This is very technical and nerdy, but I find it really cool.
Recently, Carlen and Lieb proved a very nice inequality for entanglement. Check out
by Eric A. Carlen, Elliott H. Lieb
Let me try to summarize their main result.
Consider a bipartite density matrix , 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.
One can show that
so classically, conditional spreads must be positive (or zero). However, in quantum mechanics such spreads can be negative. The Araki-Lieb inequality says that
Now, having a negative conditional spread is a bit of a dog, because spreads are not supposed to be negative.
The new Carlen/Lieb inequality teaches us that dogs can count too. If is entanglement (either entanglement of formation or squashed entanglement), then, according to Carlen/Lieb,
So a dog (i.e, a negative conditional spread ) forces the entanglement to be greater than zero by . An example of a highly influential dog.
P.S. I recently wrote some email to Profs. Lieb and Carlen asking them something about their paper. I felt infinitely dumb in their email presence, like a mouse in the presence of lions (As you probably know, The Lion and the Mouse is a beautiful fable by Aesop.) But I believe I successfully muddled and bluffed my way through the conversation. Amazing but true, it is possible to bluff a pride of lions. If you don’t believe me, watch this amazing YouTube video, entitled “BBC – Men stealing meat from lions”. In the video, 3 extremely courageous and also extremely foolish African guys, with scant weapons other than their daunting audacity, intentionally walk right into the middle of a pride of lions that is having dinner.