Finite-resource cooling of qudits beyond thermodynamic length

AI summary

Overview

Research area: Quantum thermodynamics, specifically finite-resource quantum cooling, entropy production in collision models, and information geometry (Fisher–Rao/thermodynamic length versus Finsler geometries).

Technical level: Advanced. The paper relies on relative-entropy expansions, the Kubo–Mori metric, Birkhoff's theorem, convex-hull gauges, second-order cone programming, and Finsler geometry. The core physical argument, however, is simple enough to state in a few sentences.

Scope: The paper re-derives the leading finite-N dissipation cost for cooling a d-level quantum system (qudit) with N fresh thermal machines, and shows that for d ≥ 3 the cost is governed by a Finsler geometry built from resonant permutations of energy eigenstates rather than by thermodynamic length.

What This Paper Is About

Cooling a quantum system with a finite number N of thermal machines always dissipates more heat than Landauer's bound allows in principle. For the standard "swap" collision model, the excess was known to be set by thermodynamic length — the geodesic distance between the initial and final Gibbs states in the Fisher–Rao metric — with total dissipation (L*)^2/2N + O(N^-2). Taranto et al. had proved this optimal for qubits and their numerics for qutrits and ququarts suggested it might hold generally. This paper shows that it does not: for qudits, resonances allow a single collision to move population through several system–machine channels at once, and because dissipation is quadratic in the flow, splitting the flow lowers the cost — the way a current shared by two identical resistors dissipates half the heat.

Key Contributions

1. A new collision geometry. The authors identify a classical Finsler geometry, generated by resonant permutations of energy eigenstates, that sets the leading-order finite-resource cooling cost for qudits. It reduces to thermodynamic length for qubits but differs for every d ≥ 3.

2. Theorem 1 (coherence does not help). Any collision close to a dissipationless one is bounded below, to leading order, by a mixture of resonant permutations of the product eigenbasis. The proof splits the generator into intra- and inter-eigenspace parts and uses monotonicity of the Kubo–Mori metric under the partial trace plus Birkhoff's theorem.

3. Theorem 2 (maximal gain at infinite temperature). With d_M-level machines, at the maximally mixed state every near-dissipationless collision satisfies Σ ≥ (1/(d_M−1))·(1/2)‖δp‖_F² + O(ε³), so the gain over swaps is at most a factor d_M − 1. Transfers between two levels attain this bound.

4. Explicit protocols and benchmarks. Concrete permutation-based protocols with 41 machines are constructed and benchmarked against the global optimum over all swap protocols (a jointly convex problem), showing dissipation reductions of 22–83%.

Main Findings

• Thermodynamic length is not the qudit limit. The leading cost still scales as 1/N, but its prefactor is set by the Finsler norm of the collision geometry, not by L*. The asymptotic optimality suggested by the qutrit and ququart numerics of Ref. [22] fails for every d ≥ 3 and every spectrum, even with machines of the same dimension as the system. Theorem 1 of Ref. [22], which concerns swap protocols, is unaffected.

• Improvement near infinite temperature. The new geometry improves on thermodynamic length near infinite temperature for every spectrum, with leading coefficients at most 0.50, 0.65, 0.70 and 0.73 times the thermodynamic-length value for d = 3, 4, 5 and 6.

• Improvement near the ground state. Close to zero temperature and for a non-degenerate first excited level, the cost of the dominant transfer is 1/(d − 1) of the swap value, i.e. the gain approaches a factor d − 1. Exact protocols cooling from βω = 8 to 16 save 63–83% for d = 4, …, 8.

• Qutrit mechanism at infinite temperature. All nine configurations have weight 1/9 and every permutation is resonant. A swap transfers M at cost 3M²; the permutation π_A (|0,0⟩ ↔ |2,2⟩, |0,1⟩ ↔ |2,0⟩) routes the transfer through two channels and costs (3/2)M², half the swap cost. A dual certificate over all 9! permutations shows mixtures and alternations do not do better.

• Qutrit resonance manifold. Of the 9! qutrit permutations, 575 are resonant at a generic p (360 of which move the system); on the manifold p₁² = p₀p₂ — which contains every Gibbs state of the equally spaced qutrit H_S = ω diag(0,1,2) — 234 further permutations become resonant, and the hull extends beyond the Fisher ellipse. On the Gibbs curve, 809 permutations are resonant (764 with a unique resonant machine state, 45 ranging over a line).

• Where swaps remain locally optimal. For the equally spaced qutrit cooled between βω = 1 and 3, the hull at interior points of the Fisher–Rao geodesic coincides with the Fisher ellipse in the direction of motion, so swaps are locally optimal along that path — consistent with Ref. [22]'s numerics. The computed cost ratio is below 1 except for 1.3 ≲ βω ≲ 2, and tends to 1/√2 (half the dissipation) at both high and low temperature.

• Exact protocol benchmarks (41 machines, equally spaced systems). For d = 3: 0.01→0.05 saves 46.8% (13.001 vs 6.911, in units of 10^-6); 0.1→0.5 saves 21.7% (1235.9 vs 967.5); 3→9 saves 21.6% (2288.5 vs 1794.1); 8→16 saves 49.7% (16.288 vs 8.193). For d = 4: 24.3%, 43.1%, 62.8%. For 8→16 the reductions grow with dimension: 62.8% (d=4), 71.7% (d=5), 77.1% (d=6), 80.7% (d=7), 83.3% (d=8).

• Unequal spacings. Near infinite temperature equal spacing is unnecessary: π_ext is resonant at every p with a machine of level spacing E_{d−1} − E_0 and has cost ratio 1/(d − 1). For qutrits with unequal spacings, exact protocols cooling from β = 0.01 to 0.05 save 16–31%. Near the ground state the construction saves 46% for E = (0, 1, 2.3) and 59% for E = (0, 1, 1.7, 3.1). At intermediate temperatures the gain does depend on near-commensurability: for H_S = ω diag(0, 1, 2+δ) at βω = 0.1→0.5, the π_B, π_A pattern saves 23% at δ = 0.02 and 15% at δ = 0.1, but nothing at δ = 0.3, where π_ext still saves 31% closer to infinite temperature.

• Ququarts. Permutations conserving H_S + H_M for integer machine spectra give cost ratios between 0.72 and 0.97 at all five temperatures tested; exact protocols save 24% and 43%.

• Finiteness of the gain and a conjecture. Theorem 2 bounds the infinite-temperature gain by d_M − 1. The authors conjecture the same bound holds at every temperature; it holds in all numerical tests, including exhaustive enumeration for (d, d_M) = (2,3), (2,4) and (3,3), about 95,000 permutations for (3,4) and (4,4), and a search over about 9,700 random resonant permutations. If true, the optimal quasistatic cost lies between (L*)²/2(d_M−1)N and (L*)²/2N.

• Finite-N scaling checks. For 0.01→0.05 the resonant protocol gives NΣ_N = 2.82×10^-4 at N = 40, 80 and 160, against (L*)²/2 = 5.33×10^-4. For 0.1→0.5 it gives 0.0387 at all three N, against 0.0507. For 3→9 it gives 0.0718, 0.0710 and 0.0706, against 0.0939, 0.0927 and 0.0921 for swaps, which tend to (L*)²/2 = 0.0915.

Methodology in Plain English

The authors model cooling as a sequence of collisions: in step n the system interacts through a unitary with a fresh machine that starts in a Gibbs state of a Hamiltonian the authors are free to choose, and the cost is the energy dumped into the machines. Using the equality form of Landauer's principle, the total cost splits into a fixed entropy difference between endpoints plus a sum of non-negative dissipations, one per step.

To find the cheapest possible collisions, they first restrict to unitaries that merely permute the joint energy eigenstates, then compute the dissipation of a near-dissipationless permutation as a quadratic form in the machine-population perturbation. If a permutation is resonant — meaning there is a machine state for which nothing moves — the cost is second order and can be evaluated exactly. The local cost limit is then the geometric mean of these quadratic forms: the unit ball of the relevant norm is the convex hull of the unit sets of all permutations resonant at that state. This is the "collision geometry."

Theorem 1 justifies the restriction: any coherent collision close to a dissipationless one can be replaced, to leading order, by a probabilistic mixture of resonant permutations with no larger cost. The proof decomposes the perturbation generator and shows that coherences between different eigenspaces cost more in the joint relative entropy than they recover from the system's own relative entropy.

For complete protocols the authors work in a quasistatic regime where each collision is an O(1/N) perturbation, so the total dissipation is the squared path length in the collision geometry divided by 2N. They alternate permutations that stay resonant along the path, and verify with full exact dynamics: all machine populations are optimized (L-BFGS on exact dynamics with a penalty continuation), then re-evaluated in 40-digit arithmetic with explicit machine Hamiltonians and d² × d² permutation unitaries. The swap benchmark — minimizing Σ D(p_{n−1}‖p_n) — is jointly convex and therefore a global optimum. The hull norms in Fig. 1(a) come from a primal–dual pair of cone programs that agree to within 2×10^-7 at every plotted point (βω ≤ 12); at βω = 14 the bracket [0.70773, 0.70810] is too loose to plot.

Why This Matters

Impact on research. The paper corrects an extrapolation from qubit optimality: thermodynamic length governs swap-based cooling but not the finite-resource limit for qudits. It reduces a quantum optimal-control question to a classical convex-geometric one (Theorem 1), giving a tractable framework for computing optimal costs in higher dimensions, and it supplies a matching upper bound (Theorem 2) on how much larger machines can buy.

Real-world applications (as identified in the paper):

• Trapped-ion qudits, whose Zeeman sublevels form equally spaced ladders matching the machine gaps the protocols use.
• Bosonic modes, where equally spaced spectra arise naturally.
• Heat-bath algorithmic cooling, whose compression steps are permutations of energy eigenstates; the paper notes work there has focused on achievable polarization rather than finite-N dissipation.
• Bosonic sub-bath cooling, where exchanges of several excitations already improve on Gaussian operations.

Industry relevance. Any hardware platform pursuing qudit-based quantum computing or state preparation — trapped ions, photonics, superconducting circuits with higher-dimensional encodings — needs to know the true energy cost of resetting and cooling. Knowing that the swap-based protocol is not optimal, and by how much it can be beaten (up to 83% for eight-level systems in Table 1), sets realistic budgets for on-chip refrigeration and initialization.

Future Directions

• Prove or disprove the finite-temperature form of Theorem 2. The bound Σ ≥ (1/(d_M−1))·(1/2)‖δp‖_F² is proven only at the maximally mixed state for d_M ≥ 3 (and at every state for qubit machines). Whether it holds at all temperatures is open; the authors conjecture that it does and that it bounds the gain everywhere, as suggested by Fig. 2.

• Compute the norm for d ≥ 4 and for generic low-temperature spectra. The qutrit hull is characterized, and ququarts are sampled (8,162 mechanisms, ratios 0.845, 0.894, 0.973, 0.957 and 0.720 at βω = 0.05, 0.5, 1, 2 and 4), but the geometry for d ≥ 4 and generic spectra at low temperature is not settled.

• Exploit near-resonance. Collisions applied O(1/N) away from a resonance manifold are still dominated by permutations, but the offset could enlarge the hull. An optimistic relaxation at six temperatures along the Fig. 1(a) curve enlarges it by at most 3.6%; whether real near-resonant protocols can systematically exploit this is open.

• Transfer the analysis to heat-bath algorithmic cooling. Because its compression steps are exactly the permutations studied here, the same convex-geometric machinery could quantify its finite-N dissipation rather than only its achievable polarization.

Target Audience

Quantum thermodynamics researchers working on finite-time and finite-resource cooling, entropy production, and information geometry; theorists interested in Finsler versus Riemannian structures on state space; and experimental groups building qudit platforms (trapped ions, bosonic modes) who need realistic energy budgets for reset and initialization. Readers should be comfortable with relative entropy, the Kubo–Mori metric, and convex optimization; the paper's introduction and discussion sections are accessible to graduate students with a quantum information background.

Authors’ abstract

Cooling a quantum system with $N$ thermal machines costs more than Landauer's bound. For swap-based collision models the excess is set by thermodynamic length, which is optimal for qubits and was suggested by numerics to be optimal in general. We show that for qudits, permutations of energy eigenstates that use several resonant transitions at once do better for every spectrum, reducing the excess by up to 83% near the ground state. To second order, coherent collisions do no better, and at infinite temperature $d_M$-level machines reduce the excess by at most a factor $d_M-1$.