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Unformatted text preview: qitd421 Correlations, Ensembles, Density Operators Robert B. Griffiths Version of 2 Feb. 2010 Contents 1 Correlations, Classical and Quantum 1 2 Conditional States 2 3 Ensembles 3 4 Density Operators 4 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4.2 Partial traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4.3 Density operators of ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4.4 Other sources of density operators. Purification . . . . . . . . . . . . . . . . . . . . . 7 4.5 Time development of density operators . . . . . . . . . . . . . . . . . . . . . . . . . . 7 References: CQT = Consistent Quantum Theory by Griffiths (Cambridge, 2002) 1 Corr elat ions , Classic al and Quantum Classical correlations. Charlie prepares red R and green G slips of paper, inserts them in two opaque envelopes. One goes to Alice in Atlanta, the other to Bob in Boston. Probability that Alice has R or G is 1/2, but if Bob opens his envelope and sees R , he can conclude that Alices envelope contains G . This is an example of statistical inference based upon the use of a conditional probability: Pr( A = G ) and Pr( A = G  B = R ) are two very different things No mysterious longrange influences, wave function collapse or the like in the classical world. Or the quantum world. Probabilistic model based on Pr( A, B ), where A and B can be R or G . We have Pr( R, G ) = Pr( A = R, B = G ) = 1 / 2 = Pr( G, R ); Pr( R, R ) = 0 = Pr( G, G ) . (1) Marginal probabilities Pr( A = R ) = B Pr( A = R, B ) = 1 / 2 = Pr( A = G ) , (2) and same thing for Pr( B = ). These do not reveal the correlation. The conditional probability Pr( B  A ) = Pr( A, B ) / Pr( A ) reveals the correlations, e.g., Pr( B = R  A = R ) = 0 , Pr( B = G  A = R ) = 1 . (3) 1 Statistical correlations indicate the presence of information: by examining A (and knowing the joint probability distribution) one finds out something about B ; in this sense A contains information about B . Quantum correlations. Famous EPRBohm singlet state of two qubits  = (  01  10 ) / 2 (4) as a preprobability applied to the orbasis (short for orthonormal basis) { jk } , j, k 1 gives the same thing as (1) if we think of 0 as G , 1 as R . Exercise. Check it. Does the minus sign in (4) make a difference in this case? An entangled quantum state, regarded as a preprobability, induces correlations, whereas a product state preprobability results in statistical independence of the two systems. 2 Cond itio nal State s Consider the (possibly) entangled state  , regarded as a preprobability on the Hilbert space H = H a H b of two systems a and b you can think of two qubits, but the following also works in the more general casewith orbases { a j } and { b k } :  = jk c jk  a j b k , (5) and suppose that it is normalized: = 1....
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