Random Access Codes and a

Avraham Ben-Aroya

(Tel Aviv University)

Oded Regev

(Tel Aviv University)

Ronald de Wolf

(CWI, Amsterdam)

Random Access Codes and a

Hypercontractive Inequality for

Matrix-Valued Functions

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)

•

Proving this is easy, both in the classical and quantum cases

1 0 ? ? ? ? ? ? ? ? ? ? ? ? ? ?

n

n/8

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)

•

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 (2^n/8)

1

n/8

0 ? ? ? ? ? ? ? ? ? ? ? ? ? ?

n

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

• For any two vectors a,b R

^d

,

• Or equivalently,

a

b

a b

The Parallelogram Law

•

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R^d 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 p- norms) [Tomczak-Jaegermann74, BallCarlenLieb94]

Prelims: Fourier Transform

•

Let f be a function from {0,1}ⁿ to R^d (or ℂ^d×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 Bonami- Beckner 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 function f:{0,1}ⁿℂ^2n/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}

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.

Further Applications

Direct product theorem for one-way quantum communication complexity

•

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%.

Alice Bob

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,…,A_k for Alice and k subsets B₁,…,B_k for Bob, and Bob is

supposed to tell for each i whether A_i is disjoint from B_i

•

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

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 x_i 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

• Find other applications of the inequality

• Compare this inequality to entropy-based

techniques