Quantum Information Theory · 1. N collaborating players sitting in a room 2. 2 of them selected at random and put in separated rooms 3. N-2 remaining players announce a bit C of

Quantum Information TheoryRenato Renner

ETH Zurich

Sunday, September 4, 2011

Information vs Computation Theory

• Computation Theory:

• How much time does it take to decide whether a given number is prime?

• If we are able to break an RSA cryptosystem, can we also factor large numbers?

• Information Theory:

• How many bits can we store in a given physical device?

• How many bits can we transmit through a given channel (e.g., an optical fibre)?


Quantum vs Classical Information Theory

• Classical Information Theory

• Developed 1948 by Claude Shannon

• Mathematical theory; (apparently) treats information independently of its physical representation

• Cannot correctly describe information represented by the state of a quantum system

• Quantum Information Theory

• Overcomes this limitation


Why is the classical treatment of information insufficient?


Classical Treatment of Information

• State of a system is specified by a value X (may be probabilistic, PX)

• We encode information into a system by choosing its state X.

• We retrieve information from a system by observing its state X.

Formally, X is a random variable with a probability distribution PX.


Typical Results in Classical Information Theory

• Information can be measured in terms of entropy, H(X).

• To store a value from a source PX (using optimal compression) H(X) bits are needed.

• Noisy channel coding theorem


Definition of Entropy and Mutual Information I

• X: random variable which takes values x

• s(x): surprisal: s(x) := -log PX(x)

• H(X): Shannon entropy H(X) := E[s(x)] = -∑x PX(x) log PX(x)


Definition of Entropy and Mutual Information II

• X, Y, Z: random variables

• H(X|Y) := H(XY) - H(Y)

• I(X:Y) := H(X) - H(X|Y)

• I(X:Y|Z) := H(X|Z) - H(X|YZ)


Data Compression

• N: number of bits required to store M

• Coding theorem: N = H(M)

storage device

X Xencoding decodingM MN bits


Channel Coding

• “capacity” c defined as maximum coding rate

• c = maxPx I(X:Y)

noisy channel

PY|X

X Yencoding decodingM M


Why is Classical Information Theory Incomplete?

• Example 1: Analysis of a classical game

• Example 2: Cryptographic impossibility proof


Toy Example


Toy Example

1. N collaborating players sitting in a room


Toy Example


2. 2 of them selected at random and put in separated rooms


Toy Example



3. N-2 remaining players announce a bit C of their choice

C


Toy Example




4. separated players output bits B1 and B2

B1

B2

C


Toy Example




4. separated players output bits B1 and B2

Game is won if B1 ≠ B2.

B1

B2

C


Maximum Winning Probability

Strategies B=0 B=1 B=C B=1-C



• Each player may choose one of the following four strategies (in case he is selected).

(The strategy defines how the output B is derived from the input C.)






• The game cannot be won if the two selected players follow identical strategies.







• This happens with probability ≈1/4 (for N large).







• This happens with probability ≈1/4 (for N large).

• Hence, the game is lost with probability (at least) 1/4.



What Did We Prove?

Claim

For any possible strategy, the game is lost with probability at least ≈1/4.


What Did We Prove?

Claim

For any possible strategy, the game is lost with probability at least ≈1/4.

Additional implicit assumption

All information is encoded and processed classically.


Quantum Strategies Are Stronger

The game can be won with probability 1 if the players can use an internal quantum device.

Note: all communication during the game is still purely classical.


Quantum Strategies Are Stronger

The game can be won with probability 1 if the players can use an internal quantum device.

Note: all communication during the game is still purely classical.

B1

B2

C


Quantum Strategy


Quantum Strategy

1. N players start with correlated state Ψ =｜0〉⊗N +｜1〉⊗N


Quantum Strategy


2. keep state stored


Quantum Strategy


2. keep state stored3. all remaining players measure in diagonal basis and choose

C as the xor of their measurement results

C


Quantum Strategy


2. keep state stored3. all remaining players measure in diagonal basis and choose

C as the xor of their measurement results4. separated players determine B1 and B2 by measuring in

either the diagonal or the circular basis, depending on C.

B1

B2

C


What Can We Learn From This Example?



• Quantum mechanics allows us to win games that cannot be won in a classical world (examples known as “pseudo telepathy games”).(Telepathy is obviously dangerous from a cryptographic point of view. )




• There is no physical principle that allows us to rule out quantum strategies.




• There is no physical principle that allows us to rule out quantum strategies.

It is, in general, unavoidable to take into account quantum effects.


Shannon’s “Impossibility Result”

Theorem

For information-theoretically secure encryption, the key S needs to be at least as long as the message M.

In particular, One-Time-Pad encryption is optimal.

C = enc(M,S)

C

M = dec(C,S)


Proof of Shannon’s Theorem

Let M be a uniformly distributed n-bit message, S a secret key, and C the ciphertext.

Requirements

• H(M|SC) = 0, since M is determined by S, C.

• H(M|C) = H(M) = n, since M is indep. of C.

Hence

H(S) ≥ I(M : S|C) = H(M|C) – H(M|SC) = n.Sunday, September 4, 2011

How General Is Shannon’s Result?

Observations:

• Quantum key distribution (QKD) can be used to transmit arbitrarily long messages starting with only a short initial key.

• Hence, Shannon’s impossibility theorem does not apply to our (quantum-mechanical) world.


How General Is Shannon’s Result?

Observations:

• Quantum key distribution (QKD) can be used to transmit arbitrarily long messages starting with only a short initial key.

• Hence, Shannon’s impossibility theorem does not apply to our (quantum-mechanical) world.

Question:

• What is wrong with the proof?



Let M be a uniformly distributed n-bit message, S a secret key, and C the ciphertext..

Requirements

• H(M|SC) = 0, since M determined by S, C.

• H(M|C) = H(M) = n, since M indep. of C.

Hence

H(S) ≥ I(M : S|C) = H(M|C) – H(M|SC) = n.




Requirements



Hence

H(S) ≥ I(M : S|C) = H(M|C) – H(M|SC) = n.

H(M|SCBob)




Requirements



Hence

H(S) ≥ I(M : S|C) = H(M|C) – H(M|SC) = n.

H(M|SCBob)

H(M|CEve)




Requirements



Hence

H(S) ≥ I(M : S|C) = H(M|C) – H(M|SC) = n.

H(M|SCBob)

H(M|CEve)

No cloning: CBob ≠ CEve in general


Classical Model

XAXA YA YAinput output

XBXB YB YBinput output

Alice

Bob

CA = f(XA)

XB,CA

XA,CB

CB = f’(XB)

g

g’


Representation of Information in QIT

• State of a system represented by density operator ρ

• Encode information in system: state preparation (density operator)

• Process information:state evolution (unitary, TP CPM)

• Read information from a system: quantum measurement (basis, POVM)


Classical vs Quantum Information Processing

• Classical information processing: Y = f(X)

• Quantum information processing: ρ’ = CPM(ρX)

XX Y Ypreparation read-outprocessing

ρXX ρ’ Ypreparation read-outprocessing


Quantum Information Processing

• Specified by a TPCPM“Trace Preserving Completely Positive Map”

• Mathematically, a TPCPM is a linear mapping from density operators to density operators

• Physically, a TPCPM describes any possible evolution of the system


Classical Model

XAXA YAinput output

XBXB YBinput output

Alice

Bob

CA = f(XA)

XB,CA

XA,CB

CB = f’(XB)

g

g’


Quantum Model

ρAXA YAinput output

ρBXB YBinput output

Alice

Bob

ρC = TPCPM(ρA)

ρBC

ρAC’

ρC’ = TPCPM(ρB)

TPCPM

TPCPM


Channel Coding

• Under which conditions can we guarantee that ρ = ρ’?

noisy channelor memory

TPCPMσ σ’encoding decodingρ ρ’


Von Neumann Entropy as an Uncertainty Measure

• Definition: H(ρ) := - tr(ρ log ρ)

• Convention: write H(A) instead of H(ρA).

• compare to H(X) = E[s(x)] = -∑x PX(x) log PX(x)

• Shannon entropy is a special case of von Neumann entropy


Conditional Entropy and Mutual Information

• A,B,C: subsystems (joint system in state ρABC)

• H(A|B) := H(AB) - H(B)

• I(A:B) := H(A) - H(A|B)

• I(A:B|C) := H(A|C) - H(A|BC)


Basic Properties

• H(A|B) ≤ log |A|

• H(A|B) ≥ - log |A|

• H(A|B) ≥ 0 if ρAB separable (e.g., A or B classical)

• Data processing: H(A|B) ≤ H(A|B’)

B B’processing

A


More Properties

• H(A|B) = - H(A|E) if ρABE pure

• I(A:E) = 0 if and only if ρAB =ρA ⊗ρB

equivalently: H(A) = H(A|E) if and only if ρAB =ρA ⊗ρB

• I(A:B) ≤ H(A) if ρAB separable


Operational Interpretation 1(Communication Task)

free classical communication

Amount of entanglement needed: E = H(A|B)

initially: A

finally: nothing

initially: B

initially: B

finally: AB


Operational Interpretation 2(Thermodynamics)

erasure

Amount of work required: W = kBT H(A|B)

initially: Bρ |0>processing

B

A A


No-Cloning

ρXXρX

preparationcloning

ρX

C1

C2


Quantum Cryptography

insecure quantum channel

Eve

Alice

Alice

Bob

Assume Alice and Bob exchange a Bell stateΨ = |00> + |11>

Measure them in orthonormal bases rotated by α, β


Quantum Cryptography

insecure quantum channel

Eve

Alice

X Y

α

αβ

β

Pr[X = Y] = Cos2(α-β)

Pr[X ≠ Y] = Sin2(α-β) ≈ (α-β)2 (for small angle differences)


Questions?


Documents

Quantum Information Theory · 1. N collaborating players sitting in a room 2. 2 of them selected at random and put in separated rooms 3. N-2 remaining players announce a bit C of