State-independent consistent sets

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In May, Losada and Laura wrote a paperM. Losada and R. Laura, Annals of Physics 344, 263 (2014).a   pointing out the equivalence between two conditions on a set of “elementary histories” (i.e. fine-grained historiesGell-Mann and Hartle usually use the term “fine-grained set of histories” to refer to a set generated by the finest possible partitioning of histories in path integral (i.e. a point in space for every point in time), but this is overly specific. As far as the consistent histories framework is concerned, the key mathematical property that defines a fine-grained set is that it’s an exhaustive and exclusive set where each history is constructed by choosing exactly one projector from a fixed orthogonal resolution of the identity at each time.b  ). Let the elementary histories \alpha = (a_1, \dots, a_N) be defined by projective decompositions of the identity P^{(i)}_{a_i}(t_i) at time steps t_i (i=1,\ldots,N), so that

(1)   \begin{align*} P^{(i)}_a &= (P^{(i)}_a)^\dagger \quad \forall i,a \\ P^{(i)}_a P^{(i)}_b &= \delta_{a,b} P^{(i)}_a \quad \forall i,a,b\\ \sum_{a} P^{(i)}_a (t_i) &= I \quad \forall i,k \\ C_\alpha &= P^{(N)}_{a_N} (t_N) \cdots P^{(1)}_{a_1} (t_1) \\ I &= \sum_\alpha C_\alpha = \sum_{a_1}\cdots \sum_{a_N} C_\alpha \\ \end{align*}

where C_\alpha are the class operators. Then Losada and Laura showed that the following two conditions are equivalent

  1. The set is consistent“Medium decoherent” in Gell-Mann and Hartle’s terminology. Also note that Losada and Laura actually work with the obsolete condition of “weak decoherence”, but this turns out to be an unimportance difference. For a summary of these sorts of consistency conditions, see my round-up.c   for any state: D(\alpha,\beta) = \mathrm{Tr}[C_\alpha \rho C_\beta^\dagger] = 0 \quad \forall \alpha \neq \beta, \forall \rho.
  2. The Heisenberg-picture projectors at all times commute: [P^{(i)}_{a} (t_i),P^{(j)}_{b} (t_j)]=0 \quad \forall i,j,a,b.

However, this is not as general as one would like because assuming the set of histories is elementary is very restrictive. (It excludes branch-dependent sets, sets with inhomogeneous histories, and many more types of sets that we would like to work with.) Luckily, their proof can be extended a bit.

Let’s forget that we have any projectors P^{(i)}_{a} and just consider a consistent set \{ C_\alpha \}. Then consistency for any state, or state-independent consistency, means

(2)   \begin{align*} D(\alpha,\beta) = \mathrm{Tr}[C_\alpha \rho C_\beta^\dagger] = 0 \quad \forall \alpha \neq \beta, \forall \rho \end{align*}

Now, let’s consider the coarse-grained set that contains just a single arbitrary history C_\alpha and its converse I-C_\alpha. Coarse-graining preserves state-independent consistency, so we get that \mathrm{Tr}[C_\alpha \rho (I- C_\alpha^\dagger)] = 0 for all states \rho, and in particular for any \vert \psi \rangle with \rho = \vert \psi \rangle\langle \psi \vert: \langle \psi \vert (I- C_\alpha^\dagger) C_\alpha \vert \psi \rangle = 0. But one can check (see Losada and Laura) that the only operator that always vanishes when sandwiched between two copies of an arbitrary vector is the zero operator. Therefore, (I- C_\alpha^\dagger) C_\alpha=0 or, in other words, C_\alpha = C_\alpha^\dagger C_\alpha. The operator C_\alpha^\dagger C_\alpha is positive, and C_\alpha is one of its square roots. But a positive operator can’t be equal to its square root unless it’s a projector. (Use the singular-value decomposition to see this.) Therefore C_\alpha is a projector and, since there was nothing special about this particular history, we see that all the class operators in the original set \{ C_\alpha \} are projectors. We find, without assuming anything about the set of histories being elementary, that the following two conditions are equivalent

  1. The set is consistent for any state.
  2. The class operators are projectors.

Then, if we further assume that the set is elementary, it immediately follows that the individual projectors P^{(i)}_{a} commute.

A further question, whose answer I don’t know: do all the projectors used in constructing a state-independent set of (not necessarily elementary) homogeneous histories necessarily commute?

[Minor edits 2014-9-8]

Footnotes

(↵ returns to text)

  1. M. Losada and R. Laura, Annals of Physics 344, 263 (2014).↵
  2. Gell-Mann and Hartle usually use the term “fine-grained set of histories” to refer to a set generated by the finest possible partitioning of histories in path integral (i.e. a point in space for every point in time), but this is overly specific. As far as the consistent histories framework is concerned, the key mathematical property that defines a fine-grained set is that it’s an exhaustive and exclusive set where each history is constructed by choosing exactly one projector from a fixed orthogonal resolution of the identity at each time.↵
  3. “Medium decoherent” in Gell-Mann and Hartle’s terminology. Also note that Losada and Laura actually work with the obsolete condition of “weak decoherence”, but this turns out to be an unimportance difference. For a summary of these sorts of consistency conditions, see my round-up.↵
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