qitd114 Hilbert Space Quantum Mechanics

1 qitd114 Hilbert Space Quantum MechanicsRobert B. GriffithsVersion of 16 January 2014 Contents1 Hilbert Space .. Qubit .. Intuitive picture .. Generald.. Kets as physical properties .. 42 Definition .. Dyads and completeness .. Matrices .. Dagger or adjoint .. Normal operators .. Hermitian operators .. Projectors .. Positive operators .. Unitary operators .. 93 Bloch sphere104 Composite systems and tensor Definition .. Product and entangled states .. Operators on tensor products .. Example of two qubits .. Multiple systems. Identical particles .. 13 References:CQT =Consistent Quantum Theoryby Griffiths (Cambridge, 2002), Ch. 2; Ch. 3; Ch. 4 except forSec. ; Ch. = Quantum Computation and Quantum Informationby Nielsen and Chuang (Cambridge, 2000).Secs. , ; through ; Hilbert Space In Quantum Mechanics the state of a physical system is represented by a vector in aHilbert Space : acomplex vector Space with an inner product.

2 The term Hilbert Space is often reserved for an infinite-dimensional inner product Space having theproperty that it is complete or closed. However, the term is often used nowadays, as in these notes, in a waythat includes finite-dimensional spaces, which automatically satisfy the condition of We will useDirac notationin which the vectors in the Space are denoted by|vi, called aket, wherevis some symbol which identifies the could equally well use something like~vorv. A multiple of a vector by a complex numbercis writtenasc|vi think of it as analogous toc~vofcv. In Dirac notation the inner product of the vectors|viwith|wiis writtenhv|wi. This resembles theordinary dot product~v ~wexcept that one takes a complex conjugate of the vector on theleft, thus think of~v ~ Qubit The simplest interesting Space of this sort is two-dimensional, which means that every vector in it canbe written as a linear combination of two vectors which form abasis for the Space .

3 In Quantum informationthestandard(orcomputational) basis vectors are denoted|0iand|1i, and it is assumed that both of themarenormalizedand that they are mutuallyorthogonalh0|0i= 1 =hw|wi,h0|1i= 0 =h1|0i.(1)(Note thathv|wi=hw|vi , soh0|1i= 0 suffices.) The notation|0iand|1iis intended to suggest an analogy, which turns out to be very useful, withan ordinary bit (binary digit) that takes the value 0 or 1. In Quantum information such a two-dimensionalHilbert Space , or the system it represents, is referred to asaqubit(pronounced cubit ). However, there aredisanalogies as well. Linear combinations like |0i+ |1imake perfectly good sense in the Hilbert Space ,and have a respectable physical interpretation, but there is nothing analogous for the two possible states 0and 1 of an ordinary bit. In Quantum Mechanics a two-dimensional complex Hilbert spaceHis used for describing the angularmomentum or spin of a spin-half particle (electron, proton, neutron, silver atom), which then provides aphysical representation of a qubit.

4 The polarization of a photon (particle of light) is also described byd= 2, so represents a qubit. A state or vector|visays something about one component of the spin of the spin half particle. Theusual convention is:|0i=|z+i Sz= +1/2,|1i=|z i Sz= 1/2,(2)whereSz, thezcomponent of angular momentum is measured in units of~. Here are some other correspon-dences: 2|x+i=|0i+|1i Sx= +1/2, 2|x i=|0i |1i Sx= 1/2, 2|y+i=|0i+i|1i Sy= +1/2, 2|y i=|0i i|1i Sy= 1/2.(3)The general rule is that ifwis a direction in Space corresponding to the angles and in polar coordinates,|0i+ei tan( /2)|1i Sw= +1/2,|0i ei cot( /2)|1i Sw= 1/2,(4)but see the comments below on Intuitive picture Physics consists of more than mathematics: along with mathematical symbols one always has a physical picture, some sort of intuitive idea or geometrical construction which aids in thinking about whatis going on in more approximate and informal terms than is possible using bare mathematics.

5 Most physicists think of a spin-half particle as something like a little top or gyroscope which is spinningabout some axis with a well-defined direction in Space , the direction of the angular momentum This physical picture is often very helpful, but there are circumstances in which it can mislead, as canany attempt to visualize the Quantum world in terms of our everyday experience. So one should be aware ofits limitations. In particular, the axis of a gyroscope has a very precise direction in Space , which is what makes suchobjects useful. But thinking of the spin of a spin-half particle as having a precise direction can mislead. Abetter (though by no means exact) physical picture is to think of the spin-half particle as having an angularmomentum vector pointing in a random direction in Space , butsubject to the constraint that a particularcomponent of the angular momentum, saySz, is positive, rather than negative.

6 Thus in the case of|0i, which meansSz= +1/2, think ofSxandSyas having random values. Strictlyspeaking these quantities are undefined, so one should not think about them at all. However, it is ratherdifficult to have a mental picture of an object spinning in three dimensions, but which has only one componentof angular momentum. Thus treating one component as definiteand the other two as random, while not anexact representation of Quantum physics, is less likely to lead to incorrect conclusions than if one thinks ofall three components as having well-defined values. An example of an incorrect conclusion is the notion that a spin-half particle can carry a large amountof information in terms of the orientation of its spin axis. To specify the orientation in Space of the axis of agyroscope requires on the order of log2(1/ ) + log2(1/ ) bits, where and are the precisions withwhich the direction is specified (in polar coordinates).

7 This can be quite a few bits, and in this sense thedirection along which the angular momentum vector of a gyroscope is pointing can contain or carry alarge amount of information. By contrast, the spin degree offreedom of a spin-half particle never carries orcontains more than 1 bit of information, a fact which if ignored gives rise to various misunderstandings Generald A Hilbert spaceHof dimensiond= 3 is referred to as aqutrit, one withd= 4 is sometimes called aququart, and the generic term for anyd >2 isqudit. We will assumed < to avoid complications whicharise in infinite-dimensional Hilbert spaces. A collection of linearly independent vectors{| ji}form abasisofHprovided any| iinHcan bewritten as a linear combination:| i=Xjcj| ji.(5)The numberdof vectors forming the basis is thedimensionofHand does not depend on the choice of basis. A particularly useful case is anorthonormal basis{|ji}forj= 1,2.

8 D, with the property thathj|ki= jk:(6)the inner product of two basis vectors is 0 forj6=k, , they areorthogonal, and equal to 1 forj=k, ,they arenormalized. If we write|vi=Xjvj|ji,|wi=Xjwj|ji,(7)where the coefficientsvjandwjare given byvj=hj|vi, wj=hj|wi,(8)the inner product can be written as(|vi) |wi=hv|wi=Xjv jwj,(9)3which can be thought of as the product of a bra vectorhv|= (|vi) =Xjv jhj|(10)with the ket vector|wi. (The terminology goes back to Dirac, who referred tohv|wias abracket.) For more on the operation, see below. It is often convenient to think of|wias represented by a column vector|wi= ,(11)andhv|by a row vectorhv|= (v 1, v 2, v d).(12)The inner product (9) is then the matrix product of the row times the column vector. Of course the numbersvjandwjdepend on the basis. The inner producthv|wi, however, is independentof the choice of Kets as physical properties In Quantum Mechanics , two vectors| iandc| i, wherecis anynonzerocomplex number haveexactly the samephysical significance.

9 For this reason it is sometimes helpful to say that the physical statecorresponds not to a particular vector in the Hilbert Space ,but to theray, or one-dimensional subspace,defined by the collection of all the complex multiples of a particular vector. One can always choosec(assuming| iis not the zero vector, but that never represents any physicalsituation) in such a way that the| icorresponding to a particular physical situation isnormalized,h | i= 1ork k= 1, where thenormk kof a state| iis the positive square root ofk k2=h | i,(13)and is zero if and only if| iis the (unique) zero vector, which will be written as 0 (and isnot to be confusedwith|0i). Normalized vectors can always be multiplied by aphase factor, a complex number of the formei where is real, without changing the normalization or the physicalinterpretation, so normalization by itself doesnot single out a single vector representing a particular physical state of affairs.

10 For many purposes it is convenient to use normalized vectors, and for this reason some students of thesubject have the mistaken impression thatanyvector representing a Quantum systemmustbe that is to turn convenience into legalism. There are circumstances in which (as we shall see) it is moreconvenient not to use normalized vectors. The state of a single qubit is always a linear combination of the basis vectors|0iand|1i,| i= |0i+ |1i,(14)where and are complex numbers. When 6= 0 this can be rewritten as |0i+ |1i= |0i+ ( / )|1i = |0i+ |1i , := / .(15)Since the physical significance of this state does not changeif it is multiplied by a (nonzero) constant, wemay multiply by 1and obtain a standard (unnormalized) form|0i+ |1i(16)4characterized by a single complex number . There is then a one-to-one correspondence between differentphysical states or rays, and complex numbers , if one includes = to signify the ray generated by|1i.