Avraham BenAroya

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Random Access Codes and a Hypercontractive
Inequality for Matrix-Valued Functions
Avraham Ben-Aroya (Tel Aviv University) Oded
Regev (Tel Aviv University) Ronald de
Wolf (CWI, Amsterdam)
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Outline
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Random Access Codes
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Squeezing Information?

• Assume we are trying to store n (random) bits
  into n/8 bits or qubits
• Recovering all the n original bits is clearly
  impossible
• The best success probability is obtained by
  storing, say, the first n/8 bits and is only
  2-?(n)
• Proving this is easy, both in the classical and
  quantum cases

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Random Access Codes

• But assume we wish to recover only 1 bit of the
  original n bits with good probability. Such a
  primitive is called a random access code (RAC).
• Seems clearly impossible classically
• Not so clear what happens quantumly
• Using entropy-based arguments one can show that
  RACs dont exist AmbainisNayakTa-Shma
  Vazirani99, Nayak99
• Quantum entropy behaves a lot like classical
  entropy, so same proof applies also for quantum
  RAC

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k-out-of-n Random Access Codes

• Now assume we wish to recover some arbitrary k
  bits of x (say, klogn)
• One would expect the success probability to
  behave like 2-?(k)
• Entropy-based arguments no longer work!
• For instance, consider the encoding that given
  x?0,1n outputs x with probability 10 and 0000
  with probability 90. Then it has low entropy
  (roughly 0.1n) yet we can recover all of x
  prefectly with probability 10
• We therefore have to use the fact that the
  dimension of the encoding is low (2n/8)

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Main Result

• Thm For any k-out-of-n quantum RAC on n/8
  qubits, the success probability is 2-?(k).
• Remarks
• The classical case can be proven by combinatorial
  arguments
• See also this Friday for a related result by
  Koenig and Renner

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The New Inequality
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The Parallelogram Law
ab
a-b
b
a

• For any two vectors a,b?Rd,
• Or equivalently,

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The Parallelogram Law
ab
a-b
b
a

• This was for the ??2 norm
• What happens in the ?p norm, for 1?plt2?
• The equality no longer holds, take, e.g.,
  a(1,0),b(0,1) and p1
• But, we have the following powerful inequality
  for all a,b?Rd and 1?p?2

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The Extended Parallelogram Law

• This inequality was proven by Tomczak-Jaegermann7
  4, BallCarlenLieb94
• Originally used to prove the sharp uniform
  convexity of ?p spaces
• Implies the Bonami-Beckner hypercontractive
  inequality
• An extremely useful inequality in computer
  science (analysis of Boolean functions, hardness
  of approximation, learning theory, communication
  complexity, percolation, etc.)
• Recently used by LeeNaor04 to prove a lower
  bound on the distortion of embeddings into ?1
  spaces
• Amazingly, the same inequality also holds with
  a,b being matrices and norms being matrix p-norms
  (i.e., Schatten p-norms) Tomczak-Jaegermann74,
  BallCarlenLieb94

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Prelims Fourier Transform

• Let f be a function from 0,1n to Rd (or Cdd)
• Then we define its Fourier transform as
• So, e.g.,

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The New Hypercontractive Ineq.

• Thm For any vector- or matrix-valued f on 0,1n
  and 1?p?2,
• Remark This is the extension of the
  Bonami-Beckner inequality to vector/matrix-valued
  functions

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The New Hypercontractive Ineq.

• Thm For any vector- or matrix-valued f on 0,1n
  and 1?p?2,
• Proof By induction on n.
• The case n1 is exactly the BCL94 inequality
  with af(0), bf(1)
• For simplicity, lets see how to get the n2
  case.
• This involves four matrices, af(00), bf(01),
  cf(10), df(11)

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The New Inequality (cont.)

• Using the induction hypothesis (case n1) we get

• By averaging the two inequalities, we get

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The New Inequality (cont.)

• Using the case n1, the left side is at least

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Proof of the Main Theorem
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Main Theorem (again)

• Thm For any k-out-of-n quantum RAC on n/8
  qubits, the success probability is 2-?(k).
• Proof
• For simplicity, lets prove the case k1
• kgt1 case is similar
• So assume by contradiction that there exists a
  function f0,1n?C2n/82n/8 mapping each
  x?0,1n to a density matrix on n/8 qubits, with
  the property that for all i?1,,n

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Proof

• Let us apply the inequality to f
• Since f(x) is a density matrix, we have
• therefore the RHS is at most 1, and we obtain
• Choosing p14/n yields a contradiction.

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Further Applications
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Direct product theorem for one-way quantum
communication complexity
Alice
Bob

• Consider the Disjointness problem
• Alice and Bob are each given a subset of 1,,n
  and need to decide whether their subsets are
  disjoint
• Only one message from Alice to Bob is allowed
• A naïve protocol requires n bits (Alice just
  sends her subset)
• This is essentially optimal (even quantumly)
• In other words, if Alice sends only, say, n/8
  (qu)bits, then their success probability is
  necessarily lt60.

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Direct product theorem for one-way quantum
communication complexity

• Assume now that Alice and Bob try to solve k
  independent instances of the problem
• So input consists of k subsets A1,,Ak for Alice
  and k subsets B1,,Bk for Bob, and Bob is
  supposed to tell for each i whether Ai is
  disjoint from Bi
• Clearly kn bits from Alice to Bob are enough
• We show that if Alice sends less than kn/8
  (qu)bits, then their success probability is
  2-?(k)
• Such a result is known as a direct product
  theorem

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Lower Bounds on Locally Decodable Codes

• A q-query locally decodable code (LDC) is a
  mapping f from n bits into N bits with the
  property that
• For any x?0,1n, i?1,,n, and y?0,1N that
  differs from f(x) in at most 0.01N locations, we
  can recover xi by querying only q bits in y
• For q2
• The Hadamard code is a LDC with N2n
• This is essentially optimal due to
  Kerenidis-deWolf02
• Their proof uses quantum arguments
• We can give an alternative proof using the
  hypercontractive inequality
• For q3
• Best known code uses N2n1/32582657 Yekhanin07
• Almost no lower bounds are known a huge open
  question !

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Open Questions

• Find other applications of the inequality
• Compare this inequality to entropy-based
  techniques