Chapter 9 Density Matrices - univie.ac.at

1 Chapter 9 Density MatricesIn this Chapter we want to introduce Density Matrices , also called Density operators,which conceptually take the role of the state vectors discussed so far, as they encode allthe (accessible) information about a quantum mechanical system. It turns out that the pure states, described by state vectors1| on Hilbert space, are idealized descriptionsthat cannot characterize statistical (incoherent) mixtures, which often occur in the ex-periment, in Nature. These objects are very important for the theory of quantuminformation and quantum communication. More detailed information about the densitymatrix formalism can be found in [17]. General Properties of Density MatricesConsider an observableAin the pure state| with the expectation value given by A = |A| ,( )then the following definition is obvious:Definition matrix for the pure state| is given by :=| |This Density matrix has the following properties:I) 2= projector( )II) = hermiticity( )III) Tr = 1 normalization( )IV) 0positivity( )1 Remark for experts: It is possible to find a vector representation for every given quantum mechanicalstate, even those represented by a Density matrix.

2 This can be done via the so-called GNS (Gelfand-Neumark-Segal) construction. This vector representation need not, however, be of any practical formand the concept of the Density matrix is therefore 9. Density MATRICESThe first two properties follow immediately from Definition and property III) canbe verified using the definition of the trace operation for an arbitrary operatorD:Definition an operatorDis given byTrD:= n n|D|n where{|n }is an arbitrary take the operatorD=| |and calculate its traceTrD= n n| |n = n |n n| 1 = | .( )Property IV) means that the eigenvalues of are greater or equal to zero, which canalso be expressed as | | = | | =| | |2 0,( )which is an important property because probabilities are always greater or equal to we still have to ensure is that the expectation value of an observable in the state| can be reproduced, which we will formulate in the following theorem:Theorem expectation value of an observableAin a state,represented by a Density matrix , is given by A = Tr ( A)Proof:Tr ( A) = Tr (| |A) = n n| |A|n == n |A|n n| 1 = |A| = A.

3 ( ) Pure and Mixed StatesNow we can introduce a broader class of states represented by Density Matrices , the so-calledmixed statesin contrast to the states we have considered until now, the so-calledpure PURE AND MIXED Pure StatesLet s begin with the pure states. Consider an ensemble of given objects in the states{| i }. If all the objects are in the same state, the ensemble is represented by apurestate. To make probabilistic statements the whole ensemble of identically prepared sys-tems must be the system be, , in the state| which we can expand with respect to theeigenstates of an (hermitian) operatorA| = ncn|n ,whereA|n =an|n .( )The expectation value is then given by A = n|cn|2an= nNnNan,( )where|cn|2is the probability to measure the eigenvaluean.

4 It corresponds to the frac-tionNn/N, the incidence the eigenvalueanoccurs, whereNnis the number of times thiseigenvalue has been measured out of an ensemble state is characterized by a Density matrix of the form of Definition , withthe properties I) - IV) (Eqs. ( ) - ( )), where we can combine property I) and III) toconcludeTr 2= 1.( ) Mixed StatesLet us next study the situation where not all of theNsystems (objects) of the ensemble arein the same state, are in the state| i respectively, such that Ni= probabilitypito find an individual system of the ensemble described by the state| i is then given bypi=NiN,where ipi= 1.( )We can thus write down themixed stateas a convex sum, a weighted sum with ipi= 1, of pure state Density Matrices mix= ipi purei= ipi| i i|.

5 ( )The expectation value is again given by Theorem , A mix= Tr ( mixA),( )162 Chapter 9. Density Matrices where we can express the expectation value of the mixed state as a convex sum of expec-tation values of its constituent pure states, A mix= ipi i|A| i .( )Proof:Tr ( mixA) = Tr( ipi| i i|A)== n ipi n| i i|A|n == ipi i|A n|n n| 1 i == ipi i|A| i . ( )Properties II) - IV) (Eqs. ( ) - ( )) are still valid for mixed states, but property I)does no longer hold 2mix= i jpipj| i i| j ij j|= ip2i| i i| 6= mix,( )where we, , assumed that| i and| j are orthonormal. We can then calculatethe trace of 2, which, in contrast to pure states, is no longer equal to 1 but smallerTr 2mix= n n| i jpipj| i i| j j|n == i jpipj i| j j| n|n n| i == i jpipj| i| j |2== ip2i< ipi= 1.

6 ( )The last step in this calculation is obvious, since 0 pi 1 and thereforep2i pi. Weconclude that the trace of 2is a good measure for the mixedness of a Density matrix,since it is equal to 1 for pure states and strictly smaller than 1 for mixed states. For amaximally mixed statewe have for a given dimensiondof the systemTr 2mix=1d>0.( ) TIME EVOLUTION OF Density Time Evolution of Density MatricesWe now want to find the equation of motion for the Density matrix. We start from thetime dependent Schr odinger equation and its hermitian conjugatei~ t| =H| i~ t |= |H .( )Then we differentiate the Density matrix of a mixed state (Eq. ( )) with respect totime, we multiply it byi~and combine this with Eq.

7 ( )i~ t =i~ ipi( | i i~H| i i|+| i i| i~ i|H) == ipi(H purei pureiH) == [H , ].( )Theorem Matrices satisfy thevon Neumann Equationi~ t = [H , ]The von Neumann equation is the quantum mechanical analogue to the classical Liouvilleequation, recall the substitution ( ).The time evolution of the Density matrix we can also describe by applying an unitaryoperator, thetime shift operatorU(t,t0), also calledpropagatorU(t,t0) =e i~H(t t0).( )It allows us to relate the Density matrix at a later timetto the Density matrix at someearlier timet0 (t) =U(t,t0) (t0)U (t,t0).( )Furthermore, it helps us to prove, for instance, that themixednessTr 2of a densitymatrix is time independentTr 2(t) = Tr (U (t0)U U 1 (t0)U ) = Tr ( (t0) (t0)U U 1) = Tr 2(t0),( )where we used the cyclicity of the trace 9.

8 Density MATRICESE xample: Density matrix for spin12 Generally, this will be a 2 2 matrix that can be written as linear combination of theidentity1and the Pauli Matrices x, yand z, as =12(1+~a~ ).( )The coefficient~ais named theBloch vectorand can be calculated as the expectationvalue of the Pauli Matrices ~a= Tr ( ~ ) = ~ .( )All spin12density Matrices lie on or within the so-calledBloch sphere(with radius~a= 1) and are determined by the Bloch vector~a. The length of the Bloch vector thustells us something about the mixedness, the polarization of an ensemble, of a beamof spin12particles, electrons or neutrons. We say the beam is polarized ifai= 1 andcompletely unpolarized ifai= 0 , for alli.

9 This means that pure and mixed states canbe characterized via the Bloch vector in the following waypure state 2= |~a|= 1( )mixed state 26= |~a|<1.( )A totally mixed state (ai= 0 for alli) can then be written as mix=12(| |+| |) =121,( )such thatTr mix= 1 and Tr 2mix=12.( )Remark:Note, the decomposition ( ) into up| |and down| |states isby no means unique, we can achieve the totally mixed state mix=121in many differentways.