Encrypting with entanglement matthias christandl

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


Measurement:North or south pole




50%



50%

50%



50%

50%

Bob‘s state=antipodal pointjust like a coin

Entangled Qubits

Measurement:Madrid or Wellington

Source

Entangled Qubits

50%


Source

Entangled Qubits

50%

50%


Source

Entangled Qubits

50%

50%

Bob‘s state=antipodal pointfor every measurement

„spooky action at a distance“


Source

Übersicht

• Entanglement

• Determinism?


• Test for Entanglement


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




Bell‘s Inequality


100%100%100%0%

a=0 or 1

y=0 or 1

b=0 or 1

x=0 or 1





Bell‘s Inequality

Source

a=0 or 1

y=0 or 1

b=0 or 1

x=0 or 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%






Bell‘s Inequality

Source

a=0 or 1

y=0 or 1

b=0 or 1

x=0 or 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

• 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 ⇒





?

keyrepetition ⇒





• 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?





• 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


• Theory



0 0 0 00 0.5 −0.5 00 −0.5 0.5 00 0 0 0


• Theory


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


• Theory


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


• Theory



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


• Theory



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


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


Bob 1

Bob 2

Bob k

Alice


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?




Summary and Outlook

• Entanglement

• Determinism?




Summary and Outlook

• Entanglement

• Determinism?




Fundamental PhenomenaUncertainty Relation

Pauli Principle

Summary and Outlook

• Entanglement

• Determinism?





Pauli Principle

Summary and Outlook

• Entanglement

• Determinism?





Pauli Principle

Philosophical Consequences?Locality?

Summary and Outlook

• Entanglement

• Determinism?




Pauli Principle


Summary and Outlook

• Entanglement

• Determinism?



New TechnologiesQuantum Simulator (2020?)Quantum Computer (2040?)


Pauli Principle


Summary and Outlook

• Entanglement

• Determinism?





Pauli Principle


Summary and Outlook

• Entanglement

• Determinism?





Pauli Principle


Mathematical ToolsStatistics of the Quanta

Symmetries of the Quanta

Technology

Encrypting with entanglement matthias christandl