Quantifying Memory Capacity as a Quantum Thermodynamic Resource

The information-carrying capacity of a memory is known to be a thermodynamic resource facilitating the conversion of heat to work. Szilard’s engine explicates this connection through a toy example involving an energy-degenerate two-state memory. We devise a formalism to quantify the thermodynamic value of memory in general quantum systems with nontrivial energy landscapes. Calling this the thermal information capacity, we show that it converges to the nonequilibrium Helmholtz free energy in the thermodynamic limit. We compute the capacity exactly for a general two-state (qubit) memory away from the thermodynamic limit, and find it to be distinct from known free energies. We outline an explicit memory-bath coupling that can approximate the optimal qubit thermal information capacity arbitrarily well.

Figure 1

Thermally passive memory: the pertinent information variable $K\equiv \{(p_k,k)\}$ (contained in a classical system $\mathbf{C}$) is recorded on the quantum memory $\mathbf{M}$ through a $k$-dependent energy-conserving interaction of $\mathbf{M}$ with an auxiliary system $\mathbf{A}$ in its thermal state ${\gamma }_{\mathbf{A}}$, transforming the memory’s initial (“blank tape”) state $\rho$ to the ensemble $\{(p_k,{\sigma }^{(k)})\}$ of quantum state-valued codewords. No free energy is used in this process, except that already present in $\rho$.

Figure 2

Bloch visualization of the set of states accessible by qubit thermal operations from a pure initial state; an informationally maximal code constructed from the accessible set comprises the three indicated extremal states as codewords.

Figure 3

Thermal information capacity (TIC) over different blank-memory states in the $X^{+}Z$ section of the Bloch ball, for a qubit memory (with energy gap $\mathrm{\Delta }E$) at various temperatures. The TIC of the Gibbs state $\gamma$ is zero, and is higher for states further away from $\gamma$. The zero-temperature limit behaviour persists at temperatures as high as $0.1\mathrm{\Delta }E/k_{\mathrm{B}}$; significant variation ensues in the $\mathcal{O}(\mathrm{\Delta }E/k_{\mathrm{B}})$ temperature range, while the high-temperature limit resembles the information landscape of a non–energy-degenerate qubit memory.

Figure 4

Scatter plots of the thermal information capacity vs nonequilibrium Helmholtz free energy of qubit memory states: while the two resources are correlated in their state dependence, they are distinct, particularly at lower temperatures. In each plot, the top-right point of maximum capacity corresponds to the initial state $\rho =|1⟩⟨1|$, the pure excited state. The maxima occurring to the left of this point correspond to initial states along the equator, e.g., $\rho =|+⟩⟨+|$. In the $T\to \infty$ limit, the two maximal regions get more and more similar in their free energy, as the latter converges to the purity (or “negentropy”) of $\rho$.

Figure 5

Time taken by a Jaynes-Cummings coupling to approximate the optimal qubit thermal information capacity to various efficiencies, vs bath temperature. The speed-efficiency tradeoff is reminiscent of a heat engine’s performance.