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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

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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

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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)

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

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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

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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

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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

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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.

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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:

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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:

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

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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.

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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.

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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)

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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!)

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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.

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