The proof of Theorem 4 (http://planetmath.org/3distributeddynamicalsystems#Thmthm4) showed the structure presheaf has non-unique descent, reflecting the fact that measuring devices do not necessarily reduce to products of subdevices. Similarly, as we will see, measurements do not in general decompose into independent submeasurements. Entanglement, , quantifies how far a measurement diverges in bits from the product of its submeasurements. It turns out that is necessary for a system to generate more information than the sum of its components: non-unique descent thus provides “room at the top” to build systems that perform more precise measurements collectively than the sum of their components.
Entanglement over partition of is
where and .
Projecting via marginalizes onto the subspace . Entanglement thus compares the measurement performed by the entire system with submeasurements over the decomposition of the source occasions into partition .
Theorem 9 (effective information decomposes additively when entanglement is zero).
Proof: Follows from the observations that (i) if and only if ; (ii) ; and (iii) the uniform distribution on is a tensor of uniform distributions on subsystems of .
The theorem shows the relationship between effective information and entanglement. If a system generates more information “than it should” (meaning, more than the sum of its subsystems), then the measurements it generates are entangled. Alternatively, only indecomposable measurements can be more precise than the sum of their submeasurements.
Theorem 10 (entanglement and effective information for ).
From the same propositions it follows that equals
Entanglement quantifies how far the size of the pre-image of deviates from the sizes of its and slices as and are varied.
By Corollary 8 (http://planetmath.org/4measurement#Thmthm8) entanglement also equals . In Diagram (11) (http://planetmath.org/4measurement#id2) entanglement is the vertical arrow minus both arrows at the bottom, or the difference between opposing diagonal arrows. Note that the diagonal arrows from left to right are constructed by adding edge to the null system and the subsystem respectively. Entanglement is the difference between the information generated by the diagonal arrows. It quantifies the difference between the information generates in two different contexts.
Corollary 11 (characterization of disentangled set-valued functions).
Function performs a disentangled measurement when outputting iff
for any such that .
Proof: By Theorem 10 entanglement is zero iff
for any such that . This implies the desired result since .
Thus, the measurement generated by is disentangled iff its pre-image satisfies a strong geometric “rectangularity” constraint: that the pre-image decomposes into the product of its and slices for all pairs of slices intersecting . The categorizations performed within a disentangled measuring device have nothing to do with each other, so that the device is best considered as two (or more) distinct devices that happen to have been grouped together for the purposes of performing a computation.
An XOR-gate outputting 0 generates an entangled measurement. The pre-image is so the XOR-gate generates 1 bit of information about occasions and . However, the bit is not localizable. The measurement generates no information about occasion taken singly, since its output could have been 0 or 1 with equal probability; and similarly for .
Finally, and unsurprisingly, a function is completely disentangled across all its measurements iff it is a product of two simpler functions:
Corollary 12 (completely disentangled functions are products).
If is surjective, then
for all iff decomposes into for and .
and similarly for .
- 1 David Balduzzi & Giulio Tononi (2009): Qualia: the geometry of integrated information. PLoS Comput Biol 5(8), p. e1000462, doi:10.1371/journal.pcbi.1000462.
|Date of creation||2014-04-22 23:06:40|
|Last modified on||2014-04-22 23:06:40|
|Last modified by||rspuzio (6075)|