On the Dimension of Subspaces with Bounded Schmidt Rank

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On the Dimension of Subspaces with Bounded
Schmidt Rank
Toby Cubitt, Ashley Montanaro, Andreas Winterand
also Aram Harrow, Debbie Leung
(who says there's no blackboard at AQIS?!)?
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On the Dimension of Subspaces with Bounded
Schmidt Rank
Squeezing
Renyi entropy
Entanglement cost
Tangle
Squashable entanglement
Concurrence
Entanglement of formation
Relative entropy of entanglement
2 2 3?
Schmidt rank
von Neumann entropy
Localizable entanglement
Correlation function
Distillable entanglement
Entanglement of assistance
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Previously...
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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. WinterOn the
dimension of subspaces with bounded Schmidt
rank, arXiv0706.0705
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Upper bound proof outline
(1) Characterize states with Schmidt rank lt 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.
reminder dA ? dB bipartite system, subspace S,
min Schmidt rank r
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(1) Characterize Schmidt rank lt r states
dA ? dB matrix
order3 minor
Solutions to set of simultaneous polynomials
reminder dA ? dB bipartite system, subspace S,
min Schmidt rank r
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(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 -)?
Raid algebraic geometry literature . . .
reminder dA ? dB bipartite system, subspace S,
min Schmidt rank r
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(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)

QED
reminder dA ? dB bipartite system, subspace S,
min Schmidt rank r
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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 n1 zeroelements (proof
  pick them fromcolumns of an l ? l
  totallynon-singular matrix).

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Lower bound construction (1)?
k length of kth diag.
Label diagonals of dA ? dB state matrix
totally non-singular
Pick k r 1 length k vectorslin. comb.
r non-zero elements
reminder dA ? dB bipartite system, subspace S,
min Schmidt rank r
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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 r??r submatrix with non-zero
elements on its main diagonal? non-zero order r
minor? rank r
QED
reminder dA ? dB bipartite system, subspace S,
min Schmidt rank r
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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
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Minimum output Renyi p-entropy

• Can't solve additivity for interesting case
  p1(simply not clever enough...yet!).
• Try to solve it for other values of pUntil
  recently, known to be non-additive for p gt
  4.72... Very recent progress, now known to be
  non-additive for p gt 2, 1 lt p 2 go to
  Andreas' talk!
• Final frontier p lt 1.

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p0 counterexample
IdeaPick two channels with full output rank,
but arrange for conspiracy in product channel,
leading to cancellation and non-full output rank.
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Channels with full output rank
Output is full rank for all inputs
Choi-Jamiolkowski state has no product vectors in
orthogonal complement of its support
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Product channel without full output rank
Product state in orthogonal complement
Vanishes if ?? and have orthogonal support
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p0 counterexample construction (1)?
Wanted ??? ? supported on orthogonal subspaces
whose orthogonal complements contain no product
states.
Simplify by taking supports of to be
orthogonal complements, both containing no
product states.
Use 2??2 and 3??3 QFT matrices to construct two
orthogonal subspaces with dA 4, dB 3, r 2.
Take Choi-Jamiolkowski states tobe projectors
onto these subspaces.
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p0 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.
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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 lt 0.1).

and violated AQIS presentation guidelines by
using a blackboard!