Playing games with bounded entropy

Edward Jiménez

Seth Douglas, Nidhal Mghirbi

Does playing a nonlocal game more precisely require more heat? We study devices that start and finish in pure product blanks, receive thermal answer carriers, and release classical answers. The answer depends on when the devices may communicate again. If they reset locally first, their minimum cyclic work differs from the game's minimum mutual information by a bounded alphabet cost and is at least twice its one-way distillable entanglement demand. Published entanglement bounds therefore give fixed finite games with local work $2\alpha\log_2(1/\varepsilon)+O(\log\log(1/\varepsilon))$, for a dense set of coefficients $\alpha$, including transcendental values. Quantum reunion before reset keeps the work bounded. With coherent block recovery allowing $q$ qubits and $c$ dephased bits per play, the leading cost is $(\alpha L-q)_++(\alpha L-q-c)_+$, where $L=\log_2(1/\varepsilon)$. A relative-entropy potential proves the lower bound; compression proves the upper bound. An exact entropy-production identity records arbitrary schedules without solving their optimization. For exactly matched controls, a sector allocation theorem evaluates the entanglement and correlation costs. A W-state example shows why knowledge of accessible marginals cannot replace operational access. Perfect magic-square play has cyclic local work exactly four units. Work and bath heat agree at the zero-energy endpoints used here; one unit is $k_BT\ln2$.

Seth Douglas, Nidhal Mghirbi

A finite rational binary-scoring game can require infinite dimension for perfect play while admitting finite energy and finite entanglement entropy. Our game has ten Alice questions, eleven Bob questions, at most six answers, and 52 uniform question pairs. It is spatially perfect and never finitely perfect, with sharp dimension profile $D_\Gamma(\varepsilon)=10\log_2(1/\varepsilon)+O(\log\log(1/\varepsilon))$. Among pure perfect strategies, the displayed strategy attains the least entanglement entropy, $5.362740278498934$ ebits, and least mean occupation, $275/18$ per oscillator. We classify all perfect normal type-I marginal spectra and prove that this spectrum is the unique majorization maximum. These constants are also the small-loss limits of the entropy and one-sided oscillator energy prices. In the massive free scalar field we construct a pure perfect state in two strictly separated bounded regions, equal to the vacuum outside a bounded containing region and with every polynomial energy moment finite. In contrast, every fixed faithful split state has a positive loss floor; the complementary-wedge vacuum has supremum one without attainment. Energy-analytic states, bounded-region invertible vacuum preparations and product-Kraus preparations cannot be perfect. Every perfect pure spatial strategy has infinite Schmidt rank; finite entropy is attained but not forced. In every type-II$_1$ factor the normalized trace has a loss floor, while bounded faithful prepared densities win perfectly with sharp trace-entropy deficit $\Delta_* = \ln(7/6)-\ln(2)/7$. Type-II$_\infty$ permits every finite trace entropy. A graded modular ladder gives a normal semifinite hosting criterion. Its scalar fingerprint has sharp capacity and entropy-gap bounds; the capacity floor is approached but not attained in type I, and attained in type II.

Esteban Guevara Hidalgo

Seth Douglas, Nidhal Mghirbi

A finite rational binary-scoring game can require infinite dimension for perfect play while admitting finite energy and finite entanglement entropy. Our game has ten Alice questions, eleven Bob questions, at most six answers, and 52 uniform question pairs. It is spatially perfect and never finitely perfect, with sharp dimension profile $D_\Gamma(\varepsilon)=10\log_2(1/\varepsilon)+O(\log\log(1/\varepsilon))$. Among pure perfect strategies, the displayed strategy attains the least entanglement entropy, $5.362740278498934$ ebits, and least mean occupation, $275/18$ per oscillator. We classify all perfect normal type-I marginal spectra and prove that this spectrum is the unique majorization maximum. These constants are also the small-loss limits of the entropy and one-sided oscillator energy prices. In the massive free scalar field we construct a pure perfect state in two strictly separated bounded regions, equal to the vacuum outside a bounded containing region and with every polynomial energy moment finite. In contrast, every fixed faithful split state has a positive loss floor; the complementary-wedge vacuum has supremum one without attainment. Energy-analytic states, bounded-region invertible vacuum preparations and product-Kraus preparations cannot be perfect. Every perfect pure spatial strategy has infinite Schmidt rank; finite entropy is attained but not forced. In every type-II$_1$ factor the normalized trace has a loss floor, while bounded faithful prepared densities win perfectly with sharp trace-entropy deficit $\Delta_* = \ln(7/6)-\ln(2)/7$. Type-II$_\infty$ permits every finite trace entropy. A graded modular ladder gives a normal semifinite hosting criterion. Its scalar fingerprint has sharp capacity and entropy-gap bounds; the capacity floor is approached but not attained in type I, and attained in type II.

Ramón Alonso‐Sanz