1. On the Dimension of Subspaces with Bounded Schmidt Rank Toby Cubitt, Ashley Montanaro, Andreas Winter and also Aram Harrow, Debbie Leung (who says there's no blackboard at AQIS?!)‏

2. On the Dimension of Subspaces with Bounded Schmidt Rank von Neumann entropy Relative entropy of entanglement Concurrence Tangle Entanglement of formation Entanglement cost Localizable entanglement Entanglement of assistance Distillable entanglement Squashable entanglement Squeezing Correlation function Schmidt rank Renyi entropy 2 + 2 = 3?

3. Previously... ANSWER: ~d A d B (almost the entire space!)‏ P. Hayden, D. Leung, A. Winter, “Aspects of Generic Entanglement”, Comm. Math. Phys 265:1, pp. 95–117 (2006)‏ What is the maximum dimension of a subspace S in a d A  d B bipartite system such that every state in S has entropy of entanglement “close” to maximum?

4. The Question What is the maximum dimension of a subspace S in a d A  d B bipartite system such that every state in S has Schmidt rank at least r? T. Cubitt, A. Montanaro, A. Winter “On the dimension of subspaces with bounded Schmidt rank”, arXiv:0706.0705

5. Upper bound: proof outline reminder: d A  d B bipartite system, subspace S, min Schmidt rank r (1) Characterize states with Schmidt rank < r (the ones we don't want in S). (2) Calculate the “dimension” of this set of states. (3) Dimension counting argument to bound largest S that avoids this set.

6. (1) Characterize Schmidt rank < r states iff all order–r matrix minors = 0 order–3 minor Solutions to set of simultaneous polynomials: reminder: d A  d B bipartite system, subspace S, min Schmidt rank r d A  d B matrix

7. (2) Calculate dimension “Variety” = space of solutions of set of simultaneous polynomial equations Variety defined by order– r minors of a d A  d B matrix: “determinantal variety” Oh look! That's exactly what we have :-)‏ reminder: d A  d B bipartite system, subspace S, min Schmidt rank r Raid algebraic geometry literature...

8. (3) Dimension counting argument Intersection Lemma: if V and W are projective varieties in P d such that, then d A  d B bipartite space of (unnormalized) states: Projective variety of low Schmidt-rank states to avoid: Subspace S (= linear projective variety): QED reminder: d A  d B bipartite system, subspace S, min Schmidt rank r

9. Lower bound: preliminaries Definition: a “totally non-singular” matrix has only non- zero minors. Lemma: there exist totally non-singular matrices of any size (proof: Vandermonde matrices; random matrices). Lemma: there exist sets of n vectors of any length l such that any linear combination of them contains at most n–1 zero elements (proof: pick them from columns of an l  l totally non-singular matrix).

10. Lower bound: construction (1)‏ Label diagonals of d A  d B state matrix |k| = length of k th diag. totally non-singular reminder: d A  d B bipartite system, subspace S, min Schmidt rank r Pick |k| – r + 1 length |k| vectors: lin. comb. r non-zero elements

11. Lower bound: construction (2)‏ Linear combination of S k has non-zero order– r minor → rank r Any linear combination of S has an lower-triangular r  r submatrix with non-zero elements on its main diagonal → non-zero order– r minor → rank r QED reminder: d A  d B bipartite system, subspace S, min Schmidt rank r

12. Additivity: 2 + 2 = 3? Does quantum information do bulk discounts? Entanglement of formation: can two copies of a state be created from less than twice the entanglement required for a single copy? Channel capacity: can two copies of a quantum channel transmit information at more than twice the rate of a single copy? Additivity of minimum output entropy: for p = 1

13. Minimum output Renyi p-entropy Can't solve additivity for interesting case p=1 (simply not clever enough...yet!). Try to solve it for other values of p: Until recently, known to be non-additive for p > ~4.72... Very recent progress, now known to be non-additive for p > 2, 1 < p 2 – go to Andreas' talk! Final frontier: p < 1. (recall Renyi entropy: )‏

14. p=0 counterexample Idea: Pick two channels with full output rank, but arrange for “conspiracy” in product channel, leading to cancellation and non-full output rank.

15. Channels with full output rank Output is full rank for all inputs Choi-Jamiołkowski state has no product vectors in orthogonal complement of its support

16. Product channel without full output rank Product state in orthogonal complement Vanishes if  and have orthogonal support

17. p=0 counterexample: construction (1)‏ Wanted:   supported on orthogonal subspaces whose orthogonal complements contain no product states. Use 2  2 and 3  3 QFT matrices to construct two orthogonal subspaces with d A = 4, d B = 3, r = 2. totally non-singular and unitary Take Choi-Jamiołkowski states to be projectors onto these subspaces. Simplify by taking supports of to be orthogonal complements, both containing no product states.

18. p=0 counterexample: construction (2)‏ Supplement construction with maximally entangled states in corners, to ensure orthogonal complement contains no product states. Argument by lower- triangular submatrix no longer works, but turns out subspaces still contain no product states.

19. Conclusions Question of dimension of subspaces with lower- bounded Schmidt-rank fully solved. Also solved question of dimensions of subspaces with upper-bounded Schmidt-rank (not discussed here; interestingly, question of subspaces containing only states with Schmidt-rank = r is not solved in general...)‏ Applied construction to give counter- example to additivity conjecture for p = 0, and by continuity for small p (numerically p < ~0.1). and violated AQIS presentation guidelines by using a blackboard!