Reconstructing thermal states using dimensionally limited probes : A Model for Limited Control & Memory in Quantum Thermodynamics

TL;DR

This work reframes knowledge in quantum thermodynamics as the unitary reconstruction of a thermal state's diagonal in a fixed basis using dimensionally constrained probes, linking information acquisition to thermodynamic costs. It introduces a two-step protocol—information extraction with a low-dimensional probe and estimate generation into a memory—so that coarse-grained information can be unitarily mapped into a diagonal memory state. By exploring symmetric distribution and asymmetric concentration of multiple estimates, the paper derives fidelity and majorisation conditions that determine when complete or partial recovery of the diagonal is possible, and applies these ideas to a toy extended Szilard engine to show how probe dimensionality and symmetrisation influence work extraction. The results connect coarse-grained state estimation, unitary representations of measure-and-prepare channels, and thermodynamic tasks, offering insights into the resource costs of acquiring and utilizing quantum knowledge for thermodynamic tasks.

Abstract

Whilst the complexity of acquiring knowledge of a quantum state has been extensively studied in the fields of quantum tomography and quantum learning, a physical understanding of its operational role and cost in quantum thermodynamics is lacking. Knowledge is central to thermodynamics, as exemplified by Maxwell's demon thought experiment, where a demonic agent is able to extract paradoxical amounts of work -- reconciled by the thermodynamic costs of acquiring this knowledge. In this work, we address this gap by extending unitary models of measurement to incorporate the resources available to an agent. We view an agent's knowledge of a quantum state as their ability to reconstruct it unitarily given access to states with partial knowledge of the true state. In our model, an agent correlates an unknown $d$-dimensional system, with copies of a $k$-dimensional probe ($k\leq d$), which are then used to unitarily reconstruct an estimate state in $d$-dimensional memories. We find that this framework is a unitary representation of coarse-grained POVMs. As an application, we investigate the role of knowledge in an extended Szilard Engine scenario.

Figures (8)

• Figure 1: Summary. The main message of this work is the reformulation of a scenario where an agent with access to only limited POVMs attempts to estimate the state of an unknown system, as a unitary protocol allowing us to connect to thermodynamics. Using lower-dimensional probes, an agent creates correlations between their probe and the system to extract information, which is then used to reconstruct the unknown state in a quantum memory. This creates a contrast between two exemplary situations in state reconstruction. Situation \ref{['B:motivating-example']}, the agent has access to full POVM elements and is able to perfectly reconstruct a thermal state. Situation \ref{['B:motivating-example']}, the agent has access to coarse-grained POVMs, allowing them to only partially distinguish between certain energy levels of the system, resulting in an approximate reconstruction of the state. The focus of this work is to study how limited resources such as those in Situation \ref{['B:motivating-example']} impede an agent's ability to reconstruct the state of an unknown system.
• Figure 2: Fidelity & measurement settings. For a thermal state $\rho_{\beta}$ described by an equidistant energy spectrum, we plot the fidelity as a function of $\beta$ for (a) two measurement settings $\vec{t} = [1,2,3]$ (red solid curve) using qutrit probes and $\vec{t}_2 = [1,5]$ (dashed blue curve) using qubit probes. We observe that the former is superior as the higher dimensionality allows the agent to couple less levels of the unknown system to more levels of the probe system. Focusing on qubit probes (b) for the general measurement $\vec{t} = [1, d - 1]$, we notice that the fidelity decreases with the dimension of the estimated thermal state. In this case, the information is spread over the entries of the estimate, making it challenging to reconstruct $\rho_{\beta}$.
• Figure 3: Majorisation inequalities. For a three-level thermal state, we plot the marginal majorisation inequalities described above were $p_i$ are the populations of $\rho_\beta$ and $q_i$ are the eigenvalues of $\overline{\omega}^\downarrow_0$ for the estimates given in Eq. \ref{['eq:qutrit_estimate']} in Appendix \ref{['sec:warm_up']}, as a function of $\beta$.
• Figure 4: Extend Szilard engine with limited memory. An illustration of the scenario considered in this section. An agent carries out a different quench changing the potential in a 1D box to extract work from a particle as it relaxes to a thermal state $\rho_\beta$. The quench carried out depends on the knowledge the agent has of the position distribution of the particle. A more coarse-grained understanding (pink) can lead to imperfect work extraction.
• Figure 5: Work Extraction by knowledge dependent quenches. Dissipative work $W_\text{Out}(\omega)$ extracted by the agent during relaxation for different quenches $H_\text{Quench}(\omega)$ carried out on the system based on their estimate of the system $\omega$, as the initial temperature of the system decreases $\beta$. In this plot we assume the particle is trapped in a box with a potential well described by 8 positions $H_S = \sum^7_{i=0} E_i \ketbra{E_i}{E_i}$ with $E_i \in [0,7]$, the particle is driven under $H_\text{Shift}$ for $t_1 = 0.25$. The extracted work is shown as a ratio $W_\text{Out}(\omega)/W_\text{Out}(\rho')$ to the work extractable using a quench protocol with knowledge of the actual state $\rho'$. We plot random estimates $\omega^*_{\vec{t}}$ and symmetrised estimates $\widetilde{\omega}_{\vec{t}}$ of a given coupling type $\vec{t}$ where we see that as the dimension of the agent's probe increases, they are able to extract more work from the system. The code to obtain these numerics may be obtained at code.

Theorems & Definitions (7)

• Proposition 4.1
• Corollary 4.1
• Theorem 4.1
• Lemma D.1
• proof
• Corollary D.1
• proof