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Limitations of Semidefinite Programs for Separable States and Entangled Games

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Abstract

Semidefinite programs (SDPs) are a framework for exact or approximate optimization with widespread application in quantum information theory. We introduce a new method for using reductions to construct integrality gaps for SDPs, meaning instances where the SDP value is far from the true optimum. These are based on new limitations on the sum-of-squares (SoS) hierarchy in approximating two particularly important sets in quantum information theory, where previously no \({\omega(1)}\) -round integrality gaps were known:

  1. 1.

    The set of separable (i.e. unentangled) states, or equivalently, the \({2 \rightarrow 4}\) norm of a matrix.

  2. 2.

    The set of quantum correlations; i.e. conditional probability distributions achievable with local measurements on a shared entangled state.

Integrality gaps for the \({2\rightarrow 4}\) norm had previously been sought due to its connection to Small-Set Expansion (SSE) and Unique Games (UG).

In both cases no-go theorems were previously known based on computational assumptions such as the Exponential Time Hypothesis (ETH) which asserts that 3-SAT requires exponential time to solve. Our unconditional results achieve the same parameters as all of these previous results (for separable states) or as some of the previous results (for quantum correlations). In some cases we can make use of the framework of Lee–Raghavendra–Steurer (LRS) to establish integrality gaps for any SDP extended formulation, not only the SoS hierarchy. Our hardness result on separable states also yields a dimension lower bound of approximate disentanglers, answering a question of Watrous and Aaronson et al.

These results can be viewed as limitations on the monogamy principle, the PPT test, the ability of Tsirelson-type bounds to restrict quantum correlations, as well as the SDP hierarchies of Doherty–Parrilo–Spedalieri, Navascues–Pironio–Acin, and Berta–Fawzi–Scholz. Indeed a wide range of past work in quantum information can be described as using an SDP on one of the above two problems and our results put broad limits on these lines of argument.

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Acknowledgement

AWH and AN were funded by NSF Grant CCF-1629809 and AWH was funded by NSF grant CCF-1452616. XW was funded by the NSF Waterman Award of Scott Aaronson. All three authors (AWH, AN, and XW) were funded by ARO contract W911NF-12-1-0486. We are grateful to an anonymous STOC reviewer for pointing out a mistake in an earlier version.

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  1. Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    Aram W. Harrow

  2. Department of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA, 91125, USA

    Anand Natarajan

  3. Department of Computer Science, Institute for Advanced Computer Studies, and Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD, 20742, USA

    Xiaodi Wu

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Harrow, A.W., Natarajan, A. & Wu, X. Limitations of Semidefinite Programs for Separable States and Entangled Games. Commun. Math. Phys. 366, 423–468 (2019). https://doi.org/10.1007/s00220-019-03382-y

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