorem 1: Moderate deviation rate
• proof
• Theorem 1: Large deviation rate, low-error
• Theorem 1: Large deviation rate, high-error
• Theorem 1: Zero-error rate
• Theorem 1: Extremely high-error rate