4 research outputs found
Monogamy of highly symmetric states
We study the question of how highly entangled two particles can be when also
entangled in a similar way with other particles on the complete graph for the
case of Werner, isotropic and Brauer states. In order to do so we solve
optimization problems motivated by many-body physics, computational complexity
and quantum cryptography. We formalize our question as a semi-definite program
and then solve this optimization problem analytically, using tools from
representation theory. In particular, we determine the exact maximum values of
the projection to the maximally entangled state and antisymmetric Werner state
possible, solving long-standing open problems. We find these optimal values by
use of SDP duality and representation theory of the symmetric and orthogonal
groups, and the Brauer algebra.Comment: Submitted to QIP202
Gelfand-Tsetlin basis for partially transposed permutations, with applications to quantum information
We study representation theory of the partially transposed permutation matrix
algebra, a matrix representation of the diagrammatic walled Brauer algebra.
This algebra plays a prominent role in mixed Schur-Weyl duality that appears in
various contexts in quantum information. Our main technical result is an
explicit formula for the action of the walled Brauer algebra generators in the
Gelfand-Tsetlin basis. It generalizes the well-known Gelfand-Tsetlin basis for
the symmetric group (also known as Young's orthogonal form or Young-Yamanouchi
basis).
We provide two applications of our result to quantum information. First, we
show how to simplify semidefinite optimization problems over
unitary-equivariant quantum channels by performing a symmetry reduction.
Second, we derive an efficient quantum circuit for implementing the optimal
port-based quantum teleportation protocol, exponentially improving the known
trivial construction. As a consequence, this also exponentially improves the
known lower bound for the amount of entanglement needed to implement unitaries
non-locally.
Both applications require a generalization of quantum Schur transform to
tensors of mixed unitary symmetry. We develop an efficient quantum circuit for
this mixed quantum Schur transform and provide a matrix product state
representation of its basis vectors. For constant local dimension, this yields
an efficient classical algorithm for computing any entry of the mixed quantum
Schur transform unitary
Iterative quantum amplitude estimation
We introduce a variant of Quantum Amplitude Estimation (QAE), called Iterative QAE (IQAE), which does not rely on Quantum Phase Estimation (QPE) but is only based on Grover’s Algorithm, which reduces the required number of qubits and gates. We provide a rigorous analysis of IQAE and prove that it achieves a quadratic speedup up to a double-logarithmic factor compared to classical Monte Carlo simulation with provably small constant overhead. Furthermore, we show with an empirical study that our algorithm outperforms other known QAE variants without QPE, some even by orders of magnitude, i.e., our algorithm requires significantly fewer samples to achieve the same estimation accuracy and confidence level