Composition of decoherence functionals

Cite this problem as Problem 36.


Consider a physical system where a number of (possibly incompatible) experiments can be in principle conducted. Each experiment x=1,...,m allows estimating a property with value a_x\in A_x, and any vector of possible values (a_1,...,a_x)\in \Omega, with \Omega\equiv A_1\times ...\times A_m is called a history. The above is called a joint measurement framework [1].

Given the set of all histories \Omega, define a Hilbert space H_\Omega by assigning to each a\in\Omega an element |a\rangle of an orthonormal basis \{|b\rangle:b\in\Omega\}. For any A\subseteq \Omega, define the vector |A\rangle\equiv\sum_{a\in A}|a\rangle. To model experiments on this system within the framework of Sorkin’s quantum measure theory [2], we assume that there exists a complex-valued operator D:H_\Omega\to H_\Omega, dubbed decoherence functional, with the following properties:

  1. Hermiticity: D=D^\dagger.
  2. Weak positivity: \langle A|D|A\rangle\geq 0, for all A\subseteq \Omega.
  3. Normalization: \langle\Omega|D|\Omega\rangle=1.
  4. Strong decoherence: let \{A_k\}_{k} be a partition of \Omega. If there exists a feasible experiment to determine which A_k describes the system, then \langle A_k|D|A_j\rangle=P(A_k)\delta_{kj}, where P(A_i) is the probability that the history of the system belongs to A_i.

Note that, given that there exist experimental setups to measure the basic properties x=1,...,m, for any x, the partition of \Omega \{A^a_x\}_a, with


is measurable and by axiom 4 thus satisfies \langle A^a_x|D|A^{a'}_x\rangle=P(a|x)\delta_{a,a'}, where P(a|x) is the probability to obtain the result a had we decided to conduct experiment x.

To describe two or more independent systems, with history sets \Omega_1,\Omega_2,..., we consider the set of joint histories \Omega=\Omega_1\times\Omega_2\times ... and take the decoherence functional D:H_\Omega\to H_\Omega describing the joint system as D=D_1\otimes D_2\otimes.... It is observed in [3], though, that the tensor product of two valid decoherence functionals may not satisfy the axiom of weak positivity. That is, even when D_1,D_2 satisfy all the axioms above, there may be a vector |v\rangle\in \{0,1\}^{dim(H_\Omega)} such that \langle v|D_1\otimes D_2|v\rangle<0. As noted in [3], this is not the case when, instead of weak positivity, we demand the stronger axiom of strong positivity, i.e., the requirement that, when viewed as a matrix, D is positive semidefinite (D\geq 0). In fact, the set of strongly positive decoherence functionals is closed under tensor products.

Problem: identify a set of decoherence functionals that is closed under tensor products and that contains elements which are not strongly positive. Equivalently, find a joint measure framework and a decoherence functional D describing it such that D\not\geq 0 and, for all n, D^{\otimes n} satisfies the axiom of weak positivity.


The problem of reconciling quantum theory with general relativity is as old as quantum mechanics itself. One approach to achieve this unification, pioneered by Sorkin [2] and independently by Hartle [4], begins with a reformulation of quantum theory in terms of histories, rather than instantaneous states. It is within this approach that Sorkin proposed his history-based framework of quantum measure theory, which can accommodate, not only classical and quantum mechanics, but also some physical theories beyond the quantum formalism. In [1], [3] it is raised the problem of which set of decoherence functionals to consider, and the strongly positive set is argued on the basis of composability. By adopting strong positivity as a fundamental axiom, though, we may be limiting unnecessarily the space of possible physical theories which quantum measure theory can handle.

Partial results

In [5] it is proven that, for any n, there exists a joint measurement scenario and a decoherence functional D describing it such that D^{\otimes n-1} satisfies weak positivity, but D^{\otimes n} does not. It is also shown that the set of strongly positive decoherence functionals cannot be enlarged without losing the property of being closed under under tensor products. In fact, in [5] the authors identify a family F of quantum decoherence functionals with the property that any set of decoherence functionals closed under tensor products that contains F can only contain strongly positive elements.


[1] F. Dowker, J. Henson and P. Wallden, New J. Phys. 16, 033033 (2014).

[2] R. D. Sorkin, Quantum Mechanics as Quantum Measure Theory, Mod. Phys. Lett. A9 3119-3128 (1994).

[3] X. Martin, D. OConnor and R.D. Sorkin, Random walk in generalized quantum theory, Phys. Rev. D 71, 024029 (2005).

[4] J. B. Hartle, The quantum mechanics of cosmology Dec 1989, Lectures at Winter School on Quantum Cosmology and Baby Universes, Jerusalem, Israel, Dec 27, 1989 – Jan 4, 1990.

[5] P. Boes and M. Navascues, Phys. Rev. A 95, 022114 (2017).