Abstract

Consider the n! different unitaries that permute n d-dimensional quantum systems. If d>=n, then these are linearly independent. In this talk, I’ll explain a sense in which they are approximately orthogonal if d >> n^2. This simple fact turns out to make life much easier when working with multipartite quantum states that are invariant under collective unitary rotation. After describing the basic idea, I’ll discuss some subset of the following five applications:

1. There is no efficient product test (in the sense of my previous work with Ashley Montanaro) that uses only LOCC measurements between the different copies of the state to be tested.

2. Random maximally entangled states have similar moments to fully random states.

3. Random quantum circuits on n qubits with poly(n) gates are approximate poly(n)-designs. (Joint work with Fernando Brandao and Michal Horodecki).

4. An alternate proof of the Hastings result that random unitaries give quantum expanders.

5. The N-party data-hiding scheme of Eggeling and Werner can be achieved with only poly(N) local dimension.