Quantum Information Theory · 1. N collaborating players sitting in a room 2. 2 of them selected at...
Transcript of Quantum Information Theory · 1. N collaborating players sitting in a room 2. 2 of them selected at...
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)?
Sunday, September 4, 2011
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
Sunday, September 4, 2011
Why is the classical treatment of information insufficient?
Sunday, September 4, 2011
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.
Sunday, September 4, 2011
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
Sunday, September 4, 2011
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)
Sunday, September 4, 2011
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)
Sunday, September 4, 2011
Data Compression
• N: number of bits required to store M
• Coding theorem: N = H(M)
storage device
X Xencoding decodingM MN bits
Sunday, September 4, 2011
Channel Coding
• “capacity” c defined as maximum coding rate
• c = maxPx I(X:Y)
noisy channel
PY|X
X Yencoding decodingM M
Sunday, September 4, 2011
Why is Classical Information Theory Incomplete?
• Example 1: Analysis of a classical game
• Example 2: Cryptographic impossibility proof
Sunday, September 4, 2011
Toy Example
Sunday, September 4, 2011
Toy Example
1. N collaborating players sitting in a room
Sunday, September 4, 2011
Toy Example
1. N collaborating players sitting in a room
2. 2 of them selected at random and put in separated rooms
Sunday, September 4, 2011
Toy Example
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 their choice
C
Sunday, September 4, 2011
Toy Example
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 their choice
4. separated players output bits B1 and B2
B1
B2
C
Sunday, September 4, 2011
Toy Example
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 their choice
4. separated players output bits B1 and B2
Game is won if B1 ≠ B2.
B1
B2
C
Sunday, September 4, 2011
Maximum Winning Probability
Strategies B=0 B=1 B=C B=1-C
Sunday, September 4, 2011
Maximum Winning Probability
• 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.)
Strategies B=0 B=1 B=C B=1-C
Sunday, September 4, 2011
Maximum Winning Probability
• 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.
Strategies B=0 B=1 B=C B=1-C
Sunday, September 4, 2011
Maximum Winning Probability
• 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).
Strategies B=0 B=1 B=C B=1-C
Sunday, September 4, 2011
Maximum Winning Probability
• 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).
• Hence, the game is lost with probability (at least) 1/4.
Strategies B=0 B=1 B=C B=1-C
Sunday, September 4, 2011
What Did We Prove?
Claim
For any possible strategy, the game is lost with probability at least ≈1/4.
Sunday, September 4, 2011
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.
Sunday, September 4, 2011
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.
Sunday, September 4, 2011
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
Sunday, September 4, 2011
Quantum Strategy
Sunday, September 4, 2011
Quantum Strategy
1. N players start with correlated state Ψ =|0〉⊗N +|1〉⊗N
Sunday, September 4, 2011
Quantum Strategy
1. N players start with correlated state Ψ =|0〉⊗N +|1〉⊗N
2. keep state stored
Sunday, September 4, 2011
Quantum Strategy
1. N players start with correlated state Ψ =|0〉⊗N +|1〉⊗N
2. keep state stored3. all remaining players measure in diagonal basis and choose
C as the xor of their measurement results
C
Sunday, September 4, 2011
Quantum Strategy
1. N players start with correlated state Ψ =|0〉⊗N +|1〉⊗N
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
Sunday, September 4, 2011
What Can We Learn From This Example?
Sunday, September 4, 2011
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. )
Sunday, September 4, 2011
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.
Sunday, September 4, 2011
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.
It is, in general, unavoidable to take into account quantum effects.
Sunday, September 4, 2011
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)
Sunday, September 4, 2011
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.
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.
Question:
• What is wrong with the proof?
Sunday, September 4, 2011
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 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.
Sunday, September 4, 2011
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 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.
H(M|SCBob)
Sunday, September 4, 2011
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 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.
H(M|SCBob)
H(M|CEve)
Sunday, September 4, 2011
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 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.
H(M|SCBob)
H(M|CEve)
No cloning: CBob ≠ CEve in general
Sunday, September 4, 2011
Classical Model
XAXA YA YAinput output
XBXB YB YBinput output
Alice
Bob
CA = f(XA)
XB,CA
XA,CB
CB = f’(XB)
g
g’
Sunday, September 4, 2011
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)
Sunday, September 4, 2011
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
Sunday, September 4, 2011
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
Sunday, September 4, 2011
Classical Model
XAXA YAinput output
XBXB YBinput output
Alice
Bob
CA = f(XA)
XB,CA
XA,CB
CB = f’(XB)
g
g’
Sunday, September 4, 2011
Quantum Model
ρAXA YAinput output
ρBXB YBinput output
Alice
Bob
ρC = TPCPM(ρA)
ρBC
ρAC’
ρC’ = TPCPM(ρB)
TPCPM
TPCPM
Sunday, September 4, 2011
Channel Coding
• Under which conditions can we guarantee that ρ = ρ’?
noisy channelor memory
TPCPMσ σ’encoding decodingρ ρ’
Sunday, September 4, 2011
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
Sunday, September 4, 2011
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)
Sunday, September 4, 2011
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
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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
Sunday, September 4, 2011
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
Sunday, September 4, 2011
Operational Interpretation 2(Thermodynamics)
erasure
Amount of work required: W = kBT H(A|B)
initially: Bρ |0>processing
B
A A
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No-Cloning
ρXXρX
preparationcloning
ρX
C1
C2
Sunday, September 4, 2011
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
Sunday, September 4, 2011
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)
Sunday, September 4, 2011
Questions?
Sunday, September 4, 2011