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

2. Outline Main result: k-out-of-n random access codes Proof:
A new hypercontractive inequality
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)

3. Random Access Codes

4. 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 1 ? n
n/8

5. 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

6. 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)
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)
n/8 1 ? n

7. 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

8. The New Inequality

9. The Parallelogram Law a+b a-b b a For any two vectors a,bRd,
Or equivalently,

10. The Parallelogram Law a+b a-b 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:

11. 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 p- norms) [Tomczak-Jaegermann74, BallCarlenLieb94]

12. Prelims: Fourier Transform
Let f be a function from {0,1}n to Rd (or ℂd×d)
Then we define its Fourier transform as
So, e.g.,

13. 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 Bonami- Beckner inequality to vector/matrix-valued functions

14. 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)

15. The New Inequality (cont.)
Using the induction hypothesis (case n=1) we get
By averaging the two inequalities, we get

16. The New Inequality (cont.)
Using the case n=1, the left side is at least

17. Proof of the
Main Theorem

18. 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 function f:{0,1}nℂ2n/8×2n/8 mapping each x{0,1}n to a density matrix on n/8 qubits, with the property that for all i{1,…,n}

19. 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 p=1+4/n yields a contradiction.

20. Further Applications

21. 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 <60%.

22. 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

23. 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/ [Yekhanin07]
Almost no lower bounds are known; a huge open question !

24. Open Questions Find other applications of the inequality
Compare this inequality to entropy-based techniques