More Quantum Noise and Distance Measures for Quantum Information (Some of Ch8 and Ch 9)OutlineBackground – The Bloch SphereExamples of Quantum Noise – Bit FlipSlide 5Examples of Quantum Noise - Phase FlipExamples of Quantun Noise – Phase FlipExamples of Quantum Noise – Bit-phase FlipSlide 9Examples of Quantum Noise – Depolarizing ChannelExamples of Quantum Noise – Depolarizing ChannelSlide 12Examples of Quantum Noise – Amplitude DampingSlide 14Slide 15Slide 16Examples of Quantum Noise – Phase DampingSlide 18Box 8.4 – Why Shrodinger’s Cat Doesn’t WorkDistance measures for Probability DistributionsSlide 21Distance measures for Quantum StatesTrace Preserving Quantum Operations are ContractiveFidelity of Two Quantum StatesUhlmann’s TheoremTurning Fidelity into a MetricMonotonicity of FidelityConclusionsAdvanced Computer Architecture LabUniversity of MichiganQuantum Noise and DistancePatrick CasslemanMore Quantum Noise and Distance More Quantum Noise and Distance Measures for Quantum Information Measures for Quantum Information (Some of Ch8 and Ch 9)(Some of Ch8 and Ch 9)Patrick CasslemanEECS 59811/29/01Advanced Computer Architecture LabUniversity of MichiganQuantum Noise and DistancePatrick CasslemanOutlineOutline•Types of Quantum Noise–Bit Flip–Phase Flip–Bit-phase Flip–Depolarizing Channel–Amplitude Damping–Phase Damping•Distance measures for Probability Distributions•Distance measures for Quantum StatesAdvanced Computer Architecture LabUniversity of MichiganQuantum Noise and DistancePatrick CasslemanBackground – The Bloch SphereBackground – The Bloch Sphere•Remember :•The numbers  and  define a point on the unitthree-dimensional sphere12sin02cosie01Advanced Computer Architecture LabUniversity of MichiganQuantum Noise and DistancePatrick CasslemanExamples of Quantum Noise – Bit FlipExamples of Quantum Noise – Bit Flip•A bit flip channel flips the state of a qubit from |0> to |1> with probability 1-p•Operation Elements: 011011100110pXpEpIpEAdvanced Computer Architecture LabUniversity of MichiganQuantum Noise and DistancePatrick CasslemanExamples of Quantum Noise – Bit FlipExamples of Quantum Noise – Bit Flip•Bloch sphere representation:–Before -AfterxzyAdvanced Computer Architecture LabUniversity of MichiganQuantum Noise and DistancePatrick CasslemanExamples of Quantum Noise - Phase FlipExamples of Quantum Noise - Phase Flip•Corresponds to a measurement in the |0>, |1> basis, with the result of the measurement unknown•Operation Elements: 100111100110pZpEpIpEAdvanced Computer Architecture LabUniversity of MichiganQuantum Noise and DistancePatrick CasslemanExamples of Quantun Noise – Phase FlipExamples of Quantun Noise – Phase Flip•Bloch vector is projected along the z axis, and the x and y components of the Bloch vector are lostAdvanced Computer Architecture LabUniversity of MichiganQuantum Noise and DistancePatrick CasslemanExamples of Quantum Noise – Bit-phase FlipExamples of Quantum Noise – Bit-phase Flip•A combination of bit flip and phase flip•Operation Elements:0011100110iipYpEpIpEAdvanced Computer Architecture LabUniversity of MichiganQuantum Noise and DistancePatrick CasslemanExamples of Quantum Noise – Bit-phase FlipExamples of Quantum Noise – Bit-phase Flip•Bloch vector is projected along y-axis, x and z components of the Bloch vector are lostAdvanced Computer Architecture LabUniversity of MichiganQuantum Noise and DistancePatrick CasslemanExamples of Quantum Noise – Depolarizing Examples of Quantum Noise – Depolarizing ChannelChannel•Qubit is replaced with a completely mixed state I/2 with probability p, it is left untouched with probability 1-p•The state of the quantum system after the noise is:)1(2)( ppIE Advanced Computer Architecture LabUniversity of MichiganQuantum Noise and DistancePatrick CasslemanExamples of Quantum Noise –Examples of Quantum Noise – Depolarizing Channel Depolarizing Channel•The Bloch sphere contracts uniformlyAdvanced Computer Architecture LabUniversity of MichiganQuantum Noise and DistancePatrick CasslemanExamples of Quantum Noise –Examples of Quantum Noise –Depolarizing ChannelDepolarizing Channel•Quantum Circuit Representation (1-p)|  |+p|  1 1Advanced Computer Architecture LabUniversity of MichiganQuantum Noise and DistancePatrick CasslemanExamples of Quantum Noise – Examples of Quantum Noise – Amplitude DampingAmplitude Damping•Noise introduced by energy dissipation from the quantum system–Emitting a photon•The quantum operation:1100)( EEEEEADAdvanced Computer Architecture LabUniversity of MichiganQuantum Noise and DistancePatrick CasslemanExamples of Quantum Noise – Examples of Quantum Noise – Amplitude DampingAmplitude Damping•Operation Elements:• can be thought of as the probability of losing a photon•E1 changes |1> into |> - i.e. losing energy•E leaves |> alone, but changes amplitude of |1>000100110EE2sinAdvanced Computer Architecture LabUniversity of MichiganQuantum Noise and DistancePatrick CasslemanExamples of Quantum Noise – Examples of Quantum Noise – Amplitude DampingAmplitude Damping•Quantum Circuit Representation:inout0)(yRAdvanced Computer Architecture LabUniversity of MichiganQuantum Noise and DistancePatrick CasslemanExamples of Quantum Noise –Examples of Quantum Noise – Amplitude Damping Amplitude Damping•Bloch sphere Representation:•The entire sphere shrinks toward the north pole, |0>Advanced Computer Architecture LabUniversity of MichiganQuantum Noise and DistancePatrick CasslemanExamples of Quantum Noise – Phase DampingExamples of Quantum Noise – Phase Damping•Describes the loss of quantum information without the loss of energy•Electronic states perturbed by interacting with different charges•Relative phase between energy eigenstates is lost•Random “phase kick”, which causes non diagonal elements to exponentially decay to 0•Operation elements:•  = probability that photon scattered without losing energy10010E0001EAdvanced Computer