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Quantum Key Distribution
Citation preview
Matthias ChristandlQuantum Information Theory
Institute for Theoretical Physics
ETH Zurich
Encrypting with Entanglement
Overview
• Entanglement
• Determinism?
• Quantum Cryptography
• A Test for Entanglement
message+ key----------------= cipher
Quantum Mechanics
• Theory of the smallest particles
• Big implications
Stability of matter
Fission and fusion of nuclei
Hawking radiation
�ω✿✿✿✿✿✿
Photon
Entanglement - a Quantum Mechanical Phenomenon
• Quantum mechanical correlations among two or more particles
• „spooky action at a distance“
• „entanglement is not one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought“
Albert Einstein
Erwin Schrödinger
Schrödinger 1932
The claim ... has the strange consequence that the Ψ-function of a system [System I] is changed by the performance of a measurement on a different, far separated system [System II] ...
Schrödinger Archiv, Wien
Schrödinger 1932 Schrödinger Archiv, Wien
This makes it a bit difficult to view the change in the Ψ-function as a Naturvorgang.x)
...
x) the matter becomes even more strange, if we perform a different measurement on [System II] ...
Alice and Bob
• Long distances
• Communication of measurement results
• Particles in the hands of
• Each equipped with a laboratory
Alice Bob
and
Information
• The bit = unit of information
on/off
heads/tails
north pole/ south pole
Information
• random bit
child plays with switch
toss of a coin
travel lottery
Correlated Bits
• Alice gets resultBob the opposite
• Alice heads ⇔ Bob tails
Alice tails ⇔ Bob heads
• Random, but correlated bits
Qubit, the Quantum Bit• Unit of quantum information
• Many possible states (dots on a sphere)
• Example: photon polarisation
. .
!
"
#
!"#
!$#%&%!'#
!(#%&%!)#
!*#!+#
!,#
Qubit, the Quantum Bit• Unit of quantum information
• Many possible states (dots on a sphere)
• Example: photon polarisation
. .
!
"
#
!"#
!$#%&%!'#
!(#%&%!)#
!*#!+#
!,#
Qubit• We cannot measure accurately the state
of the qubit.
• We can only measure, if the state is in one of two antipodal points:
• North or south pole?
• Madrid or Wellington?
• Bangkok or Lima?
Qubit• State: North pole
Measurement: North or south pole?Result: North pole
• State: CopenhagenMeasurement: North or south pole?Result: Nordpol (Cos2 35°/2≈91%)
• State: SingapurMeasurement: North or south pole?Result: North pole (Cos2 90°/2=50%)
•Measurement changes the state
50%
50%
SourceEntangled Qubits
SourceEntangled Qubits
Measurement:North or south pole
SourceEntangled Qubits
Measurement:North or south pole
SourceEntangled Qubits
50%
Measurement:North or south pole
SourceEntangled Qubits
50%
50%
Measurement:North or south pole
SourceEntangled Qubits
50%
50%
Bob‘s state=antipodal pointjust like a coin
Entangled Qubits
Measurement:Madrid or Wellington
Source
Entangled Qubits
50%
Measurement:Madrid or Wellington
Source
Entangled Qubits
50%
50%
Measurement:Madrid or Wellington
Source
Entangled Qubits
50%
50%
Bob‘s state=antipodal pointfor every measurement
„spooky action at a distance“
Measurement:Madrid or Wellington
Source
Übersicht
• Entanglement
• Determinism?
• Quantum Cryptography
• Test for Entanglement
message+ key----------------= cipher
Determinism?
• In classical physics, the measurement result exists before the performance of the measurement (realism)
• Is there an element of reality which determines the measurement result in quantum mechanics?
• No: Measurement results are inherently probabilistic. The world is not deterministic
?„God does not
place dice“Einstein, Podolsky and Rosen (1935)
Bell (1967)
Bell‘s Inequality
• With which probability arethe following satisfied?
• Realistic theory: probability ≤ 75%
• Quantum mechanics: probability ≈ 85%
Source
a=0 or 1
y=0 or 1
b=0 or 1
x=0 or 1
a≠b for x=0, y=0a≠b for x=0, y=1a≠b for x=1, y=0a=b for x=1, y=1
• With which probability arethe following satisfied?
• Realistic theory: probability ≤ 75%
• Quantum mechanics: probability ≈ 85%
Bell‘s Inequality
a≠b for x=0, y=0a≠b for x=0, y=1a≠b for x=1, y=0a=b for x=1, y=1
100%100%100%0%
a=0 or 1
y=0 or 1
b=0 or 1
x=0 or 1
• With which probability arethe following satisfied?
• Realistic theory: probability ≤ 75%
• Quantum mechanics: probability ≈ 85%
a≠b for x=0, y=0a≠b for x=0, y=1a≠b for x=1, y=0a=b for x=1, y=1
Bell‘s Inequality
Source
a=0 or 1
y=0 or 1
b=0 or 1
x=0 or 1
• With which probability arethe following satisfied?
• Realistic theory: probability ≤ 75%
• Quantum mechanics: probability ≈ 85%
a≠b for x=0, y=0a≠b for x=0, y=1a≠b for x=1, y=0a=b for x=1, y=1
Bell‘s Inequality
Source
a=0 or 1
y=0 or 1
b=0 or 1
x=0 or 1
Cos2 45°/2 ≈ 85%Cos2 45°/2 ≈ 85%Cos2 45°/2 ≈ 85%
1-Cos2 135°/2≈ 85%
a≠b for x=0, y=0a≠b for x=0, y=1a≠b for x=1, y=0a=b for x=1, y=1
• With which probability arethe following satisfied?
• Realistic theory: probability ≤ 75%
• Quantum mechanics: probability ≈ 85%
a≠b for x=0, y=0a≠b for x=0, y=1a≠b for x=1, y=0a=b for x=1, y=1
Bell‘s Inequality
Source
a=0 or 1
y=0 or 1
b=0 or 1
x=0 or 1
• With which probability arethe following satisfied?
• Realistic theory: probability ≤ 75%
• Quantum mechanics: probability ≈ 85%
a≠b for x=0, y=0a≠b for x=0, y=1a≠b for x=1, y=0a=b for x=1, y=1
Bell‘s Inequality
Source
a=0 or 1
y=0 or 1
b=0 or 1
x=0 or 1x=0 oder 1
must be confirmed in the experiment
each measurement must yield a result
(successful detection)
Choice of x must be independent of y
(locality)
a≠b für x=0, y=0a≠b für x=0, y=1a≠b für x=1, y=0a=b für x=1, y=1
Experimente• Photons, Aspect et al. (1982)
Locality ✗Detection ✗
• Photons, Gisin et al., Zeilinger et al.(1998)Locality ✓Detection ✗
• Superconducting QubitsWineland et al. (2001)Locality ✗Detection ✓
• Locality and detection in one experiment?
Experimente• Photons, Aspect et al. (1982)
Locality ✗Detection ✗
• Photons, Gisin et al., Zeilinger et al.(1998)Locality ✓Detection ✗
• Superconducting QubitsWineland et al. (2001)Locality ✗Detection ✓
• Locality and detection in one experiment?
Indeterminism of the world!
Security of quantum cryptography!
Overview
• Entanglement
• Determinism?
• Quantum Cryptography
• A Test for Entanglement
message+ key----------------= cipher
Quantum Cryptography
• Measurement result is random and correlated
• In principle, the measurement result has not existed before the measurement
• Only Alice and Bob know the resultAlice and Bob have a secret bit
?
many secret bitsrepetition ⇒
Quantum Cryptography
• Measurement result is random and correlated
• In principle, the measurement result has not existed before the measurement
• Only Alice and Bob know the resultAlice and Bob have a secret bit
?
keyrepetition ⇒
Quantum Cryptography
• Measurement result is random and correlated
• In principle, the measurement result has not existed before the measurement
• Only Alice and Bob know the resultAlice and Bob have a secret bit
• Encrypting with Entanglement
?
Ekert (1991)
keyrepetition ⇒
Perfectly Secret Communication
• Vernam (1926) Shannon (1949)
message+ key----------------= cipher cipher
- key----------------= message
00101 10100 01000+ 10011 01010 11010-----------------------------= 10010 11110 10010 10010 11110 10010
- 10011 01010 11010 -----------------------------=00101 10100 01000
Perfectly Secret Communication
• Vernam (1926) Shannon (1949)
• Perfect secrecy
• Commercial: idQuantique, MagiQ Technologies
Overview
• Entanglement
• Determinism?
• Quantum Cryptography
• A Test for Entanglement
message+ key----------------= cipher
Encrypting with Entanglement
• Theory
• Experiment - Noise
• Is the state entangled?Can we generate a key?
0 0 0 00 0.5 −0.5 00 −0.5 0.5 00 0 0 0
Encrypting with Entanglement
• Theory
• Experiment - Noise
• Is the state entangled?Can we generate a key?
0 0 0 00 0.5 −0.5 00 −0.5 0.5 00 0 0 0
Encrypting with Entanglement
• Theory
• Experiment - Noise
0.07 −0.04 0.01 0.03−0.04 0.44 −0.39 −0.010.01 −0.39 0.43 0.050.03 −0.01 0.05 0.06
0 0 0 00 0.5 −0.5 00 −0.5 0.5 00 0 0 0
Encrypting with Entanglement
• Theory
• Experiment - Noise
0.07 −0.04 0.01 0.03−0.04 0.44 −0.39 −0.010.01 −0.39 0.43 0.050.03 −0.01 0.05 0.06
0 0 0 00 0.5 −0.5 00 −0.5 0.5 00 0 0 0
Encrypting with Entanglement
• Theory
• Experiment - Noise
• Is the state entangled?Can we generate a key?
0.07 −0.04 0.01 0.03−0.04 0.44 −0.39 −0.010.01 −0.39 0.43 0.050.03 −0.01 0.05 0.06
0 0 0 00 0.5 −0.5 00 −0.5 0.5 00 0 0 0
Encrypting with Entanglement
• Theory
• Experiment - Noise
• Is the state entangled?Can we generate a key?
Test for Entanglement
0.07 −0.04 0.01 0.03−0.04 0.44 −0.39 −0.010.01 −0.39 0.43 0.050.03 −0.01 0.05 0.06
0 0 0 00 0.5 −0.5 00 −0.5 0.5 00 0 0 0
Monogamy of Entanglement
Alice strongly entangled with Bob 1 ➭ Alice little entangled with Bob 2
Bob 1
Bob 2
Alice
Monogamy of Entanglement
Alice strongly entangled with Bob 1 ➭ Alice little entangled with Bob 2 . . . ➭ Alice little entangled with Bob k
Bob 1
Bob 2
Bob k
Alice
Monogamy of Entanglement
Bob 1
Bob 2
Bob k
Alice
Monogamy of Entanglement
Bob 1
Bob 2
Bob k
Alice
Given: State of Alice and Bob 1
Question: Can Alice be entangled with k Bobs in equal fashion?
Answer: Yes: State is almost not entangled (almost not = )
No: State is entangled
1√k
Mathematical Formulation
extendible to k Bobs
Frobenius (Euclidian) norm
3
n = |A|2|B|O�
log |A|�2
�
= eO(�−2 log |A| log |B|)
eO(�−2 log |A| log |B|)
eO(|A|2|B|2 log �−1)
eO(|A|2|B|2)
eO(log |A| log |B|)
||X|| :=√trX†X
||X||1 := tr√X†X
ρAB = |Ψ��Ψ|AB =
1 0 0 00 0 0 00 0 0 00 0 0 0
ρAB =
1 0 0 00 0 0 00 0 0 00 0 0 0
ρAB = |Ψ��Ψ|AB =
12 0 0 1
20 0 0 00 0 0 012 0 0 1
2
ρAB =
12 0 0 1
20 0 0 00 0 0 012 0 0 1
2
� > 0
⇒
�⇐
number of Alice‘s
qubits
minσAB
||ρAB − σAB || ≤ c
�q
k
Fernando Brandão, Matthias Christandl und Jon Yard (2010)
not entangled
Algorithm: Extendible to k Bobs? Yes ⇒ almost not entangled
No ⇒ entangled
Result: Algorithm is fast
2 Bobs
not entangled Bobs
3 Bobs
k Bobs
2
Measure Esq ED KD EC EF ER E∞R EN
normalisation y y y y y y y y
faithfulness y n ? y y y y n
LOCC monotonicity y y y y y y y y
asymptotic continuity y ? ? ? y y y n
convexity y ? ? ? y y y n
strong superadditivity y y y ? n n ? ?
subadditivity y ? ? y y y y y
monogamy y ? ? n n n n ?
TABLE I: If no citation is given, the property either follows directly from the definition or was derived by
the authors of the main reference. Many recent results listed in this table have significance beyond the study
of entanglement measures, such as Hastings’ counterexample to the additivity conjecture of the minimum
output entropy [76] which implies that entanglement of formation is not strongly superadditive [79].
I. INTRODUCTION
≈ R|AB|2−1
�����
11
|AB|
�min(|A|,|B|i,j |ii��jj|
|AB|
|00�
O
�log |A|�2
�
∞−
Squashed entanglement is the quantum analogue of the intrinsic information, which is defined
as
I(X;Y ↓Z) := infPZ̄|Z
I(X;Y |Z̄),
for a triple of random variables X,Y, Z [16]. The minimisation extends over all conditional prob-
ability distributions mapping Z to Z̄. It has been shown that the minimisation can be restricted
to random variables Z̄ with size |Z̄| = |Z|[17]. This implies that the minimum is achieved and in
all quantum states
√q/k}in practice(semidefinite programming)
in theory (quasipolynomial-time)
Fernando Brandão,
Matthias Christandl
und Jon Yard
(2010)
Summary and Outlook
• Entanglement
• Determinism?
• Quantum Cryptography
• A Test for Entanglement
message+ key----------------= cipher
Summary and Outlook
• Entanglement
• Determinism?
• Quantum Cryptography
• A Test for Entanglement
message+ key----------------= cipher
Summary and Outlook
• Entanglement
• Determinism?
• Quantum Cryptography
• A Test for Entanglement
message+ key----------------= cipher
Fundamental PhenomenaUncertainty Relation
Pauli Principle
Summary and Outlook
• Entanglement
• Determinism?
• Quantum Cryptography
• A Test for Entanglement
message+ key----------------= cipher
Fundamental PhenomenaUncertainty Relation
Pauli Principle
Summary and Outlook
• Entanglement
• Determinism?
• Quantum Cryptography
• A Test for Entanglement
message+ key----------------= cipher
Fundamental PhenomenaUncertainty Relation
Pauli Principle
Philosophical Consequences?Locality?
Summary and Outlook
• Entanglement
• Determinism?
• Quantum Cryptography
• A Test for Entanglement
Fundamental PhenomenaUncertainty Relation
Pauli Principle
Philosophical Consequences?Locality?
Summary and Outlook
• Entanglement
• Determinism?
• Quantum Cryptography
• A Test for Entanglement
New TechnologiesQuantum Simulator (2020?)Quantum Computer (2040?)
Fundamental PhenomenaUncertainty Relation
Pauli Principle
Philosophical Consequences?Locality?
Summary and Outlook
• Entanglement
• Determinism?
• Quantum Cryptography
• A Test for Entanglement
New TechnologiesQuantum Simulator (2020?)Quantum Computer (2040?)
Fundamental PhenomenaUncertainty Relation
Pauli Principle
Philosophical Consequences?Locality?
Summary and Outlook
• Entanglement
• Determinism?
• Quantum Cryptography
• A Test for Entanglement
New TechnologiesQuantum Simulator (2020?)Quantum Computer (2040?)
Fundamental PhenomenaUncertainty Relation
Pauli Principle
Philosophical Consequences?Locality?
Mathematical ToolsStatistics of the Quanta
Symmetries of the Quanta