Quantum State Tomography Finite Dimensional Infinite Dimensional (Homodyne) Quantum Process...
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Transcript of Quantum State Tomography Finite Dimensional Infinite Dimensional (Homodyne) Quantum Process...
QUANTUM TOMOGRAPHY WITH AN APPLICATION TO A CNOT GATE
OUTLINE
Quantum State Tomography Finite Dimensional Infinite Dimensional (Homodyne)
Quantum Process Tomography (SQPT) Application to a CNOT gate Related topics
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
QUANTUM STATE TOMOGRAPHY
Simply put:
Do this a lot
FINITE DIMENSIONAL SPACE
Typically easier to work with Know a priori how many coefficients to
expect The value of n is known
FINITE DIMENSIONAL SPACE
Easily approached via linear inversion Ei is a particular measurement outcome
projector S and T are linear operators
.
FINITE DIMENSIONAL SPACE
Use measured probabilities and invert to obtain density matrix Sometimes leads to nonphysical density
matrix!
.
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
QST FOR ONE QUBIT
Example from class: 1 qubit
Repeatedly measure sigma x
FINITE DIMENSIONAL SPACE
FOUND r1!
INFINITE DIMENSIONAL SPACE
The value of n is unknown!
Make multiple homodyne measurements Obtain Wigner function
Find density matrix
HOMODYNE MEASUREMENTS
Analogous to constructing 3d image from multiple 2d slices
Goal is to determine the marginal distribution of all quadratures
QUANTUM PROCESS TOMOGRAPHY
In QPT, “known quantum states are used to probe a quantum process to find out how the process can be described”
QUANTUM PROCESS TOMOGRAPHY
In essence:
QUANTUM PROCESS TOMOGRAPHY
In practice:
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
QPT
Choose set of operators: Use input states:
QPT
Form linear combination
Do QST to determine each
Write them as a linear combination of basis states
QPT
Solve for lambda Now write
And solve for beta (complex)
QPT
Combine to get
Which follows that for each k:
QPT
Define kappa as the generalized inverse of beta
And show that satisfies
QPT FOR A SINGLE QUBIT
OPERATORS BASIS
QPT FOR A SINGLE QUBIT
Use input states
Now QST on output
QPT FOR A SINGLE QUBIT
Use QST to determine
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:
QPT FOR A SINGLE QUBIT
Finally arriving to:
APPLICATION TO CNOT
J.L. O’Brien et al used photons and a measurement-induced Kerr-like non-linearity to create a CNOT gate
CNOT
QPT IN PRACTICE
Φa are input states Ψb are measurement analyzer setting cab is the number of coincidence detections
RESULTS
Average gate fidelity: 0.90 Average purity: 0.83 Entangling Capability: 0.73
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
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