Quantum Noise and Quantum Operations

1. Quantum Noise and Quantum Operations Dan Ernst EECS 598 11/29/01

2. Outline • Background topics • Classical noise • Quantum operations • Closed vs. Open quantum systems • Operator-sum representation • Trace preservation • Quantum operation axioms • Freedom in the operator-sum representation

3. Density Matrix and Trace Operator • Quantum states can be expressed as a density matrix • Unitary operations on a density matrix are expressed as: • Trace of a matrix (sum of the diagonal elements) • Partial Trace (defined by linearity)

4. Classical Noise 1-p 0 0 p p 1 1 1-p

5. Quantum Operations • Quantum states transform as: • Simple Examples: • Unitary Transformation • Measurement Operation

6. Old: Closed Quantum Systems • Output of the system is determined by a unitary transformation on the input state U

7. New: Open Quantum Systems • Output of the system is determined by a unitary transformation on the principal systemand the environment. • Notice that the final state, e(r) might not be related by a unitary transformation to the initial state, r. In fact: U

8. Operator-sum Representation • We’d like a representation in terms of operators on the principal system’s Hilbert space alone. where is an operator on the principal state space.

9. Trace Preservation • In this model, the operation elements must satisfy the completeness relation: Since this relationship is true for all r it follows that:

10. Trace Preservation • This equation is satisfied by quantum operations which are trace-preserving. • When extra information about what occurred in the process is obtained by measurement, the quantum operation can be non-trace-preserving, that is:

11. Forget everything and start over

12. Axioms of Quantum Operations • We define a quantum operation e as a map from the set of density operators of the input space Q1 to the set for Q2 with the following three properties: • A1: is the probability that the process e occurs when r is the initial state. Thus, . • Note that, with this definition, the correctly normalized final state is: • This axiom makes coping with measurements easier.

13. Axioms of Quantum Operations • A2: e is a convex-linear map on the set of density matrices, that is, for probabilities {pi}, • A3: e is a completely positive map. That is, e(A) is positive for any positive operator A in Q1. Furthermore, this must hold for applying the map to any combined system RQ1.

14. The Axioms and Operator-sum Theorem 8.1: The map e satisfies axioms A1, A2, and A3if and only if: For some set of operators {Ei} which map the input Hilbert space to the output Hilbert space, and Proof: (Nielsen/Chuang, pages 368-370)

15. Unitary Freedom in Operator-sum Is operator-sum representation a unique description of an operation? (no, it’s not!)

16. Unitary Freedom in Operator-sum • When does this happen? Theorem 8.2: (Unitary Freedom in Operator-sum Representation) Let E and F be quantum operations with operation elements {E1,…,Em} and {F1,…,Fn} respectively. Fill shorter list with zeros so m=n. Then E = F if and only if: where uij is an mxm unitary matrix of complex numbers. • Can use this theorem to show that the number of elements (Ei) needed for an operator-sum representation is no more than d2, where d is the number of dimensions of the Hilbert space.