Multi-particle Entanglement in Quantum States and Evolutions
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Multi-particle Entanglement in Quantum States and Evolutions

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Slides from my first ever seminar presentation given when I was a first year Ph.D. student. University of Bristol Applied Mathematics Seminar 2001. Brief introduction to entanglement and discussion ...

Slides from my first ever seminar presentation given when I was a first year Ph.D. student. University of Bristol Applied Mathematics Seminar 2001. Brief introduction to entanglement and discussion of local unitary invariants.

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Multi-particle Entanglement in Quantum States and Evolutions Multi-particle Entanglement in Quantum States and Evolutions Presentation Transcript

  • Multi-particle Entanglement in Quantum States and Evolutions Matthew Leifer - 1st Year Ph.D., Maths Dept., University of Bristol Supervisor - Dr. Noah Linden 1. Background and Motivation 2. Physical Meaning of Entanglement 3. Quantum Mechanics 4. Entanglement in Quantum States 5. Entanglement in Quantum Evolutions 6. Further Investigations
  • 1. Background and Motivation
    • Quantum Mechanics is weird!
      • role of probability
      • measurement problem (“collapse of wave-function”)
      • non-local correlations (entanglement)
    • Quantum Mechanics is Successful!
      • Atomic Physics and Chemistry
      • Solid State Physics (semiconductors)
      • Quantum Field Theory (Particle Physics)
      • Anomalous magnetic moment of electron
    • We do not have full control of the quantum degrees of freedom in these applications.
  • What happens if we can control quantum systems?
    • Quantum Computers
      • Feynman (1982)
    • Holy Grails of Information Theory
      • Polynomial time prime factorisation - Shor (1994)
      • Perfectly secure key distribution in cryptography
    • Other discoveries
      • Teleportation - Bennett et al (1992)
      • Quantum error correction - Shor (1995)
    • These procedures use entangled states!
    Peter Shor Richard Feynman
  • 3. Quantum Mechanics Measurement
    • Quantum states, |  >, are vectors (rays) in a Hilbert space
    • Usually we normalise s.t. <  |  > = 1
    • Observables are represented by Hermitian operators (i.e Q s.t. Q † = Q)
    • If we construct an orthonormal eigenbasis{|  i >} of Q s.t. Q|  i > =  i |  i > then |  > =  a i |  i > with  |a i | 2 = 1 and a i = <  i |  >
    • The possible results of measurements of Q are its eigenvalues  i
    • The result of a measurement will be  i with probability |a i | 2
    • After obtaining a value  i , the state will become |  i >
  • 3. Quantum Mechanics Quantum Dynamics
    • States can also evolve between measurements |  >  U |  >
    • Conservation of probability => states must remain normalised: <  |U † U|  > = <  |  > => U † U = 1
    • Quantum evolutions are unitary!
    • Can also see this from Schrödinger eqn.
    In theory, can implement any unitary transformation by correct choice of H.
  • 3. Quantum Mechanics Systems and Subsystems
    • If we have 2 systems A and B, with Hilbert spaces H A and H B then the quantum state of the combined system is a vector in H A  H B
    • Example - 2 dimensional subsystems (spin-1/2 particles)
    • H A has basis {|0> A , |1> A }
    • H B has basis {|0> B , |1> B }
    • H A  H B has basis {|0> A  |0> B , |0> A  |1> B , |1> A  |0> B , |1> A  |1> B }
    • or {|00>, |01>, |10>, |11>}
    • An example vector
  • 4. Entanglement in Quantum States
    • An entangled state is one that cannot be written as |  AB > = |  A >  |  B > for any choice of basis in H A and H B
    • Specialise to n spin-1/2 particles.
    • General unitary transformation
    • Local unitary transformation
    • Each copy of U(2) acts on corresponding particle
    • Local unitaries do not change entanglement of state
  • 4. Entanglement in Quantum States # Non-Local Parameters
    • In general
    • Linear span of X T s = tangent space to orbit at v.
    • No. linearly indep. X T s gives dimension of orbit.
    • E.g. infinitesimal change under a trans. in  1 direction:
    • Write a r =c r +id r (r = 0,1) and
    • Then
    • and f(c 0 ,d 0 ,c 1 ,d 1 )  f(c 0 -  d 1 ,d 0 +  c 1 ,c 1 -  d 0 ,d 1 +  c 0 )
    • so
    • Similarly we can find u 0 ,u 2 ,u 3 . Only 3 are linearly indep. So we have 4-3 = 1 non-local parameter
  • 4. Entanglement in Quantum States Polynomial Invariants
    • Construct invariants by contracting with U(2) invariant tensors) (  ij and  ij )in all possible ways
    • Example: for 1 particle
    • For 2 particles
    • General case: Contract a’s with a * ’s using  ’s in all possible ways until we have as many functionally indep. invariants as non-local params.
  • 4. Entanglement in Quantum States Stability Groups
    • Each orbit has a stability group < U(2) n .
    • Certain states have larger stability groups than the generic case.
    • States with maximal symmetry are especially interesting.
    • Example: 3 particles
      • Generic states have no stability group.
      • Singlet  vector is invariant under SU(2)  U(1)
      • Direct products are invariant under U(1) 3
      • GHZ are invariant under U(1) 2 and discrete symmetry Z 2
  • 5. Entanglement in Quantum Evolutions
    • Consider U  V 1 UV 2 , where U  U(2 n ) and V 1 ,V 2  U(2) n
    • Does orbit space make sense?
    • Apply same ideas
      • No. invariant parameters
      • Canonical points
      • Polynomial invariants
    • 1 particle case -
      • Lie Algebra elements can now work on both sides.
    • 2 particle canonical form -
      • How are  j ’s related to polynomial invariants?
  • 6. Future Work
    • Density matrix formalism - Linden, Popescu and Sudberry 1998
    • Find canonical forms, polynomial invariants and special orbits for n particle unitaries.
    • Determine relation between non-locality in states and evolutions.
    • Allow measurements. What is the significance of
      • Carteret, Linden, Popescu and Sudberry (1998)