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# 5. Entanglement

The proof of Theorem 4
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, $\gamma$, quantifies how far a
measurement diverges in bits from the product of its
submeasurements. It turns out that $\gamma>0$ 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 relation^{} 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 ${\mathcal{P}}=\{M_{1}\ldots M_{m}\}$
of $src({\mathfrak{m}}_{{\mathbf{D}}})$ is

$\gamma({\mathfrak{m}}_{{\mathbf{D}}},{\mathcal{P}},{d_{{out}}})=H\left[{% \mathfrak{m}}_{{\mathbf{D}}}^{\natural}\circ{d_{{out}}}\Big\|\bigotimes_{{i=1}% }^{m}\pi_{j}\circ{\mathfrak{m}}_{j}^{\natural}\circ{d_{{out}}}\right]$ |

where $\pi_{j}:{\mathcal{V}}{S}^{{\mathbf{D}}}\rightarrow{\mathcal{V}}{S}^{{M_{j}}}$ and ${\mathfrak{m}}_{j}=\{(k,l)\in{\mathfrak{m}}_{{\mathbf{D}}}|k\in M_{j}\}$.

Projecting via $\pi_{j}$ marginalizes onto the subspace ${\mathcal{V}}{S}^{{M_{j}}}$. Entanglement thus compares the measurement performed by the entire system with submeasurements over the decomposition of the source occasions into partition ${\mathcal{P}}$.

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

$\gamma({\mathfrak{m}}_{{\mathbf{D}}},{\mathcal{P}},{d_{{out}}})=0\,\,\,\,\,% \implies\,\,\,\,\,ei({\mathfrak{m}}_{{\mathbf{D}}},{d_{{out}}})=\sum_{{i=1}}^{% m}ei({\mathfrak{m}}_{j},{d_{{out}}}).$ |

Proof: Follows from the observations that (i) $H[p\|p_{1}\otimes p_{2}]=0$ if and only if $p=p_{1}\otimes p_{2}$; (ii) $H[p_{1}\otimes p_{2}\|q_{1}\otimes q_{2}]=H[p_{1}\|q_{1}]+H[p_{2}\|q_{2}]$; and (iii) the uniform distribution on ${\mathbf{D}}$ is a tensor of uniform distributions on subsystems of ${\mathbf{D}}$. $\blacksquare$

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 $X\times Y\xrightarrow{g}Z$, Diagram (11). Let ${\mathcal{P}}=\{X|Y\}$.

###### Theorem 10 (entanglement and effective information for $g:X\times Y\rightarrow Z$).

$\displaystyle\gamma({\mathfrak{m}}_{{XY}},{\mathcal{P}},\delta_{z})$ | $\displaystyle=\sum_{{(x,y)\in g^{{-1}}(z)}}\frac{1}{|g^{{-1}}(z)|}\log_{2}% \frac{|g^{{-1}}(z)|}{|g^{{-1}}_{{x\times Y}}(z)|\cdot|g^{{-1}}_{{X\times Y}}(z% )|}$ | ||

$\displaystyle=ei({\mathfrak{m}}_{{XY}},\delta_{z})-ei({\mathfrak{m}}_{{X% \bullet}},\delta_{z})-ei({\mathfrak{m}}_{{\bullet Y}},\delta_{z}).$ |

Proof:
The first equality follows from Propositions^{} 5
and 6

$\gamma({\mathfrak{m}}_{{XY}},{\mathcal{P}},\delta_{z})=\sum_{{(x,y)\in g^{{-1}% }(z)}}=\sum_{{(x,y)\in g^{{-1}}(z)}}\frac{1}{|g^{{-1}}(z)|}\log_{2}\left[\frac% {1}{|g^{{-1}}(z)|}\cdot\frac{|g^{{-1}}(z)|}{|g^{{-1}}_{{x\times Y}}(z)|}\frac{% |g^{{-1}}(z)|}{|g^{{-1}}_{{X\times Y}}(z)|}\right].$ |

From the same propositions it follows that $ei({\mathfrak{m}}_{{XY}},\delta_{z})-ei({\mathfrak{m}}_{{X\bullet}},\delta_{z}% )-ei({\mathfrak{m}}_{{\bullet Y}},\delta_{z})$ equals

$\displaystyle\log_{2}\frac{|X|\cdot|Y|}{|g^{{-1}}(x)|}-\sum_{{x}}\frac{|g^{{-1% }}_{{x\times Y}}(z)|}{|g^{{-1}}(z)|}\log_{2}\frac{|X|\cdot|g^{{-1}}_{{x\times Y% }}(z)|}{|g^{{-1}}(z)|}-\sum_{y}\frac{|g^{{-1}}_{{X\times y}}(z)|}{|g^{{-1}}(z)% |}\log_{2}\frac{|Y|\cdot|g^{{-1}}_{{X\times y}}(z)|}{|g^{{-1}}(z)|}$ | ||

$\displaystyle=\log_{2}\frac{1}{g^{{-1}}(z)}-\sum_{{(x,y)\in g^{{-1}}(z)}}\frac% {1}{|g^{{-1}}(z)|}\cdot\log_{2}\frac{|g^{{-1}}_{{X\times y}}(z)|}{|g^{{-1}}(z)% |}\cdot\frac{|g^{{-1}}_{{x\times Y}}(z)|}{|g^{{-1}}(z)|}.$ |

Entanglement quantifies how far the size of the pre-image of $g^{{-1}}(z)$ deviates from the sizes of its $X\times y$ and $x\times Y$ slices as $x$ and $y$ are varied. $\blacksquare$

By Corollary 8 entanglement also equals $ei({\mathfrak{m}}_{{X\bullet}}\rightarrow{\mathfrak{m}}_{{XY}},\delta_{z})-ei(% {\mathfrak{m}}_{{\bullet Y}},\delta_{z})$. In Diagram (11) 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 $v_{Y}\rightarrow v_{Z}$ to the null system and the subsystem ${\mathfrak{m}}_{{X\bullet}}=\{v_{X}\rightarrow v_{Z}\}$ respectively. Entanglement is the difference between the information generated by the diagonal arrows. It quantifies the difference between the information $\{v_{Y}\rightarrow v_{Z}\}$ generates in two different contexts.

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

Function^{} $X\times Y\xrightarrow{g}Z$ performs a
disentangled measurement when outputting $z$ iff

$g^{{-1}}(z)=g^{{-1}}_{{x\times Y}}(z)\times g^{{-1}}_{{X\times y}}(z)$ |

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

Proof: By Theorem 10 entanglement is zero iff

$|g^{{-1}}(z)|=|g^{{-1}}_{{x\times Y}}(z)|\cdot|g^{{-1}}_{{X\times y}}(z)|$ |

for any $x,y$ such that $g(x,y)=z$. This implies the desired result since $g^{{-1}}(z)\hookrightarrow g^{{-1}}_{{x\times Y}}(z)\times g^{{-1}}_{{X\times y% }}(z)$. $\blacksquare$

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 $x\times Y$ and $X\times y$ 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 $g:X\times Y\rightarrow Z$ 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 $v_{X}$ and $v_{Y}$. However,
the bit is *not localizable*. The measurement
generates no information about occasion $v_{X}$ taken singly,
since its output could have been 0 or 1 with equal
probability; and similarly for $v_{Y}$.

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 $X\times Y\xrightarrow{g}Z$ is surjective^{}, then

$\gamma({\mathfrak{m}}_{{XY}},{\mathcal{P}},\delta_{z})=0$ for all $z\in Z$ iff $g$
decomposes into $X\times Y\xrightarrow{g_{1}\times g_{2}}Z_{1}\times Z_{2}=Z$ for $X\xrightarrow{g_{1}}Z_{1}$ and
$Y\xrightarrow{g_{2}}Z_{2}$.

Proof: The reverse implication is trivial. In the forward direction, note that $Z=\{g^{{-1}}(z)|z\in Z\}$ and, by Corollary 11, each pre-image has product structure $g^{{-1}}(z)=g^{{-1}}_{{x\times Y}}(Z)\times g^{{-1}}_{{X\times Y}}(z)$. Let $Z_{1}=\{g^{{-1}}_{{X\times y}}|y\in Y\mbox{ and }z\in Z\}$ and similarly for $Z_{2}$. Define

$g_{1}:X\rightarrow Z_{1}:x\mapsto\mbox{the unique element of form }g^{{-1}}_{{X\times y}}(z)\mbox{ containing it,}$ |

and similarly for $g_{2}$.

# References

- 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.

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