# Computing the Integrated Information of a Quantum Mechanism

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## Abstract

**:**

## 1. Introduction

## 2. Theory

#### 2.1. Classical Systems

#### 2.1.1. Cause and Effect Repertoires

#### 2.1.2. Intrinsic Difference (ID)

#### 2.1.3. Identifying Intrinsic Causes and Effects

#### 2.1.4. Disintegrating Partitions

#### 2.1.5. Mechanism Integrated Information

#### 2.2. Quantum Systems

#### 2.2.1. Quantum Cause and Effect Repertoires

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

- If ${\rho}_{t+1}^{Z|m}$ corresponds to a pure state, the purview qubits are fully determined by the mechanism qubits. Thus, ${\rho}_{t+1}^{Z|m}$ is not influenced by qubits outside of m. It follows that ${\pi}_{e}\left(Z\right|m)={\rho}_{t+1}^{Z|m}$ if the latter is pure. This is analogous to the classical case, where ${\pi}_{e}\left(Z\right|m)=p\left({Z}_{t+1}\right|{m}_{t})$ if $p\left({Z}_{t+1}\right|{m}_{t})$ is deterministic.
- Conceptually, entangled subsets are treated as indivisible units in the effect repertoire. If a purview is fully entangled, then ${\pi}_{e}\left(Z\right|m)={\rho}_{t+1}^{Z|m}$.
- Extraneous classical correlations are successfully discounted, which means they will not contribute to the integrated information of a mechanism (Figure 3).

**Definition**

**5.**

#### 2.2.2. Quantum Intrinsic Information (QID)

#### 2.2.3. Identifying Intrinsic Causes and Effects

#### 2.2.4. Disintegrating Partitions

#### 2.2.5. Quantum Mechanism Integrated Information

#### 2.2.6. The Intrinsic Structure of a Quantum System

## 3. Results

#### 3.1. CNOT

#### 3.1.1. Classical Case

#### 3.1.2. Quantum Case

#### 3.1.3. Mixed States and Extensions to Larger Systems

#### 3.1.4. Intrinsic Structure Due to Entanglement

## 4. Discussion

#### 4.1. Comparison with Previous Approaches

#### 4.2. Measurement Dynamics

#### 4.3. Formal Considerations and Limitations

#### 4.4. From Micro to Macro?

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Outline of the IIT analysis applied to a classical COPY-XOR gate. (

**a**) The COPY-XOR gate and its (deterministic) transition probability function ${\mathcal{T}}_{S}$ depicted by a probability matrix. To avoid a proliferation of subscripts, in the following we use different letters to denote inputs and outputs. For conceptual ease, $A/C$ and $B/D$ can (but do not have to) be interpreted as the same physical units before and after the update. Unit C is a copy of the input bit A, and D corresponds to an XOR function of both input bits ($A,B$). For input state $AB=10$ (also denoted by ${10}_{AB}$), the COPY-XOR gate outputs $CD=11$ (denoted by ${11}_{CD}$). (

**b**) Based on ${\mathcal{T}}_{S}$, we can identify the intrinsic effect of a mechanism M in its current state m over a purview Z as the effect state ${z}_{e}^{\prime}$ with maximal intrinsic effect information $i{i}_{e}$. For $m={10}_{AB}$ and $Z=CD$, the intrinsic effect is ${z}_{e}^{\prime}={11}_{CD}$. (

**c**) Next, we assess the irreducibility of the intrinsic effect by computing the integrated information ${\phi}_{e}(m,Z)$ over the minimum partition (MIP). (

**d**) To identify the maximally irreducible effect of a mechanism m, we compare ${\phi}_{e}(m,Z)$ across all possible effect purviews Z. Here, the maximally irreducible effect of $m={10}_{AB}$ is ${z}_{e}^{*}={11}_{CD}$ because it specifies a maximum of ${\phi}_{e}$ and is the largest purview that does so (see text for details). (

**e**) For a given system, we identify all maximally irreducible causes and effects. Given the input state $AB=10$, the classical IIT analysis identifies two irreducible effects; the first-order mechanism ${1}_{A}$ specifies the effect ${1}_{C}$, and the second-order mechanism ${10}_{AB}$ specifies the effect ${11}_{CD}$. Given the output state $CD=11$, the IIT analysis identifies three irreducible mechanisms, including mechanism ${1}_{D}$ with purview ${10}_{AB}$ or ${01}_{AB}$ (which are tied). Both intrinsic information ($ii$) and integrated information ($\phi $) are quantified in “ibit” units (see text below).

**Figure 2.**Causal marginalization. Let us assume we want to identify the effect of the input bit $B=0$ (or ${0}_{B}$) on the output $CD$ in the COPY-XOR system of Figure 1. Intuitively, by itself, ${0}_{B}$ does not have an effect on C, as it does not input into C. It also has no effect on D because, by itself, it specifies no information about the output state of the XOR D. However, simply marginalizing the input A (averaging over all possible input states of A while maintaining the common inputs from A to C and D) would result in a “spurious” correlation between the output bits that is not due to B, but instead due to the common inputs from A. Capturing the fact that ${0}_{B}$, by itself, has no effect on $CD$, requires causal marginalization (independent marginal inputs to each unit in the effect purview).

**Figure 3.**CNOT gate. The CNOT operator is shown in the top box. (

**a**) For a pure input state in the classical basis, we obtain the same results as in the classical case (Figure 1). (

**b**) For a pure input state in the Hadamard basis, the role of the “control” (here B) and “target” (here A) is reversed compared to (

**a**) (as indicated in the circuit diagram). (

**c**) The CNOT is often used to produce a “Bell state” of two maximally entangled qubits. In this exclusively quantum scenario, only the second-order mechanisms ${\left|+0\right.\u232a}_{AB}$ and ${\left|{B}^{+}\right.\u232a}_{CD}$ specify an effect or cause, respectively. None of the subsets has any cause or effect information ($\phi =0$ ibit). (

**d**) Conversely, given the input state ${\left|0+\right.\u232a}_{AB}$, all second-order mechanisms are fully reducible ($\phi =0$ ibit) and only the first-order mechanisms specify causes and effects.

**Figure 4.**Mixed states and entanglement with the environment. (

**a**) IIT analysis of the CNOT gate with a mixed input state ${\rho}^{AB}=0.5\ast (\left|00\right.\u232a\left.\u232900\right|+\left|11\right.\u232a\left.\u232911\right|)$. (

**b**) It is possible to describe the mixed state as a pure state entangled with the environment. Analyzing such an extended system for the case in (

**a**), the cause and effect of the subsystem are preserved in the larger system (gray), but we obtain additional causes and effects that span all three qubits (black). ${\left|GHZ\right.\u232a}^{\prime}$ denotes a maximally entangled superposition of states $\left|001\right.\u232a$ and $\left|110\right.\u232a$.

**Figure 5.**Intrinsic structure of three-qubit states. (

**a**) Classical states specify first-order constraints under an identity function (equivalent to three classical COPY gates). (

**b**) The maximally entangled GHZ-state only specifies a third-order constraint. (

**c**) By contrast, the W-state, which is also maximally entangled, specifies constraints of all orders. Subsets $m\subseteq s$ of the W-state are indicated by ${\rho}^{m}$. The remaining units $s\backslash m$ are traced out. $\left|{B}^{\prime +}\right.\u232a$ indicates a superposition of $\left|10\right.\u232a$ and $\left|01\right.\u232a$.

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**MDPI and ACS Style**

Albantakis, L.; Prentner, R.; Durham, I.
Computing the Integrated Information of a Quantum Mechanism. *Entropy* **2023**, *25*, 449.
https://doi.org/10.3390/e25030449

**AMA Style**

Albantakis L, Prentner R, Durham I.
Computing the Integrated Information of a Quantum Mechanism. *Entropy*. 2023; 25(3):449.
https://doi.org/10.3390/e25030449

**Chicago/Turabian Style**

Albantakis, Larissa, Robert Prentner, and Ian Durham.
2023. "Computing the Integrated Information of a Quantum Mechanism" *Entropy* 25, no. 3: 449.
https://doi.org/10.3390/e25030449