Deep Thermalization and Measurements of Quantum Resources

Quantum resource theories (QRTs) provide a unified framework for characterizing useful quantum phenomena subject to physical constraints, but are notoriously hard to assess in experimental systems. In this letter, we introduce a unified protocol for quantifying the resource-generating power (RGP) of arbitrary quantum evolutions applicable to multiple QRTs. It is based on deep thermalization (DT), which has recently gained attention for its role in the emergence of quantum state designs from partial projective measurements. Central to our approach is the use of projected ensembles, recently employed to probe DT, together with new twirling identities that allow us to directly infer the RGP of the underlying dynamics. These identities further reveal how resources build up and thermalize in generic quantum circuits. Finally, we show that quantum resources themselves undergo deep thermalization at the subsystem level, offering a complementary and another experimentally accessible route to infer the RGP. Our work connects deep thermalization to resource quantification, offering a new perspective on the essential role of various resources in generating state designs.

💡 Research Summary

This paper establishes a unified experimental protocol for quantifying the resource‑generating power (RGP) of arbitrary quantum dynamics across multiple quantum resource theories (QRTs) by leveraging the concept of deep thermalization (DT). DT describes the refined statistical relaxation of many‑body quantum systems, wherein the post‑measurement ensemble of a subsystem converges to well‑defined distributions such as state designs or finite‑temperature Scrooge ensembles. The authors focus on projected ensembles—collections of pure states obtained after a global unitary evolution followed by a measurement on a large subsystem—and show that these ensembles encode the resource content of the underlying dynamics.

The central technical tool is a set of twirling identities. For a given QRT, the authors define a free state ensemble ρ_f (or ρ_s for stabilizer‑type resources) and consider the average over random free unitaries F of the k‑fold tensor product (F U)⊗k ρ_f (F U)†⊗k, where k=2 for Z₂‑asymmetry and k=4 for non‑stabilizerness. The twirling identities take the form

 ⟨(F U)⊗k ρ_f (F U)†⊗k⟩_F = (1‑R_p(U)) ρ_f + R_p(U) Π_k,

where Π_k is the normalized Haar‑random k‑fold moment and R_p(U) is the linear‑entropic RGP associated with the unitary U in the chosen QRT. This decomposition shows that the resource generated by U is mixed with a Haar‑random component in proportion to its RGP.

Exploiting this relation, the authors propose an experimentally feasible protocol: (i) prepare a random pure free state |ψ⟩, (ii) apply the target unitary U, (iii) apply a random free operation F, (iv) measure a subsystem B in the computational basis and record the outcome probabilities p_b. Repeating the experiment over many random free states and free unitaries yields the average ⟨p_b^t⟩, which is linearly related to R_p(U) via constants k₁ and k₂ that depend only on the dimensions of the full system (2^N) and the measured subsystem (2^{N_A}). The explicit formula

 R_p(U) = (k₂ – ⟨p_b^t⟩)/(k₂ – k₁)

allows direct extraction of the RGP without full state tomography or correlation measurements.

A second key insight is that the projected states themselves carry the same resource information. For a fixed measurement outcome |b⟩, the projected pure state |ϕ_b⟩ = ⟨b|F U|ψ⟩/√p_b has an average resource ⟨R_PS⟩ that, to leading order, satisfies

 ⟨R_PS⟩ ≈ Ω R_A k₁/(Ω k₂ + (1‑Ω) k₁),

with Ω = R_p(U)/R (R being the Haar‑averaged RGP) and R_A the average resource of Haar‑random states. When the measured subsystem size approaches the full system (N_A → N), k₁ ≈ k₂ and the ratio ⟨R_PS⟩/R_A converges to R_p(U)/R, providing an alternative route to quantify the RGP using only subsystem measurements.

The third major result concerns the thermalization dynamics of resources in circuits that interleave free and non‑free operations. Defining a circuit U(t) = F_t U F_{t‑1} U … F₁ U, where each free unitary F_j is drawn independently from the free group, the authors prove that the averaged RGP obeys

 ⟨R_p