Trading athermality for nonstabiliserness

Abstract

Nonstabiliserness is a fundamental resource for quantum advantage, capturing how much a quantum state breaks the symmetries that would make it classically simulable. Can nonstabiliserness be generated from stabiliser states simply by coupling them to a heat bath? We explore the thermodynamic limits of nonstabiliserness under minimal assumptions and derive a necessary and sufficient condition for when such a process can create it from an initial stabiliser state. This provides an analytic characterisation of the nonstabiliser states that are reachable in this way, together with quantitative bounds on their degree of nonstabiliserness. Our framework also identifies optimal regimes, specifying Hamiltonians that maximise nonstabiliserness generation and the critical temperatures at which it emerges.

Figures (6)

• Figure 1: Thermal magician. This paper shows that a heat bath is not just a source of noise, it can also brew magic from stabiliser states. We determine exactly when thermal processes create magic, how much they produce, and which temperatures and Hamiltonians are optimal.
• Figure 2: Critical parameters. For a fixed population $p=0.3$ and three Hamiltonians with Bloch directions $\hat{\boldsymbol n}\in\{\tfrac{1}{\sqrt{2}}(1,1,0),\tfrac{1}{\sqrt{6}}(2,1,1),\tfrac{1}{\sqrt{3}}(1,1,1)\}$ (shown as blue $H_\textsf{H}$, yellow $H_{\textsf{w}}$, and red $H_\textsf{T}$), we illustrate the critical parameters for nonstabiliserness generation. Left panel: critical inverse temperature $\beta_{\rm{crt}}$ versus coherence $c$. The curves decrease monotonically with $c$. Right panel: critical coherence $c_{\rm{crt}}$ versus inverse temperature $\beta$. Lowering the temperature decreases the amount of coherence required to generate nonstabiliserness.
• Figure 3: Distillability landscape over Hamiltonian orientations. For a fixed initial stabiliser state with $p=0.3$ and $c=0.1$, each point on the equirectangular map (longitude $\phi$, latitude $\lambda$) corresponds to a Hamiltonian direction. Colours show the critical inverse temperature $\beta_{\rm dist}$ at which the maximal thermally reachable fidelity first exceeds the distillation threshold [$\ket{\textsf{T}}$ orbit (left) and $\ket{\textsf{H}}$ orbit (right)]. The red lobes are the easiest orientations: they are centred on the Clifford‑equivalent directions of the relevant orbit (black dots), eight for $\textsf{T}$ and twelve for $\textsf{H}$.
• Figure 4: Reachability under thermodynamic processes. Achievable states under thermal operations at $\beta = 2$, shown from three viewing angles alongside the stabiliser polytope. Starting from a stabiliser state with $p = 0.5$, $c = 0.25$, and Hamiltonian direction $\hat{\boldsymbol n}_{\textsf{H}} = (\tfrac{1}{\sqrt{2}}, \tfrac{1}{\sqrt{2}}, 0)$, the yellow region marks the produced magic states. Right: for the same initial state (black dot), upper hemispheres of fixed Bloch-ball cross-sections are shown for three Hamiltonians: $\hat{\boldsymbol n}_{\textsf{H}}$, $\hat{\boldsymbol n}_Z = (0, 0, 1)$, and $\hat{\boldsymbol n}_{\textsf{T}} = (\tfrac{1}{\sqrt{3}}, \tfrac{1}{\sqrt{3}}, \tfrac{1}{\sqrt{3}}).$
• Figure 5: Stabiliser polytope. The stabiliser polytope under rotations by an angle $\varphi$. Each panel displays a different orientation of the polytope, highlighting its octahedral symmetry. The right panel shows a cross-section of the Bloch ball together with the projection of the stabiliser polytope onto this cross-section

Theorems & Definitions (9)

• Lemma 1: Reachable populations Lostaglio2018
• Lemma 2: Reachable coherences Lostaglio2015
• Lemma 3: Rotated-frame $\ell_1$ support over circles
• proof
• Lemma 4: Single-$q$ stabiliser-containment test
• proof
• Theorem 5: TO no-magic criterion
• Corollary 6: Energy-incoherent states
• Corollary 7: Pauli-aligned special case