1. Advanced Computer Architecture Lab University of Michigan Quantum Operations and Quantum Noise Dan Ernst Quantum Noise and Quantum Operations Dan Ernst EECS 598 11/29/01

2. Advanced Computer Architecture Lab University of Michigan Quantum Operations and Quantum Noise Dan Ernst 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. Advanced Computer Architecture Lab University of Michigan Quantum Operations and Quantum Noise Dan Ernst 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. Advanced Computer Architecture Lab University of Michigan Quantum Operations and Quantum Noise Dan Ernst Classical Noise 0 1 0 1 1-p p p

5. Advanced Computer Architecture Lab University of Michigan Quantum Operations and Quantum Noise Dan Ernst Quantum Operations Quantum states transform as: Simple Examples: –Unitary Transformation –Measurement Operation

6. Advanced Computer Architecture Lab University of Michigan Quantum Operations and Quantum Noise Dan Ernst Old: Closed Quantum Systems Output of the system is determined by a unitary transformation on the input state U

7. Advanced Computer Architecture Lab University of Michigan Quantum Operations and Quantum Noise Dan Ernst New: Open Quantum Systems Output of the system is determined by a unitary transformation on the principal system and the environment. Notice that the final state,  (  ) might not be related by a unitary transformation to the initial state, . In fact: U

8. Advanced Computer Architecture Lab University of Michigan Quantum Operations and Quantum Noise Dan Ernst 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. Advanced Computer Architecture Lab University of Michigan Quantum Operations and Quantum Noise Dan Ernst Trace Preservation In this model, the operation elements must satisfy the completeness relation : Since this relationship is true for all  it follows that:

10. Advanced Computer Architecture Lab University of Michigan Quantum Operations and Quantum Noise Dan Ernst 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. Advanced Computer Architecture Lab University of Michigan Quantum Operations and Quantum Noise Dan Ernst Forget everything and start over

12. Advanced Computer Architecture Lab University of Michigan Quantum Operations and Quantum Noise Dan Ernst Axioms of Quantum Operations We define a quantum operation  as a map from the set of density operators of the input space Q 1 to the set for Q 2 with the following three properties: – A1 : is the probability that the process  occurs when  is the initial state. Thus,. Note that, with this definition, the correctly normalized final state is: This axiom makes coping with measurements easier.

13. Advanced Computer Architecture Lab University of Michigan Quantum Operations and Quantum Noise Dan Ernst Axioms of Quantum Operations – A2 :  is a convex-linear map on the set of density matrices, that is, for probabilities {p i }, – A3 :  is a completely positive map. That is,  (A) is positive for any positive operator A in Q 1. Furthermore, this must hold for applying the map to any combined system RQ 1.

14. Advanced Computer Architecture Lab University of Michigan Quantum Operations and Quantum Noise Dan Ernst The Axioms and Operator-sum Theorem 8.1 : The map  satisfies axioms A1, A2, and A3 if and only if : For some set of operators {E i } which map the input Hilbert space to the output Hilbert space, and Proof : (Nielsen/Chuang, pages 368-370)

15. Advanced Computer Architecture Lab University of Michigan Quantum Operations and Quantum Noise Dan Ernst Unitary Freedom in Operator-sum Is operator-sum representation a unique description of an operation? (no, it’s not!)

16. Advanced Computer Architecture Lab University of Michigan Quantum Operations and Quantum Noise Dan Ernst 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 {E 1,…,E m } and {F 1,…,F n } respectively. Fill shorter list with zeros so m = n. Then E = F if and only if: where u ij is an m x m unitary matrix of complex numbers. Can use this theorem to show that the number of elements (E i ) needed for an operator-sum representation is no more than d 2, where d is the number of dimensions of the Hilbert space.