1. 8.4.2 Quantum process tomography 8.5 Limitations of the quantum operations formalism 量子輪講 2003 年 10 月 16 日 担当：徳本 晋 tokumoto@is.s.u-tokyo.ac.jp

2. Motivations How do quantum operations relate to experimentally measurable quantities? What measurements should an experimentalist do if they wish to characterize the dynamics of a quantum system? (Classical) System identification (Quantum) Quantum process tomography

3. Quantum state tomography The procedure of experimentally determining an unknown quantum state. Ex. Distinguish non-orthogonal quantum states like and with certainty. it is possible to estimate ρ if we have a large number of copies of ρ.

4. Case of a single qubit Suppose we have many copies of a single qubit density matrix ρ. The set forms an orthonormal set of matrices with respect to the Hilbert-Schmidt inner product, so ρ may be expanded as Expressions like tr(Aρ) have an interpretation as the average value of observables.

5. Average value of observations To estimate tr(Zρ) we measure the observable Z a large number of times, m, obtaining outcomes. Empirical average of these quantities,, is an estimate for the true value of tr(Zρ). It becomes approximately Gaussian with mean equal to tr(Zρ) and with standard deviation In similar way we can estimate tr(Xρ), tr(Yρ). Central limit theorem

6. Case of more than one qubit Similar to the single qubit case, an arbitrary density matrix on n qubits can be expanded as where the sum is over vectors with entries chosen from the set 0,1,2,3.

7. How can we use quantum state tomography to do quantum process tomography? dimension: d Chose d 2 pure quantum state so that the corresponding density matrices form a basis set. For each state we prepare and output from process. We use quantum state tomography to determine the state. Since the quantum operation ε is now determined by a linear extension of ε to all state, we are now done.

8. Way of determining a useful representation of ε Our goal is to determine a set of operation elements { E i } for ε, To determine the E i from measurable parameters, we consider an equivalent description of ε using a fixed set of operators, for some set of complex numbers e im. So, where. Entries of a positive Hermitian matrix χ chi matrix representation

9. χ will contain d 4 -d 2 independent real parameters, because a general linear map of d by d matrices to d by d matrices is described by d 4 independent parameters, but there are d 2 additional constraints due to the fact that ρ remains Hermitian with trace one.

10. Any d × d matrix can be written as a linear combination of the basis : fixed, linearly independent basis for the space of d × d matrices. any d × d matrix can be written as a unique linear combination of the. Input: Thus, it is possible to determine by state tomography. Linear combination of the basis

11. Calculating and Each may be expressed as a linear combination of the basis states, and since is known from the state tomography, can be determined by standard linear algebra algorithms. To proceed, we may write where are complex numbers which can be determined by standard algorithms from linear algebra.

12. Combining equations Combining the last two expressions and (8.152) we have From the linear independence of the it follows that each k, This relation is a necessary and sufficient condition for the matrix χ to give the correct quantum operation ε.

13. Calculating χ One may think of χ and λ as vectors, and β as a d 4 × d 4 matrix with columns indexed by mn, and rows by jk. To show how χ may be obtained, let κ be the generalized inverse for the matrix β, satisfying relation Most computer packages for matrix manipulation are capable of finding such generalized inverses. We now prove that χ defined by satisfies the relation (8.158).

14. Calculating E i Let the unitary matrix diagonalize χ, From this it can easily be verified that are operation elements for ε.

15. Process tomography for a single qubit (1) We use input states: output states: These correspond to, where We may determine β, and similarlydetermines λ.

16. Process tomography for a single qubit (2) However, due to the particular choice of basis, and the Pauli matrix representation of, we may express the β matrix the Kronecker product, where so that χ may be expressed conveniently as in terms of block matrices.

17. 8.5 Limitations of the quantum operations formalism Motivation: Are there interesting quantum systems whose dynamics are not described by quantum operations? In this section, we will construct an artificial example of a system whose evolution is not described by a quantum operation, and try to understand the circumstances under which this is likely to occur.

18. an artificial example of a system whose evolution is not described by a quantum operation A single qubit ρ is prepared in some unknown quantum state. The preparation of this qubit involves certain procedures to be carried out in the laboratory in which the qubit prepared. The state of the system after preparation is if ρ is a state on the bottom half of the Bloch sphere, and if ρ is a state on the top half of the Bloch sphere. This process is not an affine map acting on the Bloch sphere, and therefore it cannot be a quantum operation.

19. the circumstances under which this is likely to occur Quantum system which interacts with the degrees of freedom used to prepare that system after the preparation is complete will in general suffer a dynamics which is not adequately described within the quantum operations formalism.

20. Summary of Chapter 8 The operator-sum representation Environmental models for quantum operations Quantum process tomography Operation elements for important single qubit quantum operations