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Gross, David, Nezami, Sepehr and Walter, Michael ORCID: 0000-0002-3073-1408 (2021). Schur-Weyl Duality for the Clifford Group with Applications: Property Testing, a Robust Hudson Theorem, and de Finetti Representations. Commun. Math. Phys., 385 (3). S. 1325 - 1394. NEW YORK: SPRINGER. ISSN 1432-0916

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Abstract

Schur-Weyl duality is a ubiquitous tool in quantum information. At its heart is the statement that the space of operators that commute with the t-fold tensor powers U-circle times t of all unitaries U is an element of U(d) is spanned by the permutations of the t tensor factors. In this work, we describe a similar duality theory for tensor powers of Clifford unitaries. The Clifford group is a central object in many subfields of quantum information, most prominently in the theory of fault-tolerance. The duality theory has a simple and clean description in terms of finite geometries. We demonstrate its effectiveness in several applications: We resolve an open problem in quantum property testing by showing that stabilizerness is efficiently testable: There is a protocol that, given access to six copies of an unknown state, can determine whether it is a stabilizer state, or whether it is far away from the set of stabilizer states. We give a related membership test for the Clifford group. We find that tensor powers of stabilizer states have an increased symmetry group. Conversely, we provide corresponding de Finetti theorems, showing that the reductions of arbitrary states with this symmetry are well-approximated by mixtures of stabilizer tensor powers (in some cases, exponentially well). We showthat the distance of a pure state to the set of stabilizers can be lower-bounded in terms of the sum-negativity of its Wigner function. This gives a new quantitative meaning to the sum-negativity (and the related mana) - a measure relevant to fault-tolerant quantum computation. The result constitutes a robust generalization of the discrete Hudson theorem. We show that complex projective designs of arbitrary order can be obtained from a finite number (independent of the number of qudits) of Clifford orbits. To prove this result, we give explicit formulas for arbitrary moments of random stabilizer states.

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Gross, David UNSPECIFIED UNSPECIFIED UNSPECIFIED Nezami, Sepehr UNSPECIFIED UNSPECIFIED UNSPECIFIED Walter, Michael UNSPECIFIED orcid.org/0000-0002-3073-1408 UNSPECIFIED
URN:	urn:nbn:de:hbz:38-589325
DOI:	10.1007/s00220-021-04118-7
Journal or Publication Title:	Commun. Math. Phys.
Volume:	385
Number:	3
Page Range:	S. 1325 - 1394
Date:	2021
Publisher:	SPRINGER
Place of Publication:	NEW YORK
ISSN:	1432-0916
Language:	English
Faculty:	Unspecified
Divisions:	Unspecified
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language ERROR-CORRECTING CODES; QUANTUM STATES; KRONECKER COEFFICIENTS; FINITE; SYSTEMS Multiple languages Physics, Mathematical Multiple languages
URI:	http://kups.ub.uni-koeln.de/id/eprint/58932