# Meeting notes 11 04 06

### Announcements

Journal club is at 12 on Thursdays in CSE 503. Contact Lukas (svecl@uw) for more info.

Random matrix theory at Tuesdays in 3pm in CSE 624.

Kamil and Isaac are speaking for the next two weeks.

### Status updates

Lukas is looking at the paper Toric codes and quantum doubles from two-body Hamiltonians and will talk about it eventually at the journal club.

Kevin: In security class, we discussed the difficulty of anonymous communication. Does quantum communication help with this? maybe

Is thinking about quantum information and communication complexity.

Greg: Submitted a draft of his thesis!

Aram: Discussions with Patrick Hayden

- Bipartite state <math>\mathbf C^{d_A} \otimes \mathbf C^{d_B} \ni |\psi\rangle = \sum A_{ij} |i\rangle \otimes |j\rangle</math>. Singular values of <math>A</math> are Schmidt coefficients <math>\lambda_1 \geq \dotsb \geq \lambda_{d_A} \geq 0</math> whose squares sum to <math>1</math>. Their spread-out-ness determines how entangled <math>|\psi\rangle</math> is.

- What is the least entangled state in a subspace <math>V \subset \mathbf C^{d_A} \otimes \mathbf C^{d_B}</math>? Can we find <math>V</math> of large dimension such that every state in <math>V</math> is highly entangled (measured by <math>\frac{\sum \sqrt{\lambda_i}}{\sqrt{d_A}}</math>

- Choosing <math>V</math> to be the span of a random set of <math>\sim d_A d_B \epsilon^2</math> states gives a set whose smallest entanglement is <math>1-\epsilon</math> with high probability. This is just a reformulation of Dvoretzky's theorem.

- Can we derandomize this construction? i.e., get similar bounds on entanglement, but use much less randomness (or none at all) in picking the subspace? Yes, if we use <math>d_B \sim d_A \log d_A</math>, and then we get <math>\dim V \sim d_A</math>.

- This is connected to low distortion metric embeddings <math>\ell_2 \to \ell_1(\ell_2)</math>. Here, <math>\| X \|_{\ell_2}^2 = \sum |X_i|^2</math> and <math>\| Y \|_{\ell_1(\ell_2)} = \sum_i \|Y_i\|_{\ell_2}</math>.

- The Schatten <math>p</math>-norm is <math>\| A \|_{S_p} = \|\operatorname{singular\ values}(A)\|_{\ell_p}</math>. The above is equivalent to asking for a <math>S_2 \mapsto S_1</math> embedding.

- These are connected to uncertainty relations (look up Hayden's talk)

Johnny: Criteria for new QECCs? Should have good encoding/decoding circuits, should protect against some reasonable set of errors, etc.