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.