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--- bib:albantakis2023computing [2026/01/28 14:00] – kymki
+++ bib:albantakis2023computing [2026/05/21 06:29] (current) – [Steelman] kymki
@@ Line 17: / Line 17: @@
 ===== BibTeX =====
-<code bibtex>
+<code javascript>
 @article{albantakis2023computing,
   title={Computing the integrated information of a quantum mechanism},
@@ Line 31: / Line 31: @@
 ====== Albantakis et al. (2023) — QM re-formulation and failure modes ======
-This note rewrites the paper’s core definitions in standard quantum information language, and isolates where the framework is (i) pure relabeling, (ii) a specific intervention convention, and (iii) where it becomes physically ambiguous.
+Im reading this with minimal background in IIT and most of its concepts are unknown to me. The reading is based on trying to extract what QIT the authors lean on to understand if what they propose makes "physical sense".
+This note rewrites the paper’s core definitions in standard quantum information language, and tries to explain conclusions from that standpoint. The case may be that I completely miss subtleties due to not being well read in IIT.
 ===== 1. Translation into standard QM language =====
@@ Line 227: / Line 229: @@
 ==== 3.5 The score $\varphi(m,Z,\theta)$ ====
-They define an “integrated information for this purview under this partition” as a divergence between:
+Let (Q) be factorized and let
+$$
+T:\mathcal B(\mathcal H_Q)\to\mathcal B(\mathcal H_Q))
+$$
+be the global channel. For a chosen **input subsystem** (M) prepared in state $\rho_M$, define the **intervention-induced effective channel**
+$$
-  - the full repertoire $\pi_e(Z\mid m)$, and
+\mathcal E_{M\to Z}(X):=\operatorname{tr}*{Z^0}!\bigl(T(X\otimes \tau*{M^0})\bigr),
-  - the cut repertoire $\pi_e^{\theta}(Z\mid m)$,
+\qquad \tau_{M^0}:=\frac{I_{M^0}}{d_{M^0}},
-using their QID:
 $$
-\varphi(m,Z,\theta)
+and the corresponding **output marginal** on the chosen **target subsystem** (Z):
-\;\equiv\;
-\mathrm{QID}\Bigl(\pi_e(Z\mid m)\;\Vert\;\pi_e^{\theta}(Z\mid m)\Bigr).
 $$
-Crucial detail: with their maximally mixed baseline choice, QID behaves like a **max-eigenvalue-weighted** distinguishability, and the “intrinsic effect” is the eigenvector of $\pi_e(Z\mid m)$ with the largest eigenvalue. In practice this means the comparison is dominated by the top-eigenvalue eigenspace rather than the full density matrix structure.
+\rho_Z:=\mathcal E_{M\to Z}(\rho_M).
-(Their Eq. (31) writes this out in components and evaluates it at the maximizing eigenstate.)
+$$
-----
+For a partition $\theta\in\Theta(M,Z)$, construct the paper’s **cut / partitioned output** $\sigma_Z^{(\theta)}$ by (i) injecting maximally mixed noise across the cut and (ii) taking a tensor product across the partition blocks (optionally after the paper’s ($P^*$) “entanglement cluster” factorization step for $|Z|>1)$. Denote the resulting state by
+$$
+\sigma_Z^{(\theta)}:=\pi^{\theta}_e(Z\mid m)\quad\text{(paper’s notation)}.
+$$
+Now diagonalize
+$$
+\rho_Z=\sum_i p_i|i\rangle\langle i|,\qquad
+\sigma_Z^{(\theta)}=\sum_j q_j|j\rangle\langle j|,
+\qquad
+P_{ij}:=|\langle i|j\rangle|^2.
+$$
+**Intrinsic (selected) effect state.** The paper first selects a particular eigenstate $|i^*\rangle$ of $\rho_Z$ via an “intrinsic information vs baseline” optimization. With their maximally mixed baseline on $Z$, this selection reduces to choosing the **largest-eigenvalue eigenvector** of $\rho_Z$ (or the corresponding eigenspace if degenerate). Call that chosen eigenstate
+$$
+z'_e(m,Z):=|i^*\rangle.
+$$
+**Integrated effect information for a partition.** The paper then evaluates its QID expression **at that chosen eigenstate** (not by maximizing over (i) again):
+$$
+\phi_e(m,Z,\theta)
+:=\phi(m,z'_e,\theta)
+=
+p_{i^*}\left(\log p_{i^*}-\sum_j P_{i^*j}\log q_j\right).
+$$
+Equivalently, define the geometric mean
+$$
+\bar q_{i^*}:=\exp!\left(\sum_j P_{i^*j}\log q_j\right),
+$$
+so that
+$$
+\phi_e(m,Z,\theta)=p_{i^*}\log!\frac{p_{i^*}}{\bar q_{i^*}}.
+$$
 ==== 3.6 Optimization: MIP and “best” purview ====
-For each subset $Z$, they define a “minimum information partition” (MIP) as the partition that minimizes $\varphi$:
+For fixed $M,\rho_M$ and candidate target $Z$, the paper chooses the minimum information partition (MIP) using a normalized criterion:
 $$
-\theta^{\star}(m,Z)
+\theta'(m,Z)
-\;:=\;
+=
-\arg\min_{\theta}\;\varphi(m,Z,\theta).
+\arg\min_{\theta\in\Theta(M,Z)}
+\frac{\phi_e(m,Z,\theta)}{\max_{T'*S}\phi_e(m,Z,\theta)}.
 $$
+Here $\max*{T'_S}$ ranges over alternative systems/channels of the same dimensions; this normalization is part of the selection of $\theta$.
-Then they define the integrated information for the mechanism (for effects) by maximizing over purviews:
+Then the (reported) integrated effect information for that $Z$ is
 $$
-\varphi_e(m)
+\phi_e(m,Z):=\phi_e(m,Z,\theta'(m,Z)).
-\;:=\;
-\max_{Z\subseteq Q}\;\min_{\theta}\;\varphi(m,Z,\theta).
 $$
-(They do an analogous construction for causes and combine them in the full IIT-style definition.)
+Finally, the best target subsystem (or in paper terms: “maximally irreducible effect purview” - not sure why it is irreducible) is chosen by
+$$
+Z^**e(m)=\arg\max*{Z\subseteq Q}\phi_e(m,Z),
+\qquad
+\phi_e(m)=\max_{Z\subseteq Q}\phi_e(m,Z).
+$$
-----
 ==== 3.7 Mathematics Framework ====
@@ Line 505: / Line 556: @@
 ==== Steelman ====
-As a *defined* counterfactual causal attribution scheme, the framework:
+As a defined counterfactual causal attribution scheme, the framework:
   * avoids common-cause “spurious correlation” effects by construction (their COPY-XOR/CNOT motivation),
   * tries to preserve entanglement-generated correlations by clustering entanglement before taking products,