GEOMETRIC ALGEBRA: Imaginary Numbers Are Real Timothy F. Havel (Nuclear Engineering) On spin & spinors: LECTURE The operators for the x, y & z components of the angular momentum of a spin / particle (in units of h) are usually represented by one half the Pauli matrices: s 0 s 0 i s 3 0 0 i 0 0 These are easily seen to anticommute & square to the identity, and hence correspond to orthonormal vectors in a matrix representation for G( 3 ). The state of a spin is more commonly specified by a spinor (unit vector) yò in a -D Hilbert space H over, in which case the expected components of the angular momentum are -- y s k yò 4 -- tr( Ys k ) -- tr( Ys k + s k Y ) -- tr( ( Y s k ) ) Y s k where Y yò y, i.e. the inner product of Y, s k Œ G( 3 )! Jan. 8, 00 L ECTURE # of 8
Since Y s k is a scalar, Y itself must be the sum of a scalar & a vector, and the scalar part is YÒ 0 -- tr( Y ) -- y yò --. Using the fact that the angular momentum is quantized along its own axis to, the vector part can be written as -- Rs, 3 R i.e. as a rotation of s 3 by some spinor R Œ G + ( 3 ). It follows that Y R( or, using the correspondence between -- ( + s 3 ))R G + ( 3 ) and SU( ) matrices: Y a b* 0 a* b* a a* b* a ab* yò y b a* 0 0 b a b ba* b Thus we can identify yò with y RE +, where E + -- ( + s is 3 ) an idempotent ( E + E + ). In fact, since the nd column of SU( ) matrices is determined by the first, we can just use R! Another big advantage of representing states by sums of scalars & vectors is that this representation extends directly to statistical ensembles { Y } of spin states, as follows: Y  M ---- Y M i i -- + M  M i -------- R i s 3 R i We may write this (dropping the bar) as Y ( + s ), where s s is the ensemble s polarization. The operator Y, called the density operator, yields the ensemble average expectation value of any observable A via: M i ææ -- R s 3 R dp( R ) Æ M M i + Ú y A yò M  y A yò M  A Y i Ò 0 AY Ò 0 Jan. 8, 00 L ECTURE # of 8
Spins from Space-Time (sic) The spin vector s is the net angular momentum (in units of h), which for a classical spinning particle of charge e leads to a magnetic moment of sm 0 se ( m ) ( c ), where m 0 is the Bohr magneton. The spatial vector s corresponds to a spacetime bivector s sg 0, where g 0 defines its inertial frame. In a co-moving frame, this evolves according to the covariant form of Lorentz equation (Lecture 3) as ṡ m 0 ( F ( s g 0 )) Ÿ g 0 m 0 (( E s+ ( ib ) s )g 0 ) Ÿ g 0 m 0 s B, where we have used the facts that g 0 anticommutes with s, E, B and i, while E s is a scalar but ( ib ) s is a spatial vector. The lack of response to the electric field is a result of choosing a co-moving frame. Thus the magnetic moment of the electron, which is very nearly m 0, can be explained by a simple classical model. Like the magnetic field itself, however, spin behaves under inversion like a spatial bivector is. Jan. 8, 00 L ECTURE # 3 of 8
MULTIPARTICLE ALGEBRA More than the Sum of Its Parts? To model a system of N particles: " Take as usual the direct sum of their states (vectors); " Consider the corresponding joint Clifford algebra, i.e. ( G( 3, )) N ª G( N, 3N ); " This algebra has dimension 4N (exponential in N!). Assuming there exists a common frame of reference, i.e. a m natural choice of time-like g 0 in every particle space, then: " The even subalgebras G + ( 3, ) of different particle spaces commute, since for i, j 3 and k < l N s k m sl n m m n n n n m m n m ( g k g 0 )( g l g 0 ) ( g l g 0 )( g k g 0 ) s l sk where the superscripts are particle indices. " Thus one has an isomorphism with the tensor product of the corresponding 3-D Euclidean geometric algebras: ( G + ( 3, )) N ª ( G( 3 )) ƒn ; 3N " This algebra has dimension (still exponential!). Jan. 8, 00 L ECTURE # 4 of 8
PRODUCT OPERATORS The usual nonrelativistic theory of N spin / particles relies entirely upon the algebra of N N matrices over C, which has dimension only N +. The extra degrees of freedom in the algebra ( G + ( 3, )) N are due to the fact that it contains a different imaginary unit i m for every particle m. These extra degrees of freedom can be removed can be removed by multiplication by the correlator idempotent: C -- ( i i ) -- ( i i 3 ) ºº -- ( i i N ) This projects everything onto an ideal of the correct dimension, in which all these different imaginary units have been identified. In addition, since C C is easily seen to commute with everything in the algebra, multiplication by it is trivially a homomorphism onto this ideal (subalgebra). Henceforth we shall drop the implicit factor of C expressions, and use the single imaginary unit i i C º i N C. from all our Note the correlated product of the even subalgebras in each particle space, ( G + ( 3 )) ƒn C, is a subalgebra of dimension N (the same as that of the subspace of Hermitian matrices), which has never been recognized in the matrix algebra! Jan. 8, 00 L ECTURE # 5 of 8
MULTIPARTICLE SPINORS Just as in the one-particle case, multi-particle spinors (state vectors) yò in the N -D complex Hilbert space correspond to elements of a left-ideal obtained by multiplying through by an idempotent: EC N m E + ºE+ C m ( E± -- ( ± s 3 )). This ideal corresponds, as before, to matrices whose columns are all zero save for their left-most, and provides a carrier space for the products of all possible 3-D rotations. Similarly, multiparticle density operators may now be written (omitting now the C) as Y ---- M, M Âi Y ---- M i M Âi y ỹ ---- M i i M  R ER i i i where R i Œ ( G + ( 3 )) ƒn C. Note however that R i may not be factorizable into one particle rotations R i ºR N i, in which case the corresponding spinor y i R i E cannot be expressed as a product of one-particle spinors y i ºy N i and so is termed entangled. It may also happen that the individual terms in this sum can be factorized as R m i E m + R i m, but the overall sum is not factorizable. In this case the density operator describes a correlated, but not an entangled, statistical ensemble. Jan. 8, 00 L ECTURE # 6 of 8
THE BASICS OF QUANTUM INFORMATION PROCESSING Binary information can be stored in an array of two-state quantum systems (e.g. spin /) relative to a fixed computational basis of H( N ) (or ideal in ( G( 3 )) ƒn C ): 0Ò Ò 0 0Ò 0 E + -- ( + s 0 00 3 ) 0 Ò 00 E - -- ( s 0 3 ) Each such qubit can be placed into a coherent superposition over these two basis states, which for a, b Œ C can be written in the following equivalent ways: a 0Ò + b Ò a b ( a 0Ò + b Ò) ( ã 0 + b ) a ab ãb b ( a+ bs )E + ( ã + b s ) -- ( + ( ãb )s + ( ãb )s + ( a b )s ˆ 3 Jan. 8, 00 L ECTURE # 7 of 8
A superposition over all qubits individually expands to a superposition over all integers in { 0, º, N }, e.g. W Ò N ( 0Ò + Ò)ºº ( 0Ò + Ò) N N  j Ò j 0, N ( j Ò 0º0Ò is obtained by binary expansion of the integer j ). Ï ÔÔ Ô ÔÌ Ô ÔÔ By the linearity of quantum mechanics, any unitary operation U on such a superposition operates on every term of its expansion in parallel, i.e. (in ( G + ( 3 )) ƒn C, U acts as U W Ò W Ũ, which is also linear). Moreover, there exists a polynomially-bounded basis for the unitary group U( N ) (i.e. one that is unitarily universal). Example: All one-qubit operations, including those mapping one qubit states to superpositions, e.g. the Hadamard gate Ô Ó U W Ò N N U j Ò Â j 0 R H 0Ò ( 0Ò + Ò) R H Ò ( 0Ò Ò), together with the c-not (controlled-not) logic gate which computes the XOR of its inputs (see right). Note that this operates on bit b conditional on bit a. a b a a + b Jan. 8, 00 L ECTURE # 8 of 8
There exists a subgroup of U( N ) which: " Maps single states j Ò to single states. " Admits a polynomially bounded basis for the symmetric group S ( N ) (i.e. that is computationally universal). This is nontrivial, since unitary operations are necessarily reversible (i.e. each output corresponds to a single input). Example: The Toffoli gate has three input and three output bits. It copies the a a a & b bits, but takes the NOT of the c bit b b if the other two are both (see right). Setting the c bit to thus takes the NAND of the other two. c (a * b) + c Some of the hardest and most interesting computational problems (e.g. NP-complete) are decision problems: Given a boolean function f : { 0, } N ææ { 0, } M and an integer 0 j M, decide if the set { i f ( i) j } has some global property (e.g. is nonempty, periodic, etc.). One might think such problems could be efficiently solved on a quantum computer by applying a unitary transformation U f (which computes the function f ) to a superposition, Jan. 8, 00 L ECTURE # 9 of 8
U f ( W Ò 0 Ò) N N Â i i 0 Ò f ( i) Ò, followed by measurement of the output qubit s state. But according to von Neumann s measurement postulate: The result of a measurement on a quantum system in a superposition is a random term in the superposition, each with probability equal the modulus of its coefficient. For W Ò 0 Ò, this implies the chance of observing a given j Ò is Pr( i f ( i ) j ) { i f ( i) j } N, which may be exponentially small. Even if it is not small, some properties of { i f ( i) j } (e.g. periodicity) can only be found by enumeration all or most of it. There is a way around this: If one adjusts the phases of the terms of the input superposition appropriately, an additional transformation can be found that will amplify the magnitude of the coefficients asso-ciated with the solution(s), such that the probability of finding the sol-ution upon making a measurement becomes appreciable. Because of an analogy with light diffraction, this is usually called interference. Jan. 8, 00 L ECTURE # 0 of 8
Example: The Deutsch-Jozsa algorithm. Suppose we are given a function f : { 0, } Æ { 0, }, and ask: " Is f a constant function, i.e. ( { i f ( i) 0 } 0 ) or ( { i f ( i) } 0 ), or: " Is f a balanced function, i.e. { i f ( i) 0 } { i f ( i) } (two of the four possible functions f : { 0, } Æ { 0, } are constant, while the other two are balanced). This can be determined classically only by evaluating f on both inputs. ) Starting with 0 Ò 0 Ò Ò, apply the Hadamard gates: y Ò ( R HRH ) 0 Ò ( R H 0 Ò) ( R H Ò) where f f is the complement (or negation) of f. 3) Using the fact that?? -- ( 0Ò + Ò) ( 0Ò Ò) ) Evaluate f via its unitary transformation U f : y Ò U f y Ò -- ( U f 0 Ò 0 Ò + U f Ò 0Ò U f 0Ò Ò U f Ò Ò) -- ( 0Ò f ( 0 ) Ò + Ò f ( ) Ò 0Ò f ( 0 ) Ò Ò f ( ) Ò) f ( i) Ò f ( i) Ò Ï 0Ò Ò if f ( i) 0 Ì ý Ó Ò 0Ò if f ( i) þ ( ) f i ( ) 0 ( Ò Ò) Jan. 8, 00 L ECTURE # of 8
we can rewrite this as follows: y Ò -- (( ) f ( 0 ) 0Ò + ( ) f ( ) Ò) ( 0Ò Ò) 4) Now apply the same pair of Hadamard gates again: y 3 Ò ( R HRH ) y Ò ( since R H ( 0Ò Ò) Ò) -- (( ) f ( 0 ) ( 0Ò + Ò) + ( ) f ( ) ( 0Ò Ò) ) Ò 5) This in turn may be rewritten as y 3 Ò Ï ( ) f ( 0 ) 0Ò Ò if f ( 0 ) f ( ) Ì Ó ( ) f ( 0 ) Ò Ò if f ( 0 ) f ( ) showing the concentration of coherence via interference. 6) Thus measurement of the input (left-most) qubit gives 0Ò if f is constant and Ò if f is balanced. The remarkable thing is that this took only a single evalua-tion of the function f. This algorithm was the first hint that quantum computers are more powerful than classical. NB: A general quantum search algorithm is known which is quadratically faster than classical linear search, but very few quantum algorithms are known that are exponentially faster. Jan. 8, 00 L ECTURE # of 8
The Geometry Behind the Logic The Hadamard Gate Revisited Recall the action of the Hadamard gate on basis states: R H 0Ò ( 0Ò + Ò) R H Ò ( 0Ò Ò), The idempotents corresponding to the resulting states are: R H E + R H R H E - R H -- G + ( + s ) -- G - ( s ) We can also show that R H ( 0 Ò + i Ò) (( 0 Ò + Ò) + i( ( 0 Ò Ò) )) (( + i ) 0 Ò + ( i ) Ò) which becomes ( 0 Ò i Ò) on multiplying thru by the phase factor ( i ). The corresponding idempotents are: ( 0 Ò ± i Ò) ( 0 ± i ) -- +- i F ± ( ± s ) ± i Since ( R H ) and maps x z & y y, it follows that R H is just a rotation by p about the axis h ( s + s 3 )! Jan. 8, 00 L ECTURE # 3 of 8
Recall such rotations can be written in exponential form as exp ( i( p )h ) ih (since h ). It may be observed that ( ih ), due to a geometrically irrelevant global phase (a gauge degree of freedom); the phase corrected Hadamard gate is R H exp ( i( p )( h )) h ; similarly, the NOT operation N is a rotation by p about s, i.e. N s. The c-not Gate Revisited Recall that the c-not gate takes the NOT of one qubit if the other is Ò, e.g. S 00 Ò 00 Ò S 0 Ò 0 Ò S 0 Ò Ò S Ò 0 Ò Using the nilpotent transition operators 0 Ò & Ò 0, we can write this as S 00 Ò 00 + 0 Ò 0 + Ò 0 + 0 Ò 00Ò 00 + 0Ò 0 + s ( 0 Ò 0 + Ò ) 0 ( Ò 0 ) 0 ( Ò 0 + Ò ) + s ( Ò )( 0 Ò 0 + Ò ) 0 Ò 0 ƒ + ( Ò ƒ )s E + + E - s Jan. 8, 00 L ECTURE # 4 of 8
This has an interesting ideal-theoretic interpretation: Rotate qubit by p in the left-ideal defined by E -, but leave it alone in the complementary ideal defined by the annihilating idempotent. Using a well-known formula for the exponential of any-thing times a commuting idempotent, this may be written as exp ( i( p )E - ( s )) ( E - ) + E - exp ( i( p )( s )) E + More generally, a conditional rotation is a rotation of a qubit in the left-ideal generated by the idempotents of the given states of one or more other qubits. Noting that is 3 E ± ± ie ± so that rotations about s 3 in the ideal generated by G - are also conditional phase shifts, the form E - G - makes it clear that S can alternatively be viewed as a phase shift of the first qubit conditional on the second being along x. We can therefore write S as a phase shift E + E - s conditional on both qubits being along two Hadamards, as follows: + E -G-. z, sandwiched between R H exp ( ipe -E- ) R H exp ( ipe -RHE-RH ) ipe -G- exp ( ) Jan. 8, 00 L ECTURE # 5 of 8
Using green & red arrows for the qubit projected onto the & E + ideals, this can be represented by vector diagrams as follows: z y E - x R H P.S. R H This is actually rather close to how is implemented in NMR spectroscopy, which also makes common use of such diagrams without recognizing their ideal-theoretic connection. S The Deutsch-Jozsa Algorithm Revisited The initial superposition in the Deutsch-Jozsa algorithm is y Ò y G + G- -- ( + s 4 s s s ) If f ( 0 ) f ( ) 0, U f and so the final Hadamards just transform this back to E + E -, telling us it s constant. Jan. 8, 00 L ECTURE # 6 of 8
Now suppose we have the function f ( 0 ) 0, f ( ). Then U f S, i.e. the output qubit is flipped from 0 Ò to Ò when the input bit is Ò. How does this act on the product operators in the initial state? To answer that question by geometric reasoning only, let us go into the ideals generated by G - and G +, where we find that s s s s ( G+ + G - ) s G+ s G-. This shows that this state is that shown on the right, i.e. the two components of the first qubit point in opposite directions along x. We know that S E - G - will rotate the green one by p about z to give us the state: s G+ + s G- s. Thus the net effect of S is to do a swap, s s s ; applied to y Ò y, it swaps the signs of these terms (while it commutes with & hence has no effect on ), so we get: S y Ò y S -- ( s 4 s + s s ) G -G- The final Hadamards convert this to E - E- s as expected. Jan. 8, 00 L ECTURE # 7 of 8
CLOSING REMARKS The Future of Quantum Computing ) To date, computers have done more than theory has in enabling us to deal with complexity in the natural sciences (and elsewhere), because very simple local dynamics can model arbitrarily complex global behavior. ) The situation in multiparticle quantum systems is similar: Simple (indeed linear) local dynamics can lead to very complex large-scale behavior, including the apparent nonlinearities of the classical world. 3) Quantum computers may yield only modest advantages in combinatorial problems, but they will certainly provide a powerful means of simulating general quantum systems. 4) Thus the advent of quantum computers will greatly extend the reach of fundamental physics to the complex systems found in chemistry and molecular biology. 5) Meanwhile, there is much to be learned about how simple quantum dynamics leads to classical complexity (even chaos), which is the subject of the theory of decoherence. 6) By computing with homomorphic images in Clifford algebras, this may even enable theorems to be proved by experiment (for CaF, a dimension of 0 has been reached). Jan. 8, 00 L ECTURE # 8 of 8
THE BASICS OF NMR A macroscopic quantum system: Although the magnetic dipoles of near-by spins interact, the rapid rotational diffusion of the molecules in liquids average these interactions to zero. Hence to an excellent first-order approximation, spins in different molecules do not interact. It follows that if the state of the spins in the m -th molecule is y m Ò, then density operator of an M molecule ensemble factorizes as follows: Y ) y y ºy M Ò y y ºy M y Ò y º y M Ò y M y Ò y º y M Ò y M Y Y ºY M ª Y ƒm ( Y M y m Ò y m ) Â m Thus the kinematics is identical to that of a single molecule! The dynamics and observables are also identical, since: e its mh m Y ƒm ( )e its mh m ( e ith Ye ith ) ƒm tr Y ƒm m s Ë Ê Â m a ˆ Mtr ( Ys m ) a Jan. 8, 00 L ECTURE # 9 of 8
The weak-coupling Hamiltonian: The Hamiltonian of liquid-state NMR has the form: H k k -- w  0s3 k The first term is the Zeeman interaction (in rad/sec) with the external magnetic field (along z) as before; the second is the scalar coupling of pairs of spins across chemical bonds. It follows from first-order perturbation theory that if >> pj kl " k, l (weak coupling), the scalar coupling terms may be replaced by their secular parts pj kl k l s 3 s3. Now since H is diagonal, exp ( ith ) may be given in closed form. w 0 k w 0 l Radio-frequency fields: p  --- J kl k l k l k l + ( s s + s s + s 3 s3 ) k < l Given a strong RF-field on-resonance with the k -th spin, H RF k k k k k k k k w ( cos ( w 0t )s + sin ( w 0t ) s ) w exp ( iw 0t s3 ) k with w >> p J kl " l, it follows from the Liouville-von Neumann equation that in a co-rotating frame s k k k k k Y exp ( iw 0t s3 ) Yexp ( iw 0t s3 ), the (spin vector of) the spin rotates about the x axis according k k to exp ( iw t s ). Henceforth, all our transform-ations will be referred to such a rotating frame (w/o primes). Jan. 8, 00 L ECTURE # 0 of 8
In-Phase & Anti-phase Coherence The effect of scalar coupling: The (weak) scalar coupling propagator has the form: ipj exp ( t s 3s3 ) cos ( pj t ) i sin ( pj t )s3s3 Applied to a transverse state of e.g. spin, this yields: ipj ( t s 3s3 )s ipj exp exp ( t s 3s3 ) pj cos ( t )s + pj sin ( t )s s3 This is a rotation between in-phase ( ) and anti-phase ( ) coherence on spin. A classical interpretation is found on rotating to the z-axis, where the diagonal matrix elements in the s 3 basis ( 0Ò Ò, Ò ØÒ) are deviations from equal populations: s s s3 PO 00 PO 00Ò 0 PO 0Ò 0 PO 0Ò PO Ò s 3 s 3 s3 The anti-phase population difference between 0Ò & Ò states of spin is inverted in the subensemble where spin is down. Jan. 8, 00 L ECTURE # of 8
Spectra & vector diagrams: Thus in- / anti-phase coherence corresponds to classical ensembles, characterized by these relative populations, which are in transition between spin up & down. Moreover, spin s magnetization (population difference) precesses at the rate w 0 ( p ) ± J as spin is up or down, resp. The spectrum (real part of the Fourier transform) is, s s s3 in which the anti-phase population inversion may be seen. The rotation of into may be visualized as follows: s s z y s s3 s x s s3 Magnetization of spin in subensemble with spin up & down. Jan. 8, 00 L ECTURE # of 8
The one-bit quantum logic gates: The simplest logic gate is the NOT, which is a p-rotation about the x-axis combined with a phase shift, Êi p ; -- s k ( ) ˆ k k exp i( cos ( p ) isin ( p )s Ë ) s recalling that the density operators of the basis states of the k - th spin (with a totally mixed state everywhere else) are ( ƒ º ƒ 0Ò 0 ƒ º ƒ ) k k -- ( + s 3 ) E + k ( ƒ º ƒ Ò ƒ º ƒ ) -- ( s, 3 ) E ḵ k k we can use the anticommutivity of s & s 3 to show that this NOT gate maps the 0Ò state of the k -th spin to the Ò state: s k E+ k s k # Another important one-bit, but nonboolean, gate is the Hadamard transform HAD: k k k k s -- ( + s 3 )s ( s ) k -- ( s 3 ) E ḵ i p -- s k k exp Ê ( ( + s 3 )) ˆ Ë k k ( s + s 3 ), k k k k which can be show to map s s 3 & s s. It also maps 0Ò ( 0Ò + Ò) & Ò ( 0Ò Ò), and so can be used to prepare a uniform superposition over all spins as above. Jan. 8, 00 L ECTURE # 3 of 8
The multi-bit boolean logic gates: Interesting computations require feedback, i.e. the state of one bit must influence what happens to another, but the usual AND & OR gates are not reversible (only one output!). An important two-bit gate is the controlled-not: Êi p -- s ( )E ˆ exp Ë - s E- + E + This c-not is readily shown to flip the first spin in states where the second is down (just as we considered earlier), i.e. 00Ò 00Ò 0Ò 0Ò 0Ò Ò Ò 0Ò, or more compactly: d d Ò ( d + d mod )( d ) Ò. Thus idempotents also describe the conditionality of operations. P Another obvious gate is the SWAP of two bits, exp Êi-- p Pˆ P ; Ë -- s ( + s + s s + s 3 s3 ) is also called the particle interchange operator. These can be extended to Êi p -- s 3 ( )E -E- ˆ 3 exp s Ë E- E- N bits; e.g., the Toffoli gate is: 3 + ( E -E- ) The TOF alone is universal for boolean logic; more generally, the c-not and one-bit rotations generate all of SU( N ). Jan. 8, 00 L ECTURE # 4 of 8
NMR Implementations of Gates Radio-frequency pulse sequences: The NOT gate is easily implemented by a strong RF pulse, in phase with the x-axis, whose frequency range spans only the resonance of the target spin, and whose duration is sufficient to rotate it by p (note the global phase offset of i has no effect on the density operator). The HAD gate is similarly obtained from the pulse sequence (written in left-to-right temporal order): p k p k p k -- s 8 Æ -- s Æ -- s 8 i p 8 -- s k exp Ê ˆ Ë i p -- s k exp Ê ˆ Ë i p 8 -- s k exp Ë Ê ˆ To implement the c-not gate, we proceed as follows: i p -- s exp Ê ( )E ˆ Ë - Ê i p 4 -- s ˆ exp Ë exp ( ipe -E- ) exp i p 4 -- s Ë Ê ˆ i p 4 -- s exp Ê ˆ Ë i p 4 -- s exp Ê ( 3 + s 3 ) ˆ i p Ë 4 -- s exp Ê 3s3 ˆ i p Ë exp 4 -- s Ë Ê ˆ i i p 4 -- s exp Ê ( 3 + s 3 ) ˆ i p Ë 4 -- s exp Ê ˆ Ë i p 4 -- s exp Ê 3s3 ˆ i p Ë exp 4 -- s Ë Ê ˆ i This is a ( p )-rotation about y, a weak coupling evolution for ( J ), a ( p )-rotation about x, and a Zeeman evolution. Jan. 8, 00 L ECTURE # 5 of 8
Vector diagram description: # This pulse sequence may be depicted as follows: y x x x x $ % & ' /(4J) /(4J) /(4n ) /(4n ) In terms of Bloch diagrams, we have: z y x $ % & ' Caption: Here, red is magnetization of spin in molecules where spin is up, and green that of spin where down. Jan. 8, 00 L ECTURE # 6 of 8
PSEUDO-PURE STATES Starting from equilibrium: Since pj << w 0 << k B T, the equilibrium state of a homonuclear spin system (& its partition function Q ª N ) is: exp ( H Y Z k B T ) eq ----------------------------------------------- tr( exp ( H Z k B T )) l l  w 0s3 l exp ( k B T ) --------------------------------------------------------- Q l l (  w 0s3 k l B T ) N Ê w 0 p i iòi ˆ N ª ª Á ------------------------------- Ë k B T Here 00º0Ò is a binary expansion of iò, & p i hi () N where hi ( ) is the Hadamard weight (number of s) of i. The problem with Y eq is that logical operations performed on the spins at the microscopic level do not effect the same operations on their macroscopic polarizations; for two spins:  i r eq r eq s 3 PO 00 PO 00Ò 0 PO 0Ò 0 PO 0Ò PO Ò + s 3 0 0 E + s3 0 0 Spin s polarization, i.e. the alternating row sum, goes to 0. Jan. 8, 00 L ECTURE # 7 of 8
So we average yet more! A pseudo-pure state is one whose density operator has exactly one nondegenerate eigenvalue, e.g. Y Ê pp w 0 p 0 -------------------------- 0Ò0 ˆ N Ë k B T Note that the microscopic state associated with. r pp ( + a 0Ò0 ) N. 0Ò 00º0Ò is canonically Because the identity component is unitarily invariant, the state iò provides a spinorial representation of SU( N ): UY pp Ũ ( + a ( U0 Ò) ( 0 Ũ )) N ( U Œ SU( N )) Similarly, because the identity component does not contribute to the magnetization (population differences), the ensemble average expectation value of the observables is proportional to their ordinary expectation values: k k k N --tr( Y pp s ) ( tr( s ) + atr ( s 0Ò0 )) a N + k ------------- 0 s 0Ò Since their eigenstructures differ, Y pp must be prepared from Y eq by a nonunitary process, e.g. by averaging the popu-lations over all permutations of the states i Ò i ( i > 0) (more efficient methods exist, which rely upon magnetic gradients). Jan. 8, 00 L ECTURE # 8 of 8