Introduction to Quantum Information Theory

and Applications to Classical Computer Science

Iordanis KerenidisCNRS – Univ Paris

Quantum information and computation

Quantum information and computation

How is information encoded in nature?

What is nature’s computational power?

Quantum algorithm for Factoring [Shor 93]

Information-theoretic secure key distribution [Bennett-Brassard 84]

Quantum computers probably won’t solve NP-complete problems [BBBV94]

Quantum Information

Quantum bit (qubit): Carrier of quantum information

A quantum mechanical system, which can be in a state |0 , |1 or any

convex combination of them.

Physical Examples:

Spin of an electron, Spin of a nuclear, Photon polarization, Two-level

atom, Non-abelian anyon, …

Here, a qubit is an abstract mathematical object, i.e.

a unit vector in a 2D Hilbert space.

Probability Theory – Quantum Information I

Binary Random Var. X:

Random Variable X:

Quantum bit: unit vector in a 2-dim. Hilbert space

Quantum state: on logn qubits

Correlations Entanglement (more general than correlations!!!)

Probability Theory – Quantum Information II

Evolution by Stochastic Matrices

(preserve l1-norm)

Evolution by Unitary Matrices

(preserve l2-norm)

Measurement Measurement (Projective)

A measurement of in an orthonormal basis

is a projection onto the basis

vectors and

Pr[outcome is bi ] =

Density matrix

Mixed state: Classical distribution over pure quantum states

Density matrix: (hermitian, trace 1, positive)

1. contains all information about a measurement in any basis B={b1,…bn}.

Pr[outcome is bi ]

2. Different ensembles can have the same

Entropy of Quantum states

Shannon Entropy: randomness in the measurement

Random variable X, distribution P

Von Neumann Entropy: randomness in the best possible measurement

Mixed quantum state , Density matrix

, where E are the eigenvectors and i the eigenvalues.

Von Neumann Entropy shares many properties with the Shannon Entropy.

Properties of von Neumann Entropy

where contains m qubits.

Conditional Entropy

Mutual information

Strong subadditivity

Fano’s Inequality

Probability Theory - Quantum Information

Random Variable X

Positive probabilities

Quantum state

Complex amplitudes

Correlations Entanglement

(more general than correlations!!!)

Stochastic Matrices & Measurement Unitary Matrices & Measurements

(in different bases)

Probability Distributions Mixed states

(distribution over pure quantum states)

Shannon Entropy Von Neumann Entropy

(similar properties)

Amplitudes vs. Accessible Information

Exponential number of amplitudes

We can encode n bits into log(n) qubits

e.g.

Only indirect access to the information via measurements

No measurement can give us all the bits of x.

Holevo’s bound : n qubits encode at most n bits.

Quantum information does not compress classical information

S( ) n

Quantum vs. Classical Information

001110100011

Input x Input yGoal: Output P(x,y)

( minimum communication )

Quantum communication can be exponentially smaller than

classical communication complexity

Two-way communication [Raz99]

One-way communication [Bar-Yossef, Jayram, K.04, GKKRdW07]

Simultaneous Messages [Bar-Yossef, Jayram, K. 04]

Communication complexity

Quantum Information and Cryptography

Unconditionally Secure Key distribution [Bennett, Brassard 84]

Random number Generators [id Quantique]

Private Information Retrieval [K., deWolf 03, 04]

Message Authentication, Signatures [Barnum et al. 02, Gottesman, Chuang 01]

Quantum One-way functions [Kashefi, K. 05]

001110100011100011

Quantum cryptography in practice

Quantum Information & Classical Computer Science

Classical results via quantum arguments

Lower bound for Locally Decodable Codes and Private Information

Retrieval [K., deWolf 03]

Lattice Problems in NP \ coNP [Aharonov, Regev 04]

Matrix Rigidity [deWolf 05]

Circuit Lower Bounds [K.05]

Lower bounds for Local Search [Aaronson 03]

Locally Decodable Codes

Error Correcting Codes have numerous theoretical applications

Hash functions, expanders, list decoding, hardcore predicates

One interesting property: Local Decodability

Introduced by Katz-Trevisan.

Applications of Locally Decodable Codes

Probabilistically Checkable Proofs (PCP)

Worst-case to Average-case reductions

Private Information Retrieval

Extractors

Hardness Amplification

Polynomial Identity Testing

Locally Decodable Codes (LDC)

encode

fraction errors

a1 aq

x

C(x)

q probes

Decoding Algorithm outputs

xi with prob. >1/2+

A code is a

(q, , )-locally decodable code if one

can recover any bit xi with probability

½+ by q queries, even if fraction of

the codeword is corrupted.

Locally Decodable Codes

↕

“Smooth Codes”

↕

Private Information Retrieval

mnC }1,0{}1,0{:

Exponential Lower Bound for 2-query LDCs

Theorem: A 2-query LDC has length [Kerenidis, deWolf 2003]

eq. A binary 2-server PIR scheme has complexity (n)

)(2 nm

Known Bounds

Servers PIR

k=1 (n)

k=2, binary (n)

k =2 O(n1/3) , (log n)

k > 2 O(n1/loglogn) , (log n)

Exponential Lower Bound for 2-query LDCs

Proof:

Let be a 2-query LDC s.t. one can recover any bit xi

with probability ½+ .

Claim:

From the quantum encoding one can recover any xi with prob. ½+ .

Hence,)(2 nm

mnC }1,0{}1,0{:

Proof of the Claim

Claim:

From the quantum encoding one can

recover any xi with prob. ½+ .

Proof:

1. The Decoder w.l.o.g. is similar the Hadamard Decoder.

Decoder can tolerate a fraction of errors

The distribution of queries must be ”close” to uniform.

The decoder queries a random edge of a matching and XORs

Hadamard Decoding Algorithm:

i) Pick a random r {0,1}n and query (r, r+ei)-th bits

ii) Output the XOR: x • r + x • (r+ei) = x • ei = xi

Prob[correct decoding] >1-2

Quantum Decoding algorithm

Let be the matching for xi.

Classical Decoding Algorithm:

i) Query a random edge

ii) Output:

Prob[correct decoding] > ½+

Quantum Decoding Algorithm:

i) Measure the state

in the basis

ii) Output xi = 0 if the result is

xi = 1 if the result is

Prob[correct decoding] >½+

Proof of Correctness

The decoder uses the state

Measurement in the basis

If then

If then

Conclusions

Quantum Information and Computation

Rich Mathematical Theory

Computational power of nature

Advances in Theoretical Physics

Advances in classical Computer Science

Practical Quantum Cryptography

Advances in Experimental Physics

Why is Quantum Computation

important?