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The marginal is pretty good
Lukas Schmitt, Joseph M. Renes
TL;DR
One-shot information measures often require complicated, state-dependent optimizers. This note analyzes replacing the optimizer with the explicit marginal and proves a factor-1/α bound for Petz–Rényi divergence, with analogous results for fidelity and restricted sandwiched Rényi cases.
Problem
Optimizers in one-shot information measures can depend nonlinearly on the joint state, making the canonical marginal an attractive replacement to quantify.
Method
The paper uses variational formulas, Hölder-type arguments, operator inequalities, and data processing to compare optimization over σ_B with evaluation at the marginal.
Results
For Petz–Rényi divergence with α ∈ [1/2, 1), replacing the optimizer by ρ_B incurs at most a multiplicative factor 1/α; analogous relations hold for fidelity and for pure or quantum-classical states under sandwiched Rényi divergence.
Takeaways & Limitations
The marginal is almost as good as the optimizer in the stated Petz–Rényi, fidelity, and restricted sandwiched Rényi settings.
Takeaways & Limitations
For the sandwiched divergence, the general-state case beyond the established fidelity and pure or quantum-classical regimes is not proved; only a weaker bound is obtained there.
Abstract
from arXiv · showhide
One-shot information theory measures often require an optimization over states, but the form of these optimizers can be complicated or depend on the initial problem in nonlinear ways. In this note, we show that in many instances using the marginal instead of the optimal state is sufficiently good and only changes the result by a small factor. We prove that for the Petz-Rényi divergence of order $α\in[1/2,1)$, replacing the optimizing state on $B$ by the marginal $ρ_B$ results in a multiplicative overhead of at most $1/α$. We also show a similar relation for the fidelity, and in the case of pure or quantum-classical states for the sandwiched Rényi divergence.
1 Introduction
The note studies when the canonical marginal ρ_B can replace a complicated, state-dependent optimizer in one-shot information measures. For Petz–Rényi divergence with α ∈ [1/2, 1), this replacement incurs at most a factor 1/α, with a loss independent of dimension that vanishes as α → 1.
- Optimizers in Rényi generalizations of conditional entropies can depend nonlinearly on the joint state, whereas ρ_B is canonical and explicit.
- For Petz–Rényi divergence with α ∈ [1/2, 1), replacing the optimizer on B by ρ_B achieves the optimum up to factor 1/α.
- The resulting loss is independent of dimension and tends to vanish as α → 1.
- The proof uses a Schatten-norm variational formula and an operator inequality obtained from data processing.
- The analogous relation holds for the sandwiched divergence at α = 1/2 and for pure and quantum-classical states when α ∈ (1/2, 1).
2 Preliminaries
The preliminaries introduce the state-space and fidelity notation, the Petz–Rényi divergence, and the variational tools used to analyze optimization over σ_B. The proof framework combines Hölder’s inequality with data processing.
- The state space of a quantum system B is denoted S(B), and positive-semidefinite operator powers are defined on their supports.
- Fidelity is defined as F(ω, ξ) = ∥√ω√ξ∥1.
- The Petz–Rényi divergence is introduced for α ∈ (0, 1) as the central divergence used in the variational bounds.
- The proof establishes the main equation through two lemmas, including Hölder’s inequality with exponents p = 1/α and q = 1/(1 − α).
- Data processing of the Petz–Rényi divergence for α ∈ (0, 1) supplies the comparison needed for an arbitrary density operator ω_B.
3 Petz–R´enyi bound
The Petz–Rényi result bounds the cost of replacing the optimizing state with the marginal for finite-dimensional bipartite states, under a full-rank assumption on τ_A that extends by continuity. The bound can be tight, while the optimizer becomes the marginal as α → 1.
- Petz–Rényi bound: The variational formula rewrites the optimization over σ_B using the operator W introduced in the proof.
- Petz–Rényi bound: Operator monotonicity of x ↦ x^(1−α)/α yields the key inequality, whose direction reverses after applying the logarithm because α − 1 < 0.
- Petz–Rényi bound: The proof concludes the claimed bound for the Petz–Rényi divergence.
- Petz–Rényi bound: As α → 1, the theorem still holds and the optimizer is known to be the marginal.
- Petz–Rényi bound: The full-rank assumption on τ_A is technical, and the statement extends to arbitrary density operators τ_A ≥ 0 by continuity.
- Tightness: The bound is tight for a classical distribution with one distinguished outcome and a uniform reference distribution, where the divergence difference is zero.
- Tightness: In the tightness construction, the relevant limit vanishes as 1/n^α while the marginal component converges to p.
4 Sandwiched–R´enyi bound
The section extends the marginal-versus-optimizer comparison to the sandwiched Rényi divergence: the relation holds for arbitrary states at α=1/2 and for pure or quantum-classical states when α∈(1/2,1). The fidelity case is proved using operator Cauchy–Schwarz, while the special-state result uses pinching and norm inequalities.
- At α=1/2, the sandwiched-divergence problem is equivalent to fidelity maximization for arbitrary states.
- 4.1 Fidelity bound: The fidelity bound is established with an operator Cauchy–Schwarz inequality applied to a unitary-based variational argument.
- 4.2 Special cases: For α∈(1/2,1), the corresponding relation holds for pure states and quantum-classical states.
- 4.2 Special cases: The pure and quantum-classical cases reduce the operator comparison using pinching, diagonal σB, and norm inequalities.
5 Discussion
The discussion concludes that the marginal is almost as good as the optimizer, with the restriction α≥1/2 arising from operator monotonicity. The authors cannot establish the analogous claim for general states, though weaker bounds are available.
- Theorem 3.1 shows that using the marginal is almost as good as using the optimizer.
- The proof requires α≥1/2 because it uses operator monotonicity of x^(1−α)/α.
- For general states, the authors cannot prove that the analogous sandwiched-divergence relation holds, but weaker bounds follow from Araki–Lieb–Thirring inequalities.
- Combining the weaker general-state bound with Theorem 3.1 yields an additional bound.
A Non-full-rank states
The non-full-rank case is handled by continuity: a uniform convergence argument allows the full-rank result to extend to general states. The optimization equivalence follows from the negative sign of (α−1)^−1.
- The full-rank assumption on τA can be removed by continuity.
- Uniform convergence of Qε to Q0 over the state space supports taking the ε→0 limit.
- Because (α−1)^−1<0, minimizing the Petz–Rényi divergence is equivalent to maximizing Qε.