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arXiv:2608.10147

A Simple Algorithm for Best Separable State

Prashanti Anderson, Samuel B. Hopkins, Amit Rajaraman

cs.DSquant-phSubmitted Aug 10, 2026• Announced Aug 12, 2026• 0 citations
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Abstract

We study the best separable state problem (BSS), which asks for the maximum acceptance probability of a quantum measurement over unentangled states. In classical terms, the goal is to maximize $\langle(x \otimes y), M (x \otimes y)\rangle$ over unit vectors $x,y$ where $0 \preceq M \preceq I$; we call this value $\mathrm{BSS}(M)$. We study $\mathrm{BSS}$ in the "perfect completeness" regime, where given $M$ such that $\mathrm{BSS}(M) = 1$ the goal is to find the best possible solution $x,y$ -- this generalizes the problem of finding a rank-one matrix as close as possible to a given subspace of $\mathbb{R}^{n \times n}$ guaranteed to contain a rank-one matrix. The strongest known algorithmic guarantees for this problem are: (1) an algorithm which finds a solution with value $1-\varepsilon$ in time $\exp(\sqrt{n} (\log n)^{O(1)} / \varepsilon^2)$, due to Barak, Kothari, and Steurer, and (2) an algorithm which finds a solution with value $q/n$ in time roughly $n^{O(q)}$, due to Bhattiprolu, Ghosh, Guruswami, Lee, and Tulsiani. We give a much simpler approach to rounding the SoS relaxation, generalizing the canonical "global correlation rounding" technique, and obtain a better running time. Given $M$ with $\mathrm{BSS}(M) = 1$, our algorithm finds a solution with value $1-ε$ in time $n^{O(\sqrt{n/\varepsilon})}$, and a solution of value $q/n$ in time $n^{O(\sqrt q)}$. Using the same techniques, we prove a new variant of the "pinning lemma", a measure-decomposition theorem widely used in LP/SDP rounding, high-dimensional probability, and statistical physics, which we believe is of independent interest.

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arXiv:1701.06321•Jan 23, 2017•0 citations

An Argmax Principle for Sum-of-Squares Relaxations on the Sphere

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Quantum SDP Solvers: Large Speed-Ups, Optimality, and Applications to Quantum Learning

Fernando G. S. L. Brandão, Amir Kalev, Tongyang Li +3 more

We give two new quantum algorithms for solving semidefinite programs (SDPs) providing quantum speed-ups. We consider SDP instances with m constraint matrices, each of dimension n, rank at most r, and sparsity s. The first algorithm assumes an input model where one is given access to an oracle to the entries of the matrices at unit cost. We show that it has run time O~(s^2 (sqrt{m} epsilon^{-10} + sqrt{n} epsilon^{-12})), with epsilon the error of the solution. This gives an optimal dependence in terms of m, n and quadratic improvement over previous quantum algorithms (when m ~~ n). The second algorithm assumes a fully quantum input model in which the input matrices are given as quantum states. We show that its run time is O~(sqrt{m}+poly(r))*poly(log m,log n,B,epsilon^{-1}), with B an upper bound on the trace-norm of all input matrices. In particular the complexity depends only polylogarithmically in n and polynomially in r. We apply the second SDP solver to learn a good description of a quantum state with respect to a set of measurements: Given m measurements and a supply of copies of an unknown state rho with rank at most r, we show we can find in time sqrt{m}*poly(log m,log n,r,epsilon^{-1}) a description of the state as a quantum circuit preparing a density matrix which has the same expectation values as rho on the m measurements, up to error epsilon. The density matrix obtained is an approximation to the maximum entropy state consistent with the measurement data considered in Jaynes' principle from statistical mechanics. As in previous work, we obtain our algorithm by "quantizing" classical SDP solvers based on the matrix multiplicative weight update method. One of our main technical contributions is a quantum Gibbs state sampler for low-rank Hamiltonians, given quantum states encoding these Hamiltonians, with a poly-logarithmic dependence on its dimension, which is based on ideas developed in quantum principal component analysis. We also develop a "fast" quantum OR lemma with a quadratic improvement in gate complexity over the construction of Harrow et al. [Harrow et al., 2017]. We believe both techniques might be of independent interest.

•Jan 1, 2019•60 citations
We study the best separable state problem (BSS), which asks for the maximum acceptance probability of a quantum measurement over unentangled states. In classical terms, the goal is to maximize ⟨(x⊗y),M(x⊗y)⟩\langle(x \otimes y), M (x \otimes y)\rangle⟨(x⊗y),M(x⊗y)⟩ over unit vectors x,yx,yx,y where 0⪯M⪯I0 \preceq M \preceq I0⪯M⪯I; we call this value BSS(M)\mathrm{BSS}(M)BSS(M). We study BSS\mathrm{BSS}BSS in the "perfect completeness" regime, where given MMM such that BSS(M)=1\mathrm{BSS}(M) = 1BSS(M)=1 the goal is to find the best possible solution x,yx,yx,y -- this generalizes the problem of finding a rank-one matrix as close as possible to a given subspace of Rn×n\mathbb{R}^{n \times n}Rn×n guaranteed to contain a rank-one matrix. The strongest known algorithmic guarantees for this problem are: (1) an algorithm which finds a solution with value 1−ε1-\varepsilon1−ε in time exp⁡(n(log⁡n)O(1)/ε2)\exp(\sqrt{n} (\log n)^{O(1)} / \varepsilon^2)exp(n​(logn)O(1)/ε2), due to Barak, Kothari, and Steurer, and (2) an algorithm which finds a solution with value q/nq/nq/n in time roughly nO(q)n^{O(q)}nO(q), due to Bhattiprolu, Ghosh, Guruswami, Lee, and Tulsiani. We give a much simpler approach to rounding the SoS relaxation, generalizing the canonical "global correlation rounding" technique, and obtain a better running time. Given MMM with BSS(M)=1\mathrm{BSS}(M) = 1BSS(M)=1, our algorithm finds a solution with value 1−ε1-ε1−ε in time nO(n/ε)n^{O(\sqrt{n/\varepsilon})}nO(n/ε​), and a solution of value q/nq/nq/n in time nO(q)n^{O(\sqrt q)}nO(q​). Using the same techniques, we prove a new variant of the "pinning lemma", a measure-decomposition theorem widely used in LP/SDP rounding, high-dimensional probability, and statistical physics, which we believe is of independent interest.
For every constant ε>0, we give an exp(Õ(∞n))-time algorithm for the 1 vs 1 - ε Best Separable State (BSS) problem of distinguishing, given an n2 x n2 matrix ℳ corresponding to a quantum measurement, between the case that there is a separable (i.e., non-entangled) state ρ that ℳ accepts with probability 1, and the case that every separable state is accepted with probability at most 1 - ε. Equivalently, our algorithm takes the description of a subspace 𝒲 ⊆ 𝔽n2 (where 𝔽 can be either the real or complex field) and distinguishes between the case that contains a rank one matrix, and the case that every rank one matrix is at least ε far (in 𝓁2 distance) from 𝒲.
We prove optimal finite quantum de Finetti upper bounds. Given a bosonic state ρN∈D(SymN(Cd))ρ_N\in D(\mathrm{Sym}^N(\mathbb C^d))ρN​∈D(SymN(Cd)), there is a probability measure ννν on the unit sphere such that ∥ρN(2)−∫∣u⟩⟨u∣⊗2 dν(u)∥1≤d−1N−1. \left\| ρ_N^{(2)}-\int |u\rangle\langle u|^{\otimes 2}\,dν(u) \right\|_1 \le \frac{\sqrt{d-1}}{N-1}. ​ρN(2)​−∫∣u⟩⟨u∣⊗2dν(u)​1​≤N−1d−1​​. By purification, the bosonic theorem also gives the optimal O(d/N)O(d/N)O(d/N) upper bound for arbitrary exchangeable states. These results settle the dimension dependence left open by Christandl, König, Mitchison, and Renner (CMP 2007). The proof casts de Finetti approximation as sum-of-squares rounding and applies the argmax method of Jeronimo, Wu, and Xu (manuscript 2026). More generally, ttt-site marginals satisfy O(td/N)O(t\sqrt d/N)O(td​/N) bosonic and O(td/N)O(td/N)O(td/N) permutation-invariant bounds. Our proof formulates de Finetti approximation as the integrality gap of a symmetric-extension semidefinite program and rounds an optimum by the argmax principle. The sharp bounds have several consequences. For every fixed ε∈(0,1)\varepsilon\in(0,1)ε∈(0,1), we construct a channel with input dimension D=exp⁡(Oε(dlog⁡d))=exp⁡(o(d))D=\exp(O_\varepsilon(\sqrt d\log d))=\exp(o(d))D=exp(Oε​(d​logd))=exp(o(d)) whose outputs are ε\varepsilonε-close to separable states of local dimension ddd and whose image contains every such separable state, thereby refuting Watrous's disentangler conjecture. We also obtain deterministic exp⁡(O~(d/ε))\exp(\widetilde O(\sqrt d/\varepsilon))exp(O(d​/ε))-time algorithms for explicit Best Separable State without perfect completeness and for trace-distance separability testing. Finally, spectral truncation gives the first dimension-free bosonic de Finetti theorem in Hilbert--Schmidt distance, with the optimal rate Θ(N−1/2)Θ(N^{-1/2})Θ(N−1/2) when the dimension may grow.
We prove quantitative lower bounds on the semidefinite extension complexity of the set of separable quantum states on Cd⊗Cd\mathbb{C}^d\otimes\mathbb{C}^dCd⊗Cd. We consider semidefinite programs (SDPs) that approximate the maximum acceptance probability of a measurement over separable states, the optimization problem underlying QMA(2). In the extended-formulation model of Harrow, Natarajan, and Wu (HNW), all measurements share a common feasible region and an objective-independent embedding of product states that exactly reproduces their acceptance probabilities. For every 0<θ<2/70<θ<2/70<θ<2/7, there are constants cθ,aθ>0c_θ,a_θ>0cθ​,aθ​>0 such that, for sufficiently large ddd, any such SDP with uniform additive error 0<a≤aθ0<a\le a_θ0<a≤aθ​ has size at least dcθmin⁡{a−1/3,dθ}d^{c_θ\min\{a^{-1/3},d^θ\}}dcθ​min{a−1/3,dθ}. The bound applies at sufficiently small constant error, is superpolynomial in ddd whenever a=o(1)a=o(1)a=o(1), and becomes dΩ(dθ)d^{Ω(d^θ)}dΩ(dθ) when a≤d−3θa\le d^{-3θ}a≤d−3θ, improving HNW's quasipolynomial bound at inverse- square error. The same bound holds for any SDP-representable convex set of states that contains all separable states and lies within trace distance aaa of them, giving a quantitative counterpart to Fawzi's theorem that the separable set has no exact semidefinite representation. Our proof combines the quantitative pseudo-density theorem of Lee, Raghavendra, and Steurer with explicit block-positive operators and Chebyshev amplification. Our main results are supported by Lean proofs.
For every ε>0ε>0ε>0, we give an exp⁡(O~(n/ε2))\exp(\tilde{O}(\sqrt{n}/ε^2))exp(O~(n​/ε2))-time algorithm for the 111 vs 1−ε1-ε1−ε \emph{Best Separable State (BSS)} problem of distinguishing, given an n2×n2n^2\times n^2n2×n2 matrix M\mathcal{M}M corresponding to a quantum measurement, between the case that there is a separable (i.e., non-entangled) state ρρρ that M\mathcal{M}M accepts with probability 111, and the case that every separable state is accepted with probability at most 1−ε1-ε1−ε. Equivalently, our algorithm takes the description of a subspace W⊆Fn2\mathcal{W} \subseteq \mathbb{F}^{n^2}W⊆Fn2 (where F\mathbb{F}F can be either the real or complex field) and distinguishes between the case that W\mathcal{W}W contains a rank one matrix, and the case that every rank one matrix is at least εεε far (in ℓ2\ell_2ℓ2​ distance) from W\mathcal{W}W. To the best of our knowledge, this is the first improvement over the brute-force exp⁡(n)\exp(n)exp(n)-time algorithm for this problem. Our algorithm is based on the \emph{sum-of-squares} hierarchy and its analysis is inspired by Lovett's proof (STOC '14, JACM '16) that the communication complexity of every rank-nnn Boolean matrix is bounded by O~(n)\tilde{O}(\sqrt{n})O~(n​).
We develop an argmax principle for analyzing sum-of-squares relaxations of optimization problems over the unit sphere. Given a feasible pseudo-expectation, we form a polynomial of high-order pseudo-moments, such as Φk(u)=E~⟨x,u⟩2kΦ_k(u)=\widetilde{\mathbb E}\langle x,u\rangle^{2k}Φk​(u)=E⟨x,u⟩2k. Our guiding principle is that its maximizers are rounding candidates: their local and global optimality conditions reveal the reweighed pseudo-expectation inequalities governing SoS convergence. This viewpoint unifies several problems previously analyzed by rather different techniques. We obtain three results. First, for Best Separable State, we give a degree-O(n/ε)O(\sqrt{n/ε})O(n/ε​) SoS analysis for approximating hsep(P)h_{\mathrm{sep}}(P)hsep​(P) in the perfect-completeness regime, improving and simplifying Barak, Kothari and Steurer (STOC'17). The dependence is essentially tight for inverse-linear gap under the Exponential-Time Hypothesis, matching hardness from QMA(2)\mathrm{QMA}(2)QMA(2) protocols. Second, for the matrix 2→42\to42→4 norm, degree-O(n/ε)O(\sqrt n/ε)O(n​/ε) SoS gives a multiplicative (1+ε)(1+ε)(1+ε) approximation. Barak et al. (STOC'12) previously gave a comparable-time constant-gap decision algorithm; our result gives a multiplicative guarantee and extends to a family of p→qp\to qp→q norms with even qqq. Finally, for degree-ddd polynomial optimization, we recover the convergence theorem of Bhattiprolu et al. (FOCS'17) with a shorter, more direct proof: degree-kkk SoS gives approximation ratio Od((n/k)d/2−1)O_d((n/k)^{d/2-1})Od​((n/k)d/2−1). The paper introduces no new relaxation. Instead, the high-moment argmax gives a common way to read an SoS solution, unifying previously separate convergence analyses and yielding sharper bounds or simpler proofs.
We give two new quantum algorithms for solving semidefinite programs (SDPs) providing quantum speed-ups. We consider SDP instances with m constraint matrices, each of dimension n, rank at most r, and sparsity s. The first algorithm assumes an input model where one is given access to an oracle to the entries of the matrices at unit cost. We show that it has run time O~(s^2 (sqrt{m} epsilon^{-10} + sqrt{n} epsilon^{-12})), with epsilon the error of the solution. This gives an optimal dependence in terms of m, n and quadratic improvement over previous quantum algorithms (when m ~~ n). The second algorithm assumes a fully quantum input model in which the input matrices are given as quantum states. We show that its run time is O~(sqrt{m}+poly(r))*poly(log m,log n,B,epsilon^{-1}), with B an upper bound on the trace-norm of all input matrices. In particular the complexity depends only polylogarithmically in n and polynomially in r. We apply the second SDP solver to learn a good description of a quantum state with respect to a set of measurements: Given m measurements and a supply of copies of an unknown state rho with rank at most r, we show we can find in time sqrt{m}*poly(log m,log n,r,epsilon^{-1}) a description of the state as a quantum circuit preparing a density matrix which has the same expectation values as rho on the m measurements, up to error epsilon. The density matrix obtained is an approximation to the maximum entropy state consistent with the measurement data considered in Jaynes' principle from statistical mechanics. As in previous work, we obtain our algorithm by "quantizing" classical SDP solvers based on the matrix multiplicative weight update method. One of our main technical contributions is a quantum Gibbs state sampler for low-rank Hamiltonians, given quantum states encoding these Hamiltonians, with a poly-logarithmic dependence on its dimension, which is based on ideas developed in quantum principal component analysis. We also develop a "fast" quantum OR lemma with a quadratic improvement in gate complexity over the construction of Harrow et al. [Harrow et al., 2017]. We believe both techniques might be of independent interest.