Avraham Ben-Aroya (Tel Aviv University) Oded Regev (Tel Aviv University) Ronald de Wolf

1. 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)

2. 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) Outline

3. Random Access Codes

4. n/8 1 0 ? ? ? ? ? ? ? ? ? ? ? ? ? ? n 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

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. n/8 1 0 ? ? ? ? ? ? ? ? ? ? ? ? ? ? n 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)

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=2n1/32582657[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