quantum state tomography finite dimensional infinite dimensional (homodyne) quantum process...
TRANSCRIPT
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QUANTUM TOMOGRAPHY WITH AN APPLICATION TO A CNOT GATE
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OUTLINE
Quantum State Tomography Finite Dimensional Infinite Dimensional (Homodyne)
Quantum Process Tomography (SQPT) Application to a CNOT gate Related topics
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QUANTUM STATE TOMOGRAPHY
QST “is the process of reconstructing the quantum state (density matrix) for a source of quantum systems by measurements on the system coming from the source.”
The source is assumed to prepare states consistently
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QUANTUM STATE TOMOGRAPHY
Simply put:
Do this a lot
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FINITE DIMENSIONAL SPACE
Typically easier to work with Know a priori how many coefficients to
expect The value of n is known
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FINITE DIMENSIONAL SPACE
Easily approached via linear inversion Ei is a particular measurement outcome
projector S and T are linear operators
.
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FINITE DIMENSIONAL SPACE
Use measured probabilities and invert to obtain density matrix Sometimes leads to nonphysical density
matrix!
.
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MAXIMUM LIKELIHOOD ESTIMATION
“the likelihood of a set of a parameter values given some observed outcomes is equal to the probability of those observed outcomes given those parameter values”
The likelihood of a state is the probability that would be assigned to the observed results had the system been in that state
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QST FOR ONE QUBIT
Example from class: 1 qubit
Repeatedly measure sigma x
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FINITE DIMENSIONAL SPACE
FOUND r1!
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INFINITE DIMENSIONAL SPACE
The value of n is unknown!
Make multiple homodyne measurements Obtain Wigner function
Find density matrix
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HOMODYNE MEASUREMENTS
Analogous to constructing 3d image from multiple 2d slices
Goal is to determine the marginal distribution of all quadratures
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QUANTUM PROCESS TOMOGRAPHY
In QPT, “known quantum states are used to probe a quantum process to find out how the process can be described”
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QUANTUM PROCESS TOMOGRAPHY
In essence:
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QUANTUM PROCESS TOMOGRAPHY
In practice:
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QPT
J.L. O’Brien: “The idea of QPT is to determine a completely positive map ε, which represents the process acting on an arbitrary input state ρ”
Am are a basis for operators acting on ρ
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QPT
Choose set of operators: Use input states:
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QPT
Form linear combination
Do QST to determine each
Write them as a linear combination of basis states
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QPT
Solve for lambda Now write
And solve for beta (complex)
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QPT
Combine to get
Which follows that for each k:
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QPT
Define kappa as the generalized inverse of beta
And show that satisfies
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QPT FOR A SINGLE QUBIT
OPERATORS BASIS
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QPT FOR A SINGLE QUBIT
Use input states
Now QST on output
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QPT FOR A SINGLE QUBIT
Use QST to determine
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QPT FOR A SINGLE QUBIT
Results correspond to
Now beta and lambda can be determined, but due to the particular basis choice and the Pauli matrices:
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QPT FOR A SINGLE QUBIT
Finally arriving to:
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APPLICATION TO CNOT
J.L. O’Brien et al used photons and a measurement-induced Kerr-like non-linearity to create a CNOT gate
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CNOT
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QPT IN PRACTICE
Φa are input states Ψb are measurement analyzer setting cab is the number of coincidence detections
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RESULTS
Average gate fidelity: 0.90 Average purity: 0.83 Entangling Capability: 0.73
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RELATED TOPICS
Ancilla-Assisted Process Tomography (AAPT) d2 separable inputs can be replaced by a suitable
single input state from a d2-dimensional Hilbert space
Entanglement-Assisted Process Tomography (EAPT) Need another copy of system
Tangle
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SOURCES
“Quantum Process Tomography of a Controlled-NOT Gate” http://quantum.info/andrew/publications/
2004/qpt.pdf Quantum Computation and Quantum
Information Michael A. Nielsen & Isaac L. Chuang
Wikipedia