Matthias Christandl
Quantum Information Theory
Institute for Theoretical Physics
ETH Zurich
Encrypting with Entanglement
Overview
• Entanglement
• Determinism?
• Quantum Cryptography
• A Test for Entanglement
message
+ key
----------------
= c...
Quantum Mechanics
• Theory of the smallest particles
• Big implications
Stability of matter
Fission and fusion of nuclei
H...
Entanglement - a Quantum Mechanical
Phenomenon
• Quantum mechanical correlations
among two or more particles
• „spooky act...
Schrödinger 1932
The claim ... has the strange consequence that the
Ψ-function of a system [System I] is changed by the
pe...
Schrödinger 1932 Schrödinger Archiv,Wien
This makes it a bit difficult to view the change in the
Ψ-function as a Naturvorga...
Alice and Bob
• Long distances
• Communication of measurement results
• Particles in the hands of
• Each equipped with a l...
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 result
Bob the opposite
• Alice heads Bob tails
Alice tails Bob heads
• Random, but correlate...
Qubit, the Quantum Bit
• Unit of quantum information
• Many possible states (dots on a sphere)
• Example: photon polarisat...
Qubit, the Quantum Bit
• Unit of quantum information
• Many possible states (dots on a sphere)
• Example: photon polarisat...
Qubit
• We cannot measure accurately the state
of the qubit.
• We can only measure, if the state is in one
of two antipoda...
Qubit
• State: North pole
Measurement: North or south pole?
Result: North pole
• State: Copenhagen
Measurement: North or s...
Source
Entangled Qubits
Source
Entangled Qubits
Measurement:
North or
south pole
Source
Entangled Qubits
Measurement:
North or
south pole
Source
Entangled Qubits
50%
Measurement:
North or
south pole
Source
Entangled Qubits
50%
50%
Measurement:
North or
south pole
Source
Entangled Qubits
50%
50%
Bob‘s state=antipodal point
just 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 point
for every measurement
„spooky action at a distance“
Measurement:
Madr...
Übersicht
• Entanglement
• Determinism?
• Quantum Cryptography
• Test for Entanglement
message
+ key
----------------
= ci...
Determinism?
• In classical physics, the measurement
result exists before the performance of
the measurement (realism)
• I...
Bell‘s Inequality
• With which probability are
the following satisfied?
• Realistic theory: probability ≤ 75%
• Quantum mec...
• With which probability are
the following satisfied?
• Realistic theory: probability ≤ 75%
• Quantum mechanics: probabilit...
• With which probability are
the following satisfied?
• Realistic theory: probability ≤ 75%
• Quantum mechanics: probabilit...
• With which probability are
the following satisfied?
• Realistic theory: probability ≤ 75%
• Quantum mechanics: probabilit...
• With which probability are
the following satisfied?
• Realistic theory: probability ≤ 75%
• Quantum mechanics: probabilit...
• With which probability are
the following satisfied?
• Realistic theory: probability ≤ 75%
• Quantum mechanics: probabilit...
Experimente
• Photons,Aspect et al. (1982)
Locality ✗
Detection ✗
• Photons, Gisin et al.,
Zeilinger et al.(1998)
Locality...
Experimente
• Photons,Aspect et al. (1982)
Locality ✗
Detection ✗
• Photons, Gisin et al.,
Zeilinger et al.(1998)
Locality...
Overview
• Entanglement
• Determinism?
• Quantum Cryptography
• A Test for Entanglement
message
+ key
----------------
= c...
Quantum Cryptography
• Measurement result is random and
correlated
• In principle, the measurement result has not
existed ...
Quantum Cryptography
• Measurement result is random and
correlated
• In principle, the measurement result has not
existed ...
Quantum Cryptography
• Measurement result is random and
correlated
• In principle, the measurement result has not
existed ...
Perfectly Secret Communication
• Vernam (1926) Shannon (1949)
message
+ key
----------------
= cipher cipher
- key
-------...
00101 10100 01000
+ 10011 01010 11010
-----------------------------
= 10010 11110 10010 10010 11110 10010
- 10011 01010 11...
Overview
• Entanglement
• Determinism?
• Quantum Cryptography
• A Test for Entanglement
message
+ key
----------------
= c...
Encrypting with Entanglement
• Theory
• Experiment - Noise
• Is the state entangled?
Can we generate a key?




0 0 0 ...
Encrypting with Entanglement
• Theory
• Experiment - Noise
• Is the state entangled?
Can we generate a key?




0 0 0 ...
Encrypting with Entanglement
• Theory
• Experiment - Noise




0.07 −0.04 0.01 0.03
−0.04 0.44 −0.39 −0.01
0.01 −0.39 ...
Encrypting with Entanglement
• Theory
• Experiment - Noise




0.07 −0.04 0.01 0.03
−0.04 0.44 −0.39 −0.01
0.01 −0.39 ...
Encrypting with Entanglement
• Theory
• Experiment - Noise
• Is the state entangled?
Can we generate a key?




0.07 −...
Encrypting with Entanglement
• Theory
• Experiment - Noise
• Is the state entangled?
Can we generate a key?
Test for Entan...
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 enta...
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 B...
Mathematical Formulation
extendible to k Bobs
Frobenius (Euclidian) norm
n = |A|2
|B|
O

log |A|
2

= eO(−2 log |A| log |B...
Algorithm: Extendible to k Bobs?
Yes almost not entangled
No entangled
Result: Algorithm is fast
2 Bobs
not entangled
Bobs...
Summary and Outlook
• Entanglement
• Determinism?
• Quantum Cryptography
• A Test for Entanglement
message
+ key
---------...
Summary and Outlook
• Entanglement
• Determinism?
• Quantum Cryptography
• A Test for Entanglement
message
+ key
---------...
Summary and Outlook
• Entanglement
• Determinism?
• Quantum Cryptography
• A Test for Entanglement
message
+ key
---------...
Summary and Outlook
• Entanglement
• Determinism?
• Quantum Cryptography
• A Test for Entanglement
message
+ key
---------...
Summary and Outlook
• Entanglement
• Determinism?
• Quantum Cryptography
• A Test for Entanglement
message
+ key
---------...
Summary and Outlook
• Entanglement
• Determinism?
• Quantum Cryptography
• A Test for Entanglement
Fundamental Phenomena
U...
Summary and Outlook
• Entanglement
• Determinism?
• Quantum Cryptography
• A Test for Entanglement
New Technologies
Quantu...
Summary and Outlook
• Entanglement
• Determinism?
• Quantum Cryptography
• A Test for Entanglement
New Technologies
Quantu...
Summary and Outlook
• Entanglement
• Determinism?
• Quantum Cryptography
• A Test for Entanglement
New Technologies
Quantu...
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Encrypting with entanglement matthias christandl

  1. 1. Matthias Christandl Quantum Information Theory Institute for Theoretical Physics ETH Zurich Encrypting with Entanglement
  2. 2. Overview • Entanglement • Determinism? • Quantum Cryptography • A Test for Entanglement message + key ---------------- = cipher
  3. 3. Quantum Mechanics • Theory of the smallest particles • Big implications Stability of matter Fission and fusion of nuclei Hawking radiation ω✿✿✿✿✿✿ Photon
  4. 4. 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
  5. 5. 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
  6. 6. 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] ...
  7. 7. Alice and Bob • Long distances • Communication of measurement results • Particles in the hands of • Each equipped with a laboratory Alice Bob and
  8. 8. Information • The bit = unit of information on/off heads/tails north pole/ south pole
  9. 9. Information • random bit child plays with switch toss of a coin travel lottery
  10. 10. Correlated Bits • Alice gets result Bob the opposite • Alice heads Bob tails Alice tails Bob heads • Random, but correlated bits
  11. 11. Qubit, the Quantum Bit • Unit of quantum information • Many possible states (dots on a sphere) • Example: photon polarisation . . ! # !# !$#%%!'# !(#%%!)# !*# !+# !,#
  12. 12. Qubit, the Quantum Bit • Unit of quantum information • Many possible states (dots on a sphere) • Example: photon polarisation . . ! # !# !$#%%!'# !(#%%!)# !*# !+# !,#
  13. 13. 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?
  14. 14. Qubit • State: North pole Measurement: North or south pole? Result: North pole • State: Copenhagen Measurement: North or south pole? Result: Nordpol (Cos2 35°/2≈91%) • State: Singapur Measurement: North or south pole? Result: North pole (Cos2 90°/2=50%) •Measurement changes the state 50% 50%
  15. 15. Source Entangled Qubits
  16. 16. Source Entangled Qubits
  17. 17. Measurement: North or south pole Source Entangled Qubits
  18. 18. Measurement: North or south pole Source Entangled Qubits 50%
  19. 19. Measurement: North or south pole Source Entangled Qubits 50% 50%
  20. 20. Measurement: North or south pole Source Entangled Qubits 50% 50% Bob‘s state=antipodal point just like a coin
  21. 21. Entangled Qubits Measurement: Madrid or Wellington Source
  22. 22. Entangled Qubits 50% Measurement: Madrid or Wellington Source
  23. 23. Entangled Qubits 50% 50% Measurement: Madrid or Wellington Source
  24. 24. Entangled Qubits 50% 50% Bob‘s state=antipodal point for every measurement „spooky action at a distance“ Measurement: Madrid or Wellington Source
  25. 25. Übersicht • Entanglement • Determinism? • Quantum Cryptography • Test for Entanglement message + key ---------------- = cipher
  26. 26. 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)
  27. 27. Bell‘s Inequality • With which probability are the 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=0 a≠b for x=0, y=1 a≠b for x=1, y=0 a=b for x=1, y=1
  28. 28. • With which probability are the following satisfied? • Realistic theory: probability ≤ 75% • Quantum mechanics: probability ≈ 85% Bell‘s Inequality a≠b for x=0, y=0 a≠b for x=0, y=1 a≠b for x=1, y=0 a=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
  29. 29. • With which probability are the following satisfied? • Realistic theory: probability ≤ 75% • Quantum mechanics: probability ≈ 85% a≠b for x=0, y=0 a≠b for x=0, y=1 a≠b for x=1, y=0 a=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
  30. 30. • With which probability are the following satisfied? • Realistic theory: probability ≤ 75% • Quantum mechanics: probability ≈ 85% a≠b for x=0, y=0 a≠b for x=0, y=1 a≠b for x=1, y=0 a=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=0 a≠b for x=0, y=1 a≠b for x=1, y=0 a=b for x=1, y=1
  31. 31. • With which probability are the following satisfied? • Realistic theory: probability ≤ 75% • Quantum mechanics: probability ≈ 85% a≠b for x=0, y=0 a≠b for x=0, y=1 a≠b for x=1, y=0 a=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
  32. 32. • With which probability are the following satisfied? • Realistic theory: probability ≤ 75% • Quantum mechanics: probability ≈ 85% a≠b for x=0, y=0 a≠b for x=0, y=1 a≠b for x=1, y=0 a=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=0 a≠b für x=0, y=1 a≠b für x=1, y=0 a=b für x=1, y=1
  33. 33. Experimente • Photons,Aspect et al. (1982) Locality ✗ Detection ✗ • Photons, Gisin et al., Zeilinger et al.(1998) Locality ✓ Detection ✗ • Superconducting Qubits Wineland et al. (2001) Locality ✗ Detection ✓ • Locality and detection in one experiment?
  34. 34. Experimente • Photons,Aspect et al. (1982) Locality ✗ Detection ✗ • Photons, Gisin et al., Zeilinger et al.(1998) Locality ✓ Detection ✗ • Superconducting Qubits Wineland et al. (2001) Locality ✗ Detection ✓ • Locality and detection in one experiment? Indeterminism of the world! Security of quantum cryptography!
  35. 35. Overview • Entanglement • Determinism? • Quantum Cryptography • A Test for Entanglement message + key ---------------- = cipher
  36. 36. 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 result Alice and Bob have a secret bit ? many secret bitsrepetition
  37. 37. 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 result Alice and Bob have a secret bit ? keyrepetition
  38. 38. 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 result Alice and Bob have a secret bit • Encrypting with Entanglement ? Ekert (1991) keyrepetition
  39. 39. Perfectly Secret Communication • Vernam (1926) Shannon (1949) message + key ---------------- = cipher cipher - key ---------------- = message
  40. 40. 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
  41. 41. Overview • Entanglement • Determinism? • Quantum Cryptography • A Test for Entanglement message + key ---------------- = cipher
  42. 42. Encrypting with Entanglement • Theory • Experiment - Noise • Is the state entangled? Can we generate a key?     0 0 0 0 0 0.5 −0.5 0 0 −0.5 0.5 0 0 0 0 0    
  43. 43. Encrypting with Entanglement • Theory • Experiment - Noise • Is the state entangled? Can we generate a key?     0 0 0 0 0 0.5 −0.5 0 0 −0.5 0.5 0 0 0 0 0    
  44. 44. Encrypting with Entanglement • Theory • Experiment - Noise     0.07 −0.04 0.01 0.03 −0.04 0.44 −0.39 −0.01 0.01 −0.39 0.43 0.05 0.03 −0.01 0.05 0.06         0 0 0 0 0 0.5 −0.5 0 0 −0.5 0.5 0 0 0 0 0    
  45. 45. Encrypting with Entanglement • Theory • Experiment - Noise     0.07 −0.04 0.01 0.03 −0.04 0.44 −0.39 −0.01 0.01 −0.39 0.43 0.05 0.03 −0.01 0.05 0.06         0 0 0 0 0 0.5 −0.5 0 0 −0.5 0.5 0 0 0 0 0    
  46. 46. 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 −0.01 0.01 −0.39 0.43 0.05 0.03 −0.01 0.05 0.06         0 0 0 0 0 0.5 −0.5 0 0 −0.5 0.5 0 0 0 0 0    
  47. 47. 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 −0.01 0.01 −0.39 0.43 0.05 0.03 −0.01 0.05 0.06         0 0 0 0 0 0.5 −0.5 0 0 −0.5 0.5 0 0 0 0 0    
  48. 48. Monogamy of Entanglement Alice strongly entangled with Bob 1 ➭ Alice little entangled with Bob 2 Bob 1 Bob 2 Alice
  49. 49. 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
  50. 50. Monogamy of Entanglement Bob 1 Bob 2 Bob k Alice
  51. 51. 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
  52. 52. Mathematical Formulation extendible to k Bobs Frobenius (Euclidian) norm 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|| := √ tr X†X ||X||1 := tr √ X†X   1 0 0 0 0 0 0 0   number of Alice‘s qubits min σAB ||ρAB − σAB|| ≤ c q k Fernando Brandão, Matthias Christandl und JonYard (2010) not entangled
  53. 53. Algorithm: Extendible to k Bobs? Yes almost not entangled No entangled Result: Algorithm is fast 2 Bobs not entangled Bobs 3 Bobs k Bobs 11 |AB| min(|A|,|B| i,j |iijj| |AB| |00 O log |A| 2 ∞− entanglement is the quantum analogue of the intrinsic information, which is defined I(X;Y ↓Z) := inf P ¯Z|Z I(X; Y | ¯Z), f random variables X, Y, Z [16]. The minimisation extends over all conditional prob- butions mapping Z to ¯Z. It has been shown that the minimisation can be restricted ariables ¯Z with size | ¯Z| = |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 JonYard (2010)
  54. 54. Summary and Outlook • Entanglement • Determinism? • Quantum Cryptography • A Test for Entanglement message + key ---------------- = cipher
  55. 55. Summary and Outlook • Entanglement • Determinism? • Quantum Cryptography • A Test for Entanglement message + key ---------------- = cipher
  56. 56. Summary and Outlook • Entanglement • Determinism? • Quantum Cryptography • A Test for Entanglement message + key ---------------- = cipher Fundamental Phenomena Uncertainty Relation Pauli Principle
  57. 57. Summary and Outlook • Entanglement • Determinism? • Quantum Cryptography • A Test for Entanglement message + key ---------------- = cipher Fundamental Phenomena Uncertainty Relation Pauli Principle
  58. 58. Summary and Outlook • Entanglement • Determinism? • Quantum Cryptography • A Test for Entanglement message + key ---------------- = cipher Fundamental Phenomena Uncertainty Relation Pauli Principle Philosophical Consequences? Locality?
  59. 59. Summary and Outlook • Entanglement • Determinism? • Quantum Cryptography • A Test for Entanglement Fundamental Phenomena Uncertainty Relation Pauli Principle Philosophical Consequences? Locality?
  60. 60. Summary and Outlook • Entanglement • Determinism? • Quantum Cryptography • A Test for Entanglement New Technologies Quantum Simulator (2020?) Quantum Computer (2040?) Fundamental Phenomena Uncertainty Relation Pauli Principle Philosophical Consequences? Locality?
  61. 61. Summary and Outlook • Entanglement • Determinism? • Quantum Cryptography • A Test for Entanglement New Technologies Quantum Simulator (2020?) Quantum Computer (2040?) Fundamental Phenomena Uncertainty Relation Pauli Principle Philosophical Consequences? Locality?
  62. 62. Summary and Outlook • Entanglement • Determinism? • Quantum Cryptography • A Test for Entanglement New Technologies Quantum Simulator (2020?) Quantum Computer (2040?) Fundamental Phenomena Uncertainty Relation Pauli Principle Philosophical Consequences? Locality? Mathematical Tools Statistics of the Quanta Symmetries of the Quanta

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