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This 2 page Class Notes was uploaded by Santos Fadel on Monday October 19, 2015. The Class Notes belongs to CSCI 4967 at Rensselaer Polytechnic Institute taught by Staff in Fall. Since its upload, it has received 31 views. For similar materials see /class/224851/csci-4967-rensselaer-polytechnic-institute in ComputerScienence at Rensselaer Polytechnic Institute.
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Date Created: 10/19/15
Quantum Computationquot Lecture 10 Quantum Noise 11 Notes taken by Brandon Belew November 2007 Summary This lecture covers ve main areas 1 example of quantum noise 2 the subsystem point of View 3 the density matrix as a fundamental object 4 example of desnity matrix and quantum teleporta tion and 5 delity measures 1 Example of Quantum Noise The point of the rst section was that working with the desnity matrix saves a lot of work Working with state vectors would be complicated For instance think of applying the X gate with probability p and probability 1 7 p that the not gate fails in which case nothing happens Then we have P ijl jgtlt jl gt 2P1PXl jgtlt lePj1 Pl jgtlt jl PX X1P7 We can also ask how good a not gate this is We compare ideal output X lzbj to actual output This brings us to the notion mentioned in the outline as 5 of delity measures Fidelity is a measure of how similar two states are and is de ned as Fayb lltalbgtl This can range from O totally dissimilar to 1 the same To compare la to a Ej pjl jgtlt jl we compute the delity Faa wltalalagti Actual output is for a given state Thus we have FXl gt7El gtlt l lt lXEl gt WWW 10 1 PXWXWV lt Fidelity ranges from E for l gt 0 and 1 for l gt My 2 2 The Subsystem PointofView The density matrix characterizes a sybsystem of a larger system For instance in a multistate entangled system we can t talk about each state individually but we can assign a density matrix Summarizing the examples talked about in this section we can say that given a state l gt of systems A and B and a measurement de ned by a projector Pj that acts on A alone tTPj gtlt pA gives the measurement statistics where m E ZakzazmlWl E trBltl gtlt lgt klm Lecture Notes for a course given by Stephen F Bush at RPI is the reduced density matrix of system A Alternatively we can de ne tTBla1gtlta2l lblgtltb2lEla1gtlta2lgtltWl171gtltb2lgtltb2lblgtla1gtlta2l This de nition can be extended linearly to arbitrarily many matrices 3 The Density Matrix as a Fundamental Object This section of the lecture dealt with reformulating the Four Postulates in terms of the density matrix 4 An Example of the Density Matrix and Quantum Teleportation Using the density matrix we can show why quantum teleportation does not allow fasterthanlight FTL travel The initial state is l gt ioobgm B s initial reduced density matrix is just the reduced density matrix for a Bell state PB It was shown that Bob s nal reduced density matrix after Alice applies her transformations is the same as be fore Thus the statistics of any measurement he can do will be the same after Alice s initial measurement as before Thus there is no possible ow of information 5 Fidelity Measures This material was covered in the rst section
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