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Handbook on Semidefinite, Conic and Polynomial Optimization pp 601–634Cite as

SDP Relaxations for Non-Commutative Polynomial Optimization

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Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 166))

Abstract

We consider the problem of minimizing an arbitrary hermitian polynomial p(X) in non-commutative variables X = (X1, …, XN), where the polynomial p(X) is evaluated over all states and bounded operators (X1, …, Xn) satisfying a finite set of polynomial constraints. Problems of this type appear frequently in areas as diverse as quantum chemistry, condensed matter physics, and quantum information science; finding numerical tools to attack them is thus essential. In this chapter, we describe a hierarchy of semidefinite programming relaxations of this generic problem, which converges to the optimal solution in the asymptotic limit. Furthermore, we derive sufficient optimality conditions for each step of the hierarchy. Our method is related to recent results in non-commutative algebraic geometry and can be seen as a generalization to the non-commutative setting of well-known semidefinite programming hierarchies that have been introduced in scalar (i.e. commutative) polynomial optimization. After presenting our results, we discuss at the end of the chapter some open questions and possible directions for future research.

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Notes

  1. 1.

    To see this, take \(\{{\overline{u}}_{i}\}\) to be the canonical basis of \({\mathbb{K}}^{s}\), and note that A = B ∗ B, with \(B ={ \sum \nolimits }_{i=1}^{s}{\overline{v}}_{i} \cdot {({\overline{u}}_{i})}^{{_\ast}}\). It follows that \(\mbox{ rank}(A) = \mbox{ rank}(B) =\dim (\mbox{ span}\{{\overline{v}{}_{i}\}}_{i})\).

  2. 2.

    The proof that we give is similar to the one of Lemma 21.1. An alternative proof based on a Gelfand–Naimark–Segal like construction is given in [49].

  3. 3.

    Actually, Pál and Vértesi’s algorithms work by optimizing over the set of operators and states defined by (21.58).

  4. 4.

    In the Schrödinger representation for the operators a, a ∗ , the Hilbert space is \({\mathcal{L}}^{2}(\mathbb{R})\), and, for any \(f(x) \in {\mathcal{L}}^{2}(\mathbb{R})\), the action of a over f(x) is given by \(af(x) = [xf(x) + f^{\prime}(x)]/\sqrt{2}\).

  5. 5.

    Actually, it is not necessary to invoke unbounded operators to get situations similar to this second point: the commutation condition i[X, Y ] ≥ 0 is satisfied in a non-trivial way only in infinite-dimensional spaces (see for instance  [20]).

  6. 6.

    In fact, this condition holds for any pair of arbitrary operators O1 and O2.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Bristol, University Walk, BS8 1TW, Bristol, UK

    Miguel Navascués

  2. Laboratoire d’Information Quantique, C.P. 225, Av. F. D. Roosevelt 50, B-1050, Bruxelles, Belgium

    Stefano Pironio

  3. ICFO-Institut de Ciències Fotòniques, Mediterranean Technology Park, 08860, Castelldefels, Spain

    Antonio Acín

  4. ICREA-Institució Catalana de Recerca i Estudis Avançats, Passeig Lluís Companys 23, 08010, Barcelona, Spain

    Antonio Acín

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  1. Miguel Navascués
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  2. Stefano Pironio
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Correspondence to Miguel Navascués .

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Editors and Affiliations

  1. Department of Mathematics and Industrial, Ecole Polytechnique Montreal, Édouard-Montpetit Boulevard 2350, Montreal, QC, Canada, H3C 3A7

    Miguel F. Anjos

  2. LAAS-CNRS, Avenue du Colonel Roche 7, 31077, Toulouse Cedex 4, France

    Jean B. Lasserre

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Navascués, M., Pironio, S., Acín, A. (2012). SDP Relaxations for Non-Commutative Polynomial Optimization. In: Anjos, M.F., Lasserre, J.B. (eds) Handbook on Semidefinite, Conic and Polynomial Optimization. International Series in Operations Research & Management Science, vol 166. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0769-0_21

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