# Graduate Seminar Series

**
Tue., Oct. 24, 2023 11:30 a.m.
**

**
Location: ED 438
**

**Speaker**: Shane Crerar

**Title:** Probability of Entanglement (91 kB)

**Abstract:**

For Hilbert spaces *H* and *K*, with 2 ≤ dim *H* ≤ dim *K* < ∞, the set of isometries from *H* to the direct sum of *r* copies of *K*, denoted *V ^{r}*(

*H*;

*K*), can be associated with the set of extensions of a faithful state ω of

*B*(

*H*) to states of

*B*(

*K*⊗

*H*) that have rank at most

*r*, denoted

*E*

^{ r}(ω). More precisely, there is a natural left action of scalar

*r*×

*r*matrices on

*V*(

^{r}*H*;

*K*) such that the quotient

*V*(

^{r}*H*;

*K*)/

*U*(r) is in bijection with

*E*

^{ r}(ω). This allows probability measures on

*E*

^{ r}(ω) to be constructed from measures on

*V*(

^{r}*H*;

*K*). The construction and properties of

*V*(

^{r}*H*;

*K*) will be discussed, with particular attention given to the efficacy of using geometric methods to measure the probability that a given extension is entangled.