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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)
Outline
•
•
•
Main result:
• k-out-of-n random access codes
Proof:
1. A new hypercontractive inequality
2. The proof
Other applications of the inequality:
• Direct product theorem for one-way
communication complexity
• A new approach to lower bounds on locally
decodable codes (LDCs)
Random Access Codes
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)
n/8
1 0 ? ? ? ? ? ? ? ? ? ? ? ? ? ?
•
n
Proving this is easy, both in the classical and quantum
cases
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 don’t exist [AmbainisNayakTa-Shma
Vazirani99, Nayak99]
• Quantum entropy behaves a lot like classical
entropy, so same proof applies also for quantum
RAC
•
•
k-out-of-n Random Access Codes
Now assume we wish to recover some arbitrary k
bits of x (say, k=logn)
One would expect the success probability to
behave like 2-(k)
n/8
1 0 ? ? ? ? ? ? ? ? ? ? ? ? ? ?
•
•
n
Entropy-based arguments no longer work!
• For instance, consider the encoding that given
x{0,1}n outputs x with probability 10% and
000…0 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)
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
The New Inequality
The Parallelogram Law
b
•
a
For any two vectors a,bRd,
• Or equivalently,
The Parallelogram Law
b
•
•
a
This was for the 2 norm
What happens in the p norm, for 1p<2?
• The equality no longer holds, take, e.g.,
a=(1,0),b=(0,1) and p=1
• But, we have the following powerful inequality
for all a,bRd and 1p2:
The Extended Parallelogram Law
•
•
This inequality was proven by [Tomczak-Jaegermann74,
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 pnorms) [Tomczak-Jaegermann74, BallCarlenLieb94]
Prelims: Fourier Transform
•
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Let f be a function from {0,1}n to Rd (or ℂd×d)
Then we define its Fourier transform as
•
So, e.g.,
The New Hypercontractive Ineq.
•
Thm: For any vector- or matrix-valued f on {0,1}n
and 1p2,
•
Remark: This is the extension of the BonamiBeckner inequality to vector/matrix-valued
functions
The New Hypercontractive Ineq.
•
Thm: For any vector- or matrix-valued f on {0,1}n
and 1p2,
•
Proof: By induction on n.
• The case n=1 is exactly the [BCL94] inequality
with a=f(0), b=f(1)
•
For simplicity, let’s see how to get the n=2 case.
• This involves four matrices, a=f(00), b=f(01),
c=f(10), d=f(11)
The New Inequality (cont.)
•
Using the induction hypothesis (case n=1) we get
•
By averaging the two inequalities, we get
The New Inequality (cont.)
•
Using the case n=1, the left side is at least
Proof of the
Main Theorem
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, let’s prove the case k=1
• k>1 case is similar
• So assume by contradiction
that there exists a
n/8×2n/8
n
2
function f:{0,1} ℂ
mapping each x{0,1}n
to a density matrix on n/8 qubits, with the
property that for all i{1,…,n}
Proof
•
Let us apply the inequality to f
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Since f(x) is a density matrix, we have
therefore the RHS is at most 1, and we obtain
•
Choosing p=1+4/n yields a contradiction.
Further Applications
Direct product theorem for one-way
quantum communication complexity
•
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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 <60%.
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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•
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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{0,1}n, i{1,…,n}, and y{0,1}N that differs
from f(x) in at most 0.01N locations, we can recover xi by
querying only q bits in y
For q=2:
• The Hadamard code is a LDC with N=2n
• This is essentially optimal due to [Kerenidis-deWolf02]
• Their proof uses quantum arguments
• We can give an alternative proof using the
hypercontractive inequality
For q=3:
• Best known code uses N=2n1/32582657 [Yekhanin07]
• Almost no lower bounds are known; a huge open question !
Open Questions
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Find other applications of the inequality
Compare this inequality to entropy-based
techniques