Stabilizer entropy is trustworthy for mixed states

Technical Summary: "Stabilizer Entropy is Trustworthy for Mixed States"

Problem Statement
Quantifying non-stabilizerness (often referred to as "magic") is essential for understanding quantum advantage, chaos, and the limitations of classical simulation. While non-stabilizerness monotones exist for pure states, their extension to mixed states presents a severe computational barrier. Recent work has established that any strict non-stabilizerness monotone for mixed states lies in the complexity class $QP_2$, requiring superexponential time to compute. This renders strict monotones impractical for realistic physical applications involving noise, partial measurements, or mixed-state dynamics. Furthermore, the naive extension of Stabilizer Entropy (SE) to mixed states fails to satisfy monotonicity under free operations (specifically partial trace and dephasing), violating a core requirement of resource theories.

Methodology
The authors propose a relaxation of the strict monotonicity condition by introducing the concept of an $\eta$-resource proxy. A function $f$ is defined as an $\eta$-resource proxy for a state ensemble $\mathcal{S}$ if it is faithful (zero on free states) and the probability of it violating monotonicity under free operations decays exponentially with the system size $N$ (specifically as $\exp(-\eta N)$).

The paper focuses on the Linear Stabilizer Entropy ($\tilde{M}_{lin}$) for mixed states, defined as:
$$\tilde{M}_{lin}(\rho) = \text{Pur}(\rho) - SP(\rho)$$
where $\text{Pur}(\rho) = \text{Tr}(\rho^2)$ is the purity and $SP(\rho) = d \text{Tr}(Q \rho^{\otimes 4})$ is the stabilizer purity, with $Q$ being the projector onto the stabilizer subspace.

The authors analyze the monotonicity gap $\bar{M}_E(\rho) = \tilde{M}_{lin}(\rho) - \tilde{M}_{lin}(E(\rho))$ under two free operations:

1. Partial Trace (reducing the system).
2. Dephasing in the computational basis (modeling non-post-selected measurements).

They employ analytical techniques involving Haar averages, Clifford group averages, and properties of Matrix Product States (MPS) to compute the mean and variance of this gap. They utilize Lévy's lemma and Chebyshev's inequality to bound the probability of monotonicity violations. Additionally, they perform numerical simulations on random MPS and the ground states of the 1D quantum XY model to verify these bounds in non-random, physically relevant settings.

Key Contributions and Results

1. Definition of Resource Proxies: The paper formalizes the notion of a resource proxy, arguing that for mixed states where strict monotones are computationally intractable, a measure that behaves as a monotone with overwhelming probability is a necessary and practical alternative.

2. Analytical Guarantees in Random Ensembles:

   • Haar Random States: The authors prove that for generic Haar-random states, the violation probability of $\tilde{M}_{lin}$ under partial trace and dephasing decays as $\exp(-\eta N)$ with $\eta = O(d/\log d)$, where $d=2^N$. This indicates superexponentially rare violations.
   • Clifford Orbits: For states generated by random Clifford unitaries acting on a fixed seed, the violation probability decays as $\exp(-\eta N)$ with $\eta = O(1)$. While the ensemble is smaller than the Haar case, violations remain exponentially rare.
   • Random Matrix Product States (MPS): For random MPS with bond dimension $\chi$, the violation probability is bounded by $\exp(-\eta N)$ where $\eta = O(\chi^2)$. Even for constant $\chi$, violations are exponentially unlikely, and the typicality improves as $\chi$ increases.

3. Numerical Verification in Physical Systems:

   • XY Model: The authors tested the logarithmic mixed-state SE ($\tilde{M}_2$) on the ground states of the spin-1/2 quantum XY chain. They computed the outcome-averaged post-measurement non-stabilizerness gap for partial measurements in the computational basis. Across the entire phase diagram (including ordered and disordered phases and both parity sectors), the gap remained strictly positive. This demonstrates that $\tilde{M}_2$ (and by extension $\tilde{M}_{lin}$) does not increase on average under partial measurements in this specific many-body system.
   • Open Boundary Conditions (OBC): Numerical analysis of random MPS with OBC confirmed that both $\tilde{M}_{lin}$ and $\tilde{M}_2$ maintain positive gaps under dephasing and partial trace across various system sizes and bond dimension scalings.

Significance and Claims
The paper claims that Linear Stabilizer Entropy ($\tilde{M}_{lin}$) serves as a trustworthy, efficiently computable proxy for non-stabilizerness in mixed states. The authors argue that given the proven computational hardness of strict monotones for mixed states, the "resource proxy" framework is not merely a convenience but a necessity for practical applications.

The significance lies in enabling the reliable quantification of magic resources in:

• Noisy quantum information processes involving measurements and decoherence.
• Many-body systems where access to subsystem properties (via reduced density matrices) is required.
• Experimental settings where full state tomography is infeasible, but linear entropies are experimentally accessible.

The work suggests that while strict monotonicity cannot be guaranteed for all mixed states, the probability of violation is so low in physically relevant ensembles (Haar, Clifford, MPS, and specific Hamiltonian ground states) that Stabilizer Entropy can be used with "overwhelming probability" as a valid resource measure. The authors remain modest, noting that while their results are robust across tested ensembles, further verification is needed for other state classes (e.g., $t$-doped stabilizer states) and 2D variational ansatze (PEPS).