Quantum State and Process Measurement and Characterization

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Quantum State and Process Measurement and
Characterization
Daniel F. V. JAMES Theoretical Division T-4 Los
Alamos National Laboratory E-mail
dfvj_at_lanl.gov Universität Innsbruck 5 May 2004
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State of a Single Qubit

• Photon polarization based qubits

• Measure multiple (assumed identical) copies
  frequency of clicks gives estimate of b2

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• Find relative phase of a and b by performing a
  unitary operation before beam splitter
• e.g.
• Frequency of clicks now gives an estimate of
• Systematic way of getting all the data needed.

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Stokes Parameters
Measure intensity with four different filters
G. G. Stokes, Trans Cambridge Philos Soc 9 399
(1852)
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Stokes Parameters
These 4 parameters completely specify
polarization of beam Beam is an ensemble of
photons
Pauli matrices
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2 Qubit Quantum States

• Pure states
• Ideal case

• Mixed states
• Quantum state is random need averages and
  correlations of coefficients

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State Creation by OPDC
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2 Qubit Quantum State Tomography
Coincidence Rate measurements for two photons
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• Doesnt give the right answer.

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Maximum Likelihood Tomography

• Numerically Minimize the function

• Maximum Likelihood fit to "physical" density
  matrix
• Density matrix must be Hermitian, normalized,
  non-negative
• D. F. V. James, et al., Phys Rev A 64, 052312
  (2001).

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Example Measured Density Matrix
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Quantum State Tomography I

• sublevels of Hydrogen (partial) (Ashburn et al,
  1990)
• Optical mode (Raymer et al., 1993)
• Molecular vibrations (Walmsley et al, 1995)
• Motion of trapped ion (Wineland et al., 1996)
• Motion of trapped atom (Mlynek et al., 1997)
• Liquid state NMR (Chaung et al, 1998)
• Entangled Photons (Kwiat et al, 1999)
• Entangled ions (Blatt et al., 2002)
• A. G. White, D. F. V. James, P. H. Eberhard and
  P. G. Kwiat, Phys Rev Lett 83, 3103 (1999).

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Characterizing the State
Purity
Fidelity how close are two states?
Pure states
Mixed states doesnt work
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Measures of Entanglement

• Pure states
• How much entanglement is in this state?
• Entropy of reduced density matrix of one photon
• Concurrence
• Concurrence is equivalent to Entanglement
• C0 implies separable state
• C1 implies maximally entangled state (e.g. Bell
  states)

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Entanglement in Mixed States

• Mixed states can be de-composed into incoherent
  sums of pure states
• Average Concurrence dependent on decomposition
• Minimized Average Concurrence
• Independent of decomposition
• C0 implies separable state
• C1 implies maximally entangled state (e.g. Bell
  states)
• Analytic expression (Wootters, 98) makes things
  very convenient!

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Two Qubit Mixed State Concurrence
spin flip matrix
Transpose (in computational basis)
Eigenvalues of R (in decreasing order)
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Map of Hilbert Space
D.F.V. James and P.G .Kwiat, Los Alamos
Science, 2002
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Process Tomography

• Trace Preserving Completely Positive Maps Every
  thing that could possibly happen to a quantum
  state (without measuring it)
• 
  operator-sum
  formalism
• Kraus
  operators

set of basis matrices, e.g.
Trace orthogonality
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Decompose the Kraus operators
then-
where-
c is a Hermitian, positive 16x16 matrix(error
correlation matrix), with the constraints-
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Process Tomography

16 Input states
16 Projection states
Estimate probability from counts 16x16 256
data Recover cmn by linear inversion -
problematic in constraining positivity - close
analogy with state tomography
I. L. Chuang and M. A. Nielsen, J. Mod Op. 44,
2455 (1997) M. W. Mitchell et al., Phys Rev Lett
91, 120402 (2003).
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Maximum Likelihood Process Tomography
Numerically optimize where
(256 free parameters) Constraints on cmn -
positive - Hermitian - additional constraint
for physically allowed process
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Process Tomography of UQ Optical CNOT
Most Likely cmn matrix
Actual CNOT
J. L. OBrien, G. J. Pryde, A. Gilchrist,D. F.
V. James, N. K. Langford, T. C. Ralph and A. G.
White, Quantum process tomography of a
controlled-NOT gate, Phys Rev Lett, submitted
(2004) quant-ph/0402166.
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Characterizing Processes
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Conclusions Quantum state tomography now
reduced to practice for a number of qubit
systems. How to characterize states Entropy,
Entanglement, and all that First steps in
tomography of processes how do you characterize
a process?
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Collaborators Prof. Paul G. Kwiat
(Illinois) Dr. Andrew White (Queensland)
Dr. Bill Munro (HP Bristol) LANL Postdocs
Dr. John Grondalski (now X-1) Dr. Sergey
Ponomarenko Recent LASS Students Mr.
David Etlinger, Rochester Mr. David Hume,
Kentucky Miss. Miho Ishibashi, Salisbury
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The end