16,259 research outputs found
Schur-Weyl Duality for the Clifford Group with Applications: Property Testing, a Robust Hudson Theorem, and de Finetti Representations
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 tensor
powers of all unitaries is spanned by the permutations of the 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:
(1) 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.
(2) We find that tensor powers of stabilizer states have an increased
symmetry group. 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).
(3) 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.
(4) 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.Comment: 60 pages, 2 figure
Dissolution of nucleons in giant nuclei
We discuss the possibility that nuclei with very large baryon numbers can exist in the form of large quark blobs in their ground states. A calculation based on the picture of quark bags shows that, in principle, the appearance of such exotic nuclear states in present laboratory experiments cannot be excluded. Some speculations in connection with the recently observed anomalous positron production in heavy-ion experiments are presented
Mathematics and the Internet: A Source of Enormous Confusion and Great Potential
Graph theory models the Internet mathematically, and a number of plausible mathematically intersecting network models for the Internet have been developed and studied. Simultaneously, Internet researchers have developed methodology to use real data to validate, or invalidate, proposed Internet models. The authors look at these parallel developments, particularly as they apply to scale-free network models of the preferential attachment type
- …