On the Dimension of

Subspaces with

Bounded Schmidt

RankToby Cubitt, Ashley Montanaro, Andreas Winterand also Aram Harrow, Debbie Leung

(who says there's no blackboard at AQIS?!)

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?

Previously...

ANSWER: ~dA dB

(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 dA dB bipartite system such that every state in S has entropy of entanglement “close” to maximum?

The Question

What is the maximum dimension of a subspace S in a dA dB 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

Upper bound: proof outline

reminder: dA dB 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.

(1) Characterize Schmidt rank < r

states

iff allorder–r matrix minors = 0

order–3 minor

Solutions to set of simultaneouspolynomials:

reminder: dA dB bipartite system, subspace S, min Schmidt rank r

dA dB matrix

(2) Calculate dimension “Variety” = space of solutions of set of simultaneous

polynomial equations

Variety defined by order– r minors of a dA dB matrix: “determinantal variety”

Oh look! That's exactly what we have :-)

reminder: dA dB bipartite system, subspace S, min Schmidt rank r

Raid algebraic geometry literature . . .

(3) Dimension counting argument Intersection Lemma: if V and W are projective varieties

in Pd such that , then

dA dB bipartite space of (unnormalized) states:

Projective variety of low Schmidt-rank states to avoid:

Subspace S (= linear projective variety):

QEDreminder: dA dB bipartite system, subspace S, min Schmidt rank r

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 anylength l such that any linear combinationof them contains at most n–1 zeroelements (proof: pick them fromcolumns of an l l totallynon-singular matrix).

Lower bound: construction (1)

Label diagonals of dA dB state matrix

|k| = length of kth diag.

totally non-singular

reminder: dA dB bipartite system, subspace S, min Schmidt rank r

Pick |k| – r + 1 length |k| vectors:lin. comb. r non-zero elements

Lower bound: construction (2)

Linear combination of Sk has non-zero order– r minor → rank r

Any linear combination of S has an lower-triangular rr submatrix with non-zero elements on its main diagonal→ non-zero order– r minor→ rank r

QEDreminder: dA dB bipartite system, subspace S, min Schmidt rank r

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

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: )

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.

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

Product channel without full output

rank

Product state in orthogonal

complement

Vanishes if and have orthogonal

support

p=0 counterexample: construction (1)Wanted: supported on orthogonal subspaces whose orthogonal complements contain no product states.

Use 22 and 33 QFT matrices to construct two orthogonal subspaces with dA = 4, dB = 3, r = 2.

totally non-singular and unitary

Take Choi-Jamiołkowski states tobe projectors onto these subspaces.

Simplify by taking supports of to be orthogonal complements, both containing no product states.

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.

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!