Consistent Histories and Decoherence Functionals

A decoherence functional measures interference between sequences of quantum alternatives. Its diagonal entries are candidate history probabilities; off-diagonal entries determine whether those probabilities obey classical sum rules under coarse-graining. For the interpretation of these histories and the rules for combining them, see consistent histories.

Required background. Density operators: Use density operators to calculate probabilities and distinguish coherent from incoherent alternatives.

Class operators and normalization

Choose an initial density operator $\rho$ and exhaustive orthogonal projectors at times $t_1<\cdots<t_n$. In the Heisenberg picture, a history $\alpha=(\alpha_1,\ldots,\alpha_n)$ has class operator

$$C_\alpha=P_{\alpha_n}(t_n)\cdots P_{\alpha_1}(t_1).$$

Define

$$D(\alpha,\beta)=\operatorname{Tr}(C_\alpha\rho C_\beta^\dagger).$$

It is Hermitian in its history labels, positive as a matrix, and has nonnegative diagonal entries. Summing over a fully specified exhaustive chain of alternatives yields normalized diagonal weights. The class operators need not themselves be projectors, since projectors at different times need not commute.

Coarse-graining reveals interference

The union of two exclusive fine-grained histories is represented by $C_{\alpha\lor\beta}=C_\alpha+C_\beta$. Its candidate probability is

$$p(\alpha\lor\beta)=p(\alpha)+p(\beta) +2\operatorname{Re}D(\alpha,\beta).$$

The classical sum rule requires the last term to vanish. “Weak consistency” refers to vanishing real off-diagonal terms for the relevant family. A commonly used stronger sufficient condition, medium decoherence, sets the full off-diagonal entries to zero. Their stability properties and composition behavior are not identical.

A qubit interference calculation

Take no evolution, initial $|+\rangle$, an intermediate $Z$ alternative, and final projector $P_+=|+\rangle\langle+|$. The two class operators are $C_0=P_+P_0$ and $C_1=P_+P_1$. Acting on the initial vector,

$$C_0|+\rangle=C_1|+\rangle=\frac12|+\rangle.$$

Consequently $p(0,+)=p(1,+)=1/4$ and $D(0,+;1,+)=1/4$. Grouping the intermediate alternatives gives final probability one, not $1/2$.

This is the difference between an unmeasured coherent alternative and a performed intermediate measurement. Actually measuring $Z$ and ignoring its record changes the physical process and gives final $+$ probability $1/2$.

Records can decohere histories

If the two intermediate alternatives become correlated with orthogonal environmental records, their branch vectors become orthogonal in the enlarged Hilbert space. The corresponding off-diagonal functional vanishes. Stable records therefore support an approximate classical probability calculus for selected coarse-grained histories.

Approximate decoherence needs a quantitative tolerance. Small individual off-diagonal entries do not automatically guarantee small errors for arbitrarily large unions, because many terms can accumulate. A claim about a particular family should control the sum relevant to its observables.

Exercises

Replace perfect record orthogonality by overlap $\gamma$. What happens to the interference contribution in the qubit example?

Solution

The branch-vector cross term is multiplied by the environment overlap (or its complex conjugate, depending on the ordering convention). The grouped final probability contains $(1/2)\operatorname{Re}\gamma$ in addition to the diagonal sum $1/2$. Perfect orthogonality gives $1/2$; identical in-phase records restore probability one.

References

• R. B. Griffiths, “Consistent histories and the interpretation of quantum mechanics,” Journal of Statistical Physics 36, 219–272 (1984), doi:10.1007/BF01015734.
• J. P. Paz and W. H. Zurek, “Environment-induced decoherence, classicality, and consistency of quantum histories,” Physical Review D 48, 2728–2738 (1993), doi:10.1103/PhysRevD.48.2728.