* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download koutofn
Measurement in quantum mechanics wikipedia , lookup
Density matrix wikipedia , lookup
Interpretations of quantum mechanics wikipedia , lookup
Quantum machine learning wikipedia , lookup
Path integral formulation wikipedia , lookup
Quantum computing wikipedia , lookup
Hidden variable theory wikipedia , lookup
Quantum group wikipedia , lookup
EPR paradox wikipedia , lookup
Canonical quantization wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Quantum entanglement wikipedia , lookup
Quantum state wikipedia , lookup
Quantum teleportation wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Quantum key distribution wikipedia , lookup
Bell's theorem wikipedia , lookup
A Hypercontractive Inequality for Matrix-Valued Functions with Applications to Quantum Computing Avraham Ben-Aroya (Tel Aviv University) Oded Regev (Tel Aviv University) Ronald de Wolf (CWI, Amsterdam) Outline • The new hypercontractive inequality • Main application: k-out-of-n random access • codes Another application: direct product theorem for one-way communication complexity The New Inequality The Parallelogram Law b • a For any two vectors a,bRd, • Or equivalently, The Parallelogram Law b • • a This was for the 2 norm What happens in the p norm, for 1p<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,bRd and 1p2: 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) 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., The New Hypercontractive Ineq. • Thm: For any vector- or matrix-valued f on {0,1}n and 1p2, • 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 1p2, • 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 case n=1 we get • By averaging the two inequalities, we get The New Inequality (cont.) • Using the case n=1 again, the left side is at least Application 1: Random Access Codes Compressing Information? • Assume we are trying to store n (random) bits into n/8 bits or qubits • Recovering all of 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. Such a primitive is called a random access code (RAC). • ‘Clearly’ impossible classically… what happens quantumly? More formally: n/8 n 2 • A RAC is a function f:{0,1} R mapping each x{0,1}n to a probability distribution on n/8 bits, with the property that for all i{1,…,n} • Using entropy-based arguments one can show that RACs don’t exist [AmbainisNayakTa-ShmaVazirani99, 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 (think of 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) k-out-of-n Random Access Codes Thm: For any k-out-of-n quantum random access code on n/8 qubits, the success probability is 2-(k). Remark: The classical case can be proven by ‘brute-force’ Proof: • For simplicity, let’s prove the classical k=1 case • The quantum case is identical (using matrices instead of vectors) • k>1 case is similar • Recall that the RAC is described by a function • n/8 n 2 f:{0,1} R Let us apply the inequality to f k-out-of-n Random Access Codes • Since f(x) is a probability distribution, we have • therefore the RHS is at most 1 The LHS is at least k-out-of-n Random Access Codes • By rearranging, we get • Choosing p=1+4/n yields in contradiction. Application 2: Communication Complexity Direct product theorem for one-way quantum communication complexity • • • Alice Bob Consider the Disjointness function: • Alice and Bob are each given a subset of {1,…,n} and need to decide whether their subsets are disjoint A naïve one-way protocol requires n bits of oneway communication (Alice just sends her subset) • This is essentially optimal (even quantumly) We show that if Alice and Bob try to solve k independent instances of the problem with less than kn/2 (qu)bits of one-way communication, then their success probability is 2-(k) Open Questions • Find other applications of the inequality