On polyhedral approximations of the positive semidefinite cone

Let $D$ be the set of $n\times n$ positive semidefinite matrices of trace equal to one, also known as the set of density matrices. We prove two results on the hardness of approximating $D$ with polytopes. First, we show that if $0 < \epsilon < 1$ and $A$ is an arbitrary matrix of trace equal to one, any polytope $P$ such that $(1-\epsilon)(D-A) \subset P \subset D-A$ must have linear programming extension complexity at least $\exp(c\sqrt{n})$ where $c > 0$ is a constant that depends on $\epsilon$. Second, we show that any polytope $P$ such that $D \subset P$ and such that the Gaussian width of $P$ is at most twice the Gaussian width of $D$ must have extension complexity at least $\exp(cn^{1/3})$. The main ingredient of our proofs is hypercontractivity of the noise operator on the hypercube.

💡 Research Summary

The paper investigates how hard it is to approximate the set of n × n positive semidefinite matrices with trace one (the quantum density matrices, denoted D) by a polytope. The quality of approximation is measured in two ways. The first notion requires a polytope P to satisfy (1 − ε)(D − A) ⊆ P ⊆ D − A for some arbitrary trace‑one matrix A and a fixed ε∈(0,1). The second notion asks for a polytope P that contains D and whose Gaussian width w_G(P) is at most twice the Gaussian width of D. The complexity metric is the linear‑programming extension complexity, i.e., the smallest number of facets of a higher‑dimensional polytope that can be linearly projected onto P.

The authors prove two lower‑bound theorems. Theorem 1 shows that any polytope satisfying the first approximation condition must have extension complexity at least exp(c √n), where c>0 depends only on ε. Theorem 2 shows that any polytope satisfying the second condition must have extension complexity at least exp(c n^{1/3}) for an absolute constant c>0.

The proof strategy is unified and rests on hypercontractivity of the noise operator on the Boolean hypercube. First, the problem is reduced to the case A = (1/n)I_n by averaging over cyclic permutations; this step increases extension complexity by at most a factor n, which does not affect the exponential lower bound. The authors then consider the slack matrix of the pair ((1 − ε)C, C) where C = D − (1/n)I_n. The extreme points of C are matrices of the form xxᵀ − (1/n)I_n with x on the unit sphere, and the polar body’s extreme points are I_n − nyyᵀ. Computing the slack entries yields the kernel (1 − ε)n(xᵀy)² + ε, which is non‑negative.

To connect the slack matrix to extension complexity, Yannakakis’s theorem (generalized to arbitrary inner and outer bodies) is invoked: the extension complexity of any polytope sandwiched between the two bodies equals the non‑negative rank of the slack matrix. Hence a lower bound on the non‑negative rank yields the desired extension‑complexity lower bound.

Assuming the slack matrix can be written as a sum of N rank‑one non‑negative matrices, each term can be expressed as f_i(x)g_i(y) with non‑negative functions f_i, g_i on the hypercube H_n = {−1,1}ⁿ. By normalising the expectation of each f_i to 1, the authors apply the noise operator T_ρ (with ρ≈p⁵/n) to both sides of the decomposition. Hypercontractivity (Bonami‑Beckner inequality) guarantees that T_ρ smooths each f_i, making its higher L_q norm comparable to its L_p norm for suitable p and q. A concentration lemma (Lemma 1) is proved: for any non‑negative f with mean 1 and bounded maximum (≤e^{√n}), the probability that T_ρ f exceeds 4 is at most e^{−c√n}. Using this, the authors argue that for every point y∈H_n there must exist an index i with T_ρ f_i(y)≥4, which forces N to be at least e^{c√n}. The case where some f_i have very large maxima is handled separately by showing that then the corresponding g_i must be tiny, making their contribution negligible; the remaining indices still satisfy the concentration bound, leading again to the exponential lower bound.

For Theorem 2, the same construction is used but the Gaussian width constraint replaces the relative‑error condition. The Gaussian width of D is Θ(√n); requiring w_G(P)≤2w_G(D) forces the slack matrix to have a similar structure, and the same hypercontractivity argument with a different choice of ρ (roughly ρ≈n^{−2/3}) yields a concentration bound of order e^{−c n^{1/3}}. Consequently the non‑negative rank, and thus the extension complexity, must be at least exp(c n^{1/3}).

The paper situates its results among prior work on polyhedral approximations of convex bodies. While the Euclidean ball can be approximated within (1 − ε) using a polytope of linear (or O(n log 1/ε)) extension complexity (Ben‑Tal & Nemirovski), the density matrix set D is shown to be intrinsically harder: even allowing a constant relative error or a modest increase in Gaussian width forces exponential extension complexity. The authors also discuss related constructions (δ‑nets, ellipsoids) and note that the lower bounds are not known to be tight; the best known upper bound for a (1 − ε)‑approximation uses a δ‑net and yields exp(O(n)) facets.

In summary, the paper establishes that any polyhedral approximation of the quantum density matrix set that is either relatively accurate or has controlled Gaussian width must have extension complexity exponential in √n or n^{1/3}. The key technical contribution is the clever use of hypercontractivity on the hypercube to translate geometric approximation requirements into lower bounds on non‑negative rank, thereby linking functional analysis, quantum information geometry, and polyhedral combinatorics. This methodology may be applicable to other high‑dimensional convex sets arising in quantum theory and optimization.