Check out this great blog post in Richard Lipton’s and Ken Reagan’s wonderful blog (much better than this one). The post is about Jim Simons, an accomplished mathematician turned hedge fund manager and philanthropist. Simons had already endowed a math (and string theory) institute in Stony Brook. Now he is endowing a new institute at U.C Berkeley, one dedicated to computer science.
This is important for quantum computing because computer scientists cannot escape the lure of quantum mechanics. Can a moth resist flying towards the flame of a candle? Can a comet escape the gravitational pull of the Sun? Can flabby humans fend off the Borg?
For me, quantum computing is the byproduct that arises from the collision of two gigantic fields, computer science and quantum physics.
According to the Wikipedia article on him, Jim Simons has done a lot of good deeds with his money, including making donations to organizations concerned with health care and elementary education. It’s interesting that his string theory institute is on the east coast and his new computer science institute will be on the west coast. If he is planning a third science institute, I hope he goes for the geographical center: a quantum computing institute in Texas? (Contrary to what many people believe, not all the inhabitants of Texas are football addled, barbecue sauce smeared, gun-toting, bible thumping, mutant apes. I have a good friend in Austin that has never played American football in his life, although he does show a disturbing affinity for the barbie).
, I hope he considers starting a quantum computer company too! Or maybe he can guide or inspire others to start a Quantum SpaceX company. Quantum Space, an untapped, wild frontier.
SIT =Shannon Information Theory
denote the classical entropy of a probability distribution of the random variables 
.
. For any density matrix 
that acts on 
, let 
be its von Neumann entropy.
is a ket (pure quantum state) with density matrix given by ![\rho_{\underline{a}_1,\underline{a}_2, \ldots, \underline{a}_N} = [|\psi \rangle_{\underline{a}_1,\underline{a}_2, \ldots, \underline{a}_N}][h.c.]](http://s0.wp.com/latex.php?latex=%5Crho_%7B%5Cunderline%7Ba%7D_1%2C%5Cunderline%7Ba%7D_2%2C+%5Cldots%2C+%5Cunderline%7Ba%7D_N%7D+%3D+%5B%7C%5Cpsi+%5Crangle_%7B%5Cunderline%7Ba%7D_1%2C%5Cunderline%7Ba%7D_2%2C+%5Cldots%2C+%5Cunderline%7Ba%7D_N%7D%5D%5Bh.c.%5D&bg=ffffff&fg=000000&s=0 "\rho_{\underline{a}_1,\underline{a}_2, \ldots, \underline{a}_N} = [|\psi \rangle_{\underline{a}_1,\underline{a}_2, \ldots, \underline{a}_N}][h.c.]")
. Then it is well known (a consequence of the Schmidt decomposition or of the Araki-Lieb Inequality) that for this pure state
and 
are disjoint sets whose union equals 
, and where 
and 
. For example, if 
, then we have
and permutations
and permutations
th roots of unity. In other words, we are now setting
for 
. 
.
are two disjoint subsets of 
. Then
, and define 
and the same with 1 and 2 swapped. Suppose 
is the von Neumann entropy of 12, 
is that of 1 and 
is that of 2. Call the classical counterparts of these entropies 
, 
and 
. Call 
(respectively, 
) the quantum (resp., classical) conditional spread (or conditional variance) of 2 given 1. 
,
says that 
,
.
is entanglement (either entanglement of formation or squashed entanglement), then, according to Carlen/Lieb, 
) forces the entanglement to be greater than zero by 
. An example of a highly influential dog.
, is a real valued function with 3 slots filled by the random variables 
. (In what follows, we will indicate random variables by underlining.) In 
are the two parties that are having trouble understanding each other and 
is the mediator. 
by
,
) by
,
,
.
, 
and 
can all be expressed in terms of the CMI. Indeed,
.
. Basically, this identity is saying to me that the spread of 
) plus a conditional spread (of 
