Schur–Weyl Duality for the Clifford Group with Applications: Property Testing, a Robust Hudson Theorem, and de Finetti Representations

Abstract

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 $$U^{\otimes t}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>U</mml:mi> <mml:mrow> <mml:mo>⊗</mml:mo> <mml:mi>t</mml:mi> </mml:mrow> </mml:msup> </mml:math> of all unitaries $$U\in U(d)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>U</mml:mi> <mml:mo>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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 show that 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.

Topics & Concepts