2.  Quantum State Tomography  Finite Dimensional  Infinite Dimensional (Homodyne)  Quantum Process Tomography (SQPT)  Application to a CNOT gate  Related topics

3.  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

4. Simply put: Do this a lot

5.  Typically easier to work with  Know a priori how many coefficients to expect  The value of n is known

6.  Easily approached via linear inversion  E i is a particular measurement outcome projector  S and T are linear operators.

7.  Use measured probabilities and invert to obtain density matrix  Sometimes leads to nonphysical density matrix!.

8.  “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

9.  Example from class: 1 qubit  Repeatedly measure sigma x

10.  FOUND r 1 !

11.  The value of n is unknown!  Make multiple homodyne measurements  Obtain Wigner function  Find density matrix

12.  Analogous to constructing 3d image from multiple 2d slices  Goal is to determine the marginal distribution of all quadratures

13.  In QPT, “known quantum states are used to probe a quantum process to find out how the process can be described”

14.  In essence:

15.  In practice:

16.  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”  A m are a basis for operators acting on ρ

17.  Choose set of operators:  Use input states:

18.  Form linear combination  Do QST to determine each  Write them as a linear combination of basis states

19.  Solve for lambda  Now write  And solve for beta (complex)

20.  Combine to get  Which follows that for each k:

21.  Define kappa as the generalized inverse of beta  And show thatsatisfies

22. OPERATORS BASIS

23.  Use input states  Now QST on output

24.  Use QST to determine

25.  Results correspond to  Now beta and lambda can be determined, but due to the particular basis choice and the Pauli matrices:

26.  Finally arriving to:

27.  J.L. O’Brien et al used photons and a measurement-induced Kerr-like non-linearity to create a CNOT gate

29.  Φ a are input states  Ψ b are measurement analyzer setting  c ab is the number of coincidence detections

30.  Average gate fidelity:0.90  Average purity:0.83  Entangling Capability:0.73

31.  Ancilla-Assisted Process Tomography (AAPT)  d 2 separable inputs can be replaced by a suitable single input state from a d 2 -dimensional Hilbert space  Entanglement-Assisted Process Tomography (EAPT)  Need another copy of system  Tangle

32.  “Quantum Process Tomography of a Controlled- NOT Gate”  http://quantum.info/andrew/publications/2004/q pt.pdf  Quantum Computation and Quantum Information  Michael A. Nielsen & Isaac L. Chuang  Wikipedia