5. Entanglement

The proof of Theorem 4 (http://planetmath.org/3distributeddynamicalsystems#Thmthm4) showed the structureMathworldPlanetmath presheafPlanetmathPlanetmathPlanetmath has non-unique descent, reflecting the fact that measuring devices do not necessarily reduce to productsPlanetmathPlanetmathPlanetmath of subdevices. Similarly, as we will see, measurements do not in general decompose into independentPlanetmathPlanetmath submeasurements. Entanglement, γ, quantifies how far a measurement diverges in bits from the product of its submeasurements. It turns out that γ>0 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 has no direct relationMathworldPlanetmathPlanetmath to quantum entanglement. The name was chosen because of a formal resemblance between the two quantities, see Supplementary Information of [1].

Definition 10.

Entanglement over partition 𝒫={M1Mm} of src(𝔪𝐃) is


where πj:𝒱S𝐃𝒱SMj and 𝔪j={(k,l)𝔪𝐃|kMj}.

Projecting via πj marginalizes onto the subspace 𝒱SMj. Entanglement thus compares the measurement performed by the entire system with submeasurements over the decomposition of the source occasions into partition 𝒫.

Theorem 9 (effective information decomposes additively when entanglement is zero).

Proof: Follows from the observations that (i) H[pp1p2]=0 if and only if p=p1p2; (ii) H[p1p2q1q2]=H[p1q1]+H[p2q2]; and (iii) the uniform distributionMathworldPlanetmath on 𝐃 is a tensor of uniform distributions on subsystems of 𝐃.

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.

We conclude with some detailed computations for X×Y𝑔Z, Diagram (11) (http://planetmath.org/4measurement#id2). Let 𝒫={X|Y}.

Theorem 10 (entanglement and effective information for g:X×YZ).
γ(𝔪XY,𝒫,δz) =(x,y)g-1(z)1|g-1(z)|log2|g-1(z)||gx×Y-1(z)||gX×Y-1(z)|

Proof: The first equality follows from PropositionsPlanetmathPlanetmath 5 (http://planetmath.org/4measurement#Thmthm5) and 6 (http://planetmath.org/4measurement#Thmthm6)


From the same propositions it follows that ei(𝔪XY,δz)-ei(𝔪X,δz)-ei(𝔪Y,δz) equals


Entanglement quantifies how far the size of the pre-image of g-1(z) deviates from the sizes of its X×y and x×Y slices as x and y are varied.

By Corollary 8 (http://planetmath.org/4measurement#Thmthm8) entanglement also equals ei(𝔪X𝔪XY,δz)-ei(𝔪Y,δz). 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 vYvZ to the null system and the subsystem 𝔪X={vXvZ} respectively. Entanglement is the difference between the information generated by the diagonal arrows. It quantifies the difference between the information {vYvZ} generates in two different contexts.

Corollary 11 (characterization of disentangled set-valued functions).

Function X×Y𝑔Z performs a disentangled measurement when outputting z iff


for any x,y such that g(x,y)=z.

Proof: By Theorem 10 entanglement is zero iff


for any x,y such that g(x,y)=z. This implies the desired result since g-1(z)gx×Y-1(z)×gX×y-1(z).

Thus, the measurement generated by g is disentangled iff its pre-image g-1(z) satisfies a strong geometric “rectangularity” constraint: that the pre-image decomposes into the product of its x×Y and X×y slices for all pairs of slices intersecting g-1(z). 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.

Example 4.

An XOR-gate g:X×YZ outputting 0 generates an entangled measurement. The pre-image is g-1(0)={00,11} so the XOR-gate generates 1 bit of information about occasions vX and vY. However, the bit is not localizable. The measurement generates no information about occasion vX taken singly, since its output could have been 0 or 1 with equal probability; and similarly for vY.

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 X×Y𝑔Z is surjectivePlanetmathPlanetmath, then
γ(mXY,P,δz)=0 for all zZ iff g decomposes into X×Yg1×g2Z1×Z2=Z for Xg1Z1 and Yg2Z2.

Proof: The reverse implicationMathworldPlanetmath is trivial. In the forward direction, note that Z={g-1(z)|zZ} and, by Corollary 11, each pre-image has product structure g-1(z)=gx×Y-1(Z)×gX×Y-1(z). Let Z1={gX×y-1|yY and zZ} and similarly for Z2. Define

g1:XZ1:xthe unique element of form gX×y-1(z) containing it,

and similarly for g2.


  • 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.
Title 5. Entanglement
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