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Quantum Mechanics

by: Marjorie Hahn

Quantum Mechanics PHYSICS 137B

Marjorie Hahn

GPA 3.95

J. Moore

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J. Moore
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This 3 page Class Notes was uploaded by Marjorie Hahn on Thursday October 22, 2015. The Class Notes belongs to PHYSICS 137B at University of California - Berkeley taught by J. Moore in Fall. Since its upload, it has received 20 views. For similar materials see /class/226692/physics-137b-university-of-california-berkeley in Physics 2 at University of California - Berkeley.

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Date Created: 10/22/15
Physics 137B Quantum mechanics ll7 Fall 2007 Lecture on quantum statistical mechanics and entanglement This lecture studies some examples of the density matrix formalism for quantum statistical mechanics We will see that the familiar singlet state of two particles has a property known as entanglement that is quite surprising from a classical point of view Entanglement is one of the basic notions of quantum information 7 Lightning review of the density matrix formalism The density operator is explicitly written as N p Z lagtWaltal7 1 041 where la are some normalized states not necessarily orthogonal or complete This is now shown to reproduce the sort of statistical average discussed above Let s take an operator A and ask about its statistical expectation In a particular orthonormal basis7 the matrix representation of p is N W ltnipwgt 2ltniagtltawgtm lt2 041 Now TFPA anm An m Z We can simplify this greatly using the completeness relation for the basis completeness requires 2 WW 1 4 V L Then in the above sum7 both the sums over 71 and 71 just give unity7 leaving TrpA ZWaltozlAlozgt 5 Some simple properties of the density matrix that follow from the above de nition are Tr p 1 6 and all diagonal elements are nonnegative7 since the diagonal elements are just the probabilities of being in different pure states We also showed that for a pure state7 p2 p You might ask7 given the density matrix7 how to express the entropy of a quantum system The logical de nition is the von Neumann entropy7 de ned if we want to count entropy dimensionlessly7 in bits as 59 Trplogz p 7 For a diagonal density matrix with equal probabilities this is a mixed state this reduces to the classical entropy up to a constant Any pure quantum mechanical state has entropy 07 since a pure state can be converted by a change of basis to a matrix with diagonal elements 17 07 70 This is connected to some recent developments in the theory of entanglemen 7 of quantum systems Suppose that a quantum system is made up of two subsystems A and B and that the whole system AB is in a pure state p wow lt8 More precisely the full system s Hilbert space is a product of A and B Hilbert spaces We can de ne the reduced density matrix for subsystem A by a partial trace over subsystem B lt gt1lpAl gt2gt Zltlt gt1l X lt ilgtl gtlt lltl gt2gt X WW 9 739 Here the sum is over a basis of the B Hilbert space This reduced density matrix can give us the results of any measurement of an operator that is of the form 0A X 13 ie that can be thought of a measurement on subsystem A To see this rst note that lt0A lBgt WHOx lBl gt Zlt lltl gtigt l jgtgtltlt il lt jlgtlt A 13W 10 Here in the second step we have inserted a version of the identity operator made from the product basis states recall that for an orthonormal basis 2k 1 Applying the same process again to insert another copy of the identity operator we get 2 lt lltl gtigt l jgtgtltlt il lt 1lgtlt0A 13l gtkgt l lgtgtltlt kl lt gtzlgtl gt ijkl 20m DzAm X lt jlgtl gtlt lltl igt X 1 20mph TWOAPA 11 Lk jl Lk where we have dropped the A and B subscripts in the middle equations Note that this can be a mixed density matrix even if we started from a pure state for the whole system As an example consider the singlet state TT 7 l ll for a state of two spin half particles The reduced density matrix for either particle is found to be pi pB 132 13 12 We can con rm by a calculation do calculation for singlet that this gives 0 for a product state 11 l gt1li gt2 and 1 for a fully entangled state of two qubits quantum bits ie quantum two state systems For example a singlet H 7 l But thinking more about the singlet state we seem to have found a physically inconsistent result The entropy of the whole system is 0 because it is in a pure state but if we can only perform measurements on one spin then the density matrix describing those measurements has one bit of entropy Has the physics somehow changed because we only look at one part of the system What does it mean if a part of a system looks like a mixed state if in fact the whole system is in a pure state This type of question was rst asked by Einstein Podolsky and Rosen in a famous paper in the early days of quantum mechanics Their idea more precisely was to create a singlet pair of particles there are indeed physical processes that tend to create singlets and then spatially separate the particles The fact that their spins remain in the singlet indicates some type of correlation between the particles for example if the state of one say up or down along the z axis is known then the state of the other is known In fact one can make a stronger statement than this the standard interpretation of measurement in quantum mechanics the Copenhagen interpretation is that the state of the two spin system actually changes instantaneously as a result of measuring one spin Since the two spin system may be extended over a long region of space Einstein worried that this instantaneous action at a distance must violate the basic ideas of relativitiy A partial resolution to this puzzle can be obtained by asking if any measurement on subsystem A can tell whether a measurement of 51 has taken place on subsystem B The density matrix of the whole system changes as a result of the measurement it goes from the singlet a pure state to the mixed state with p 12 to be l TALE and p 12 to be l UTE However the reduced density matrix for subsystem A is 12 both before and after the measurement Hence density matrices give us a quick way to see that no observer of only A can tell whether B has been measured and that whatever change in the state has occured as a result of the measurement of B is not detectable locally at A


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