Building a quantum superposition of conscious states with integrated information theory

Could there be a quantum superposition of consciousness? This question was raised by Eugene Wigner in the thought experiment that is now known as “Wigner’s Friend”. Wigner imagined his friend, in a nearby sealed lab, making a quantum measurement. Wigner, who is uncertain of his friend’s result, wonders whether he should consider his friend to have entered a quantum superposition of experiencing different results. Wigner argued that this is “absurd because it implies that my friend was in a state of suspended animation”. He then concluded that “consciousness must have a different role in quantum mechanics than the inanimate measuring device” ([51, p.180]).

There has since been much speculation by physicists and philosophers over whether states of consciousness could be superposed and what that would even mean. For example, there have been many attempts to extend the Wigner’s friend scenario and the associated epistemological and metaphysical implications ([21], [14], [19], [13], [52]). There have also been many attempts to make sense of superpositions of conscious states in many worlds and many minds interpretations of quantum mechanics ([20], [44], [50], [33], [16], [5], [8], [31], [32]). However, without any well-defined criteria for determining which physical states are conscious (and to what degree), the question of whether there could be such a superposition, and what it would be like to be in one, is difficult to evaluate.

Recent neuroscience, on the other hand, has seen the rise of mathematical theories of consciousness, notably, the integrated information theory, or IIT for short ([46], [47], [40], [48], [4])). IIT associates systems with both quantitative amounts of consciousness (roughly, the amount of integrated information in the system, denoted by the symbol $\Phi$) and qualitative states of consciousness (roughly, the “shape” of the system’s integrated information, or its “Q-shape”). More recently, IIT has been extended into the quantum domain in a framework known as QIIT ([53], [27], [3]). Inspired by these results, Wigner’s suggestion that consciousness may be responsible for the collapse of the wave function has been resurrected in models that use integrated information as a criterion for collapse ([28], [17]). In comparison to standard collapse models [11], it has been claimed that IIT-based consciousness-collapse models may be much easier to experimentally test, since they can be tested by the right sorts of quantum computers, if only we could design the right sort of circuit [17].

In this paper, we propose such a circuit which, if implemented, would put a simple quantum computer into a superposition of states of conscious experience according to the IIT definition of consciousness. Following [17], we consider the simplest non-zero $\Phi$ system, a feedback dyad. Classically, the dyad has four possible states: (0,0), (1,1), (0,1), and (1,0). Each state is predicted to have a tiny amount of consciousness. This prediction is robust across successive IIT formalisms. Each dyad state has $\Phi=2$ in IIT2.0 and $\Phi=1$ in IIT3.0, as shown in [37]. Here, we show that each dyad state has $\Phi=2$ in IIT4.0 (section 3) and in QIIT (appendix B). Although these states have the same amount of consciousness, they yield different states of consciousness, because they are associated with different Q-shapes, as we show in section 4. The dyad in a superposition of two of its four possible states is therefore the simplest consciousness superposition predicted by IIT. We refer to this as “Schrödinger’s dyad” and we propose a simple quantum circuit that allows Schrödinger’s dyad to be built.

We would like to stress that we are not endorsing IIT, and so we remain agnostic on whether the dyad is conscious in any meaningful sense. IIT has been shown to be consistent with a number of important experimental results in neuroscience ([34], [15], [22], [2], [30], [29], [38]). However, many criticisms of IIT have also been proposed, and we are sympathetic with some of them ([23], [1], [12], [7], [18], [41]). Either way, what we show is that unless one drastically revises IIT (e.g. [36], [35]), then either IIT is false or the dyad is conscious and can easily be put into a superposition of conscious states. We leave it to the reader to decide between these options.

In section 5 we identify the simplest possible consciousness-collapse model, which predicts that Schrödinger’s dyad is unstable and collapses at a rate determined by a measure of difference between the superposed conscious states. We take the Q-shapes defined in section 4, and use them to define the simplest possible collapse operators. This toy model makes a number of important properties of such models transparent. We then compare our toy-model to the more general consciousness-collapse model proposed in [17].

Finally, in section 6 we propose a physical implementation of Schrödinger’s dyad, in which two photons enter into a feedback loop inside an optical cable. On the one hand, the implementation may potentially falsify the simplest versions of the IIT-based consciousness collapse models. On the other hand, the example raises a difficulty with IIT when it comes to physical implementation: IIT assumes that there is always an objective fact of the matter about what the basic causal units in a physical system are.

In addition to identifying this prediction of IIT, our analysis helps to reveal much about the structure of IIT. For example, we resolve a crucial ambiguity in IIT in which logic gates are treated as having binary states (section 2). We also identify a subtle inconsistency between the IIT4.0 description of the dyad and the axioms of IIT4.0 (appendix A).

The paper is organized as follows. Section 2 describes the classical feedback dyad. Section 3 shows how to calculate the classical dyad’s $\Phi$ using IIT4.0. Section 4 provides a simple way of describing the classical dyad’s Q-shape. Section 5 explains the simple consciousness-collapse model. Section 6 proposes a physical implementation of the dyad, which may test the model, but which also raises questions about how to understand causality in IIT.

Finally, appendix A explains IIT4.0 more generally and identifies the steps in the IIT calculus that our simple dyad allowed us to skip; appendix B shows how our analysis is consistent with QIIT as presented in [3]; and appendix C proves a general result concerning our Q-shape collapse operators.

The classical dyad is a simple system consisting of two elements or channels, A and B, that simply swap their states from one time step to the next. That is, if at some time, $t_{0}$, A is in state 1 and B is in state 0, then at the next time step, $t_{+1}$, A is in state 0 and B is in state 1. The action on these channels is equivalent to a logical SWAP gate which is given a simple diagrammatic representation in Figure 1.

The figure makes it clear that there are three distinct levels of description to the dyad: channels, channel values, and channel relationships. A and B are the channels that are related via the logical SWAP gate in such a way as to exchange their values. In the language of quantum information, the channels are systems, the channel values are states, and the channel relationships are transformations. This is a crucial point. The SWAP gate is a transformation of the states of systems A and B. The gate itself is never “in a state” on its own.

This is an important distinction because gates are frequently described as being in a state that possesses a value, especially in IIT3.0 [40]. In particular, the elements or nodes in the IIT3.0 diagrams have binary states but are also treated as being logic gates. If the nodes are understood as neurons, then they are considered as being in an “active” or an “inactive” state [9], much like a channel. Yet the neurons are also said to act like gates by only activating in response to the right combination of connections to other neurons that are themselves either active or inactive. But this is really a notational relic from the early days of Boolean networks [24, 25] that ignores what is happening at a more granular level. In the neuronal case, an “active” or “inactive” neuron really refers to whether it sends a signal via some channel, i.e. it represents an action. The difference is typically unimportant at the granularity in which it is usually considered. But when considering quantum models of these networks, this treatment breaks down. As such, it is the states of the channels that can be in superposition, not the gates themselves. In a neuronal sense, it is thus conscious states that are in superposition, not the physical neurons themselves.

Figure 1 also highlights a fundamental causal dependence in the dyad. The output of channel A causally depends on the input to channel B and vice-versa. In order to emphasize this point, we use capital letters to identify the channels or systems themselves and lowercase letters to identify the values the channels can attain, i.e. their states. One could think of the SWAP gate as a black box with the channels simply identifying the locations of the inputs and outputs of the box. Values are fed into the inputs and then produced by the outputs.

To develop a feedback system with this SWAP gate we simply feed the outputs directly back into the inputs. For simplicity we can represent this system over a series of time steps in the manner shown in Figure 3.

The output at a given time step is given as in Figure 1. For example, if the system at a given time step is given by $(a,b)\in\{0,1\}$ where the first element of the set is the state of channel A and the second is the state of channel B, if the inputs were $(a=0,b=1)\equiv(0,1)$ the evolution of the system state over time is just $(0,1)\to(1,0)\to(0,1)$.

Creating Schrödinger’s dyad then requires that we treat the channels as quantum and represent their states as such. That is, a classical state $(a,b)$ is equivalent to a pure quantum state in the so-called computational basis $\ket{a,b}$. A superposition of the $\ket{1,0}$ and $\ket{0,0}$ states can be achieved by feeding the superposition state

$$\ket{+}=\frac{1}{\sqrt{2}}\left(\ket{0}+\ket{1}\right)$$

(1)

into channel $B$ at $t_{-1}$. The input state to the dyad as a whole at $t_{-1}$ is

$$\ket{0,+}=\ket{0}\otimes\ket{+}=\frac{1}{\sqrt{2}}\left(\ket{0,0}+\ket{0,1}\right),$$

(2)

which then evolves into the following state at $t_{0}$:

$$\ket{+,0}=\ket{+}\otimes\ket{0}=\frac{1}{\sqrt{2}}\left(\ket{0,0}+\ket{1,0}\right).$$

(3)

This is not a superposition of $\Phi$ values, since all four possible classical states of the dyad have the same $\Phi$ value. It is, however, a superposition of distinct Q-shapes according to IIT3.0 and IIT4.0, as we show in section 4. And so according to IIT, the $t_{0}$ state of equation 3 represents a superposition of qualitatively distinct states of consciousness. We begin by calculating the dyad’s $\Phi$.

The general procedure for calculating $\Phi$ and Q-shape takes many steps. Fortunately, the simplicity of our dyad allows us to skip several steps and to emphasize the most important ones. We explain the more general case in appendix A.

The dyad consists of two parts, A and B. We begin by calculating the integrated cause information and the integrated effect information of each part. Integrated information concerns how much information is lost by partitioning the system, which means replacing a causal relationship with noise where the noise is represented as an equiprobable distribution over all possible states.

To illustrate, let us calculate how much integrated effect information A has, given its present state, about the next state of each of the system’s parts, A and B. The maximum of these defines A’s integrated effect information.

Given that our dyad is a SWAP gate, it is trivially true that A’s present state has zero integrated effect information about A’s next state since A’s next state is entirely determined by B’s present state. Put another way, A’s possible next states are all equally probable given its current state. So introducing a partition that induces noise between A at $t_{0}$ and A at $t_{+1}$ makes no difference. This makes sense given that there is no causal connection between them in the first place: A affects B but not itself in the next time step. A’s present state is not causally connected to A’s future state and so there is no integrated effect information.

However, A’s present state does fully determine B’s future state and so if, for example, our system’s present state is (1,0), equation 39 in [4] tells us that the integrated effect information of A’s state at time $t_{0}$ given that it is in state 1 at that instant, is

$$\phi_{e}(a_{t_{0}}=1)=p(b_{t_{+1}}=1|a_{t_{0}}=1)\log_{2}\left[\frac{p(b_{t_{+1}}=1|a_{t_{0}}=1)}{p^{\theta}(b_{t_{+1}}=1|a_{t_{0}}=\textrm{noise})}\right].$$

(4)

Here, $p(b_{t_{+1}}=1|a_{t_{0}}=1)$ is the probability that B will be in state $b=1$ at time $t_{+1}$ given that A is currently in state 1. It is trivially true that this equals 1. Likewise $p^{\theta}(b_{t_{+1}}=1|a_{t_{0}}=\textrm{noise})$ represents the probability that B will be in state 1 at time $t_{+1}$ given the partition $\theta$ which sets the value of channel A to an equiprobable distribution of the two possible states. In other words, the partition replaces the effect that A had on B with noise, which means that B’s future state is randomly determined. Since there are only two possible states, that means that $p^{\theta}(b_{t_{+1}}=1|a_{t_{0}}=\textrm{noise})=0.5$. As such, we have

$$\phi_{e}(a_{t_{0}}=1)=1\cdot\log_{2}\left[\frac{1}{0.5}\right]=1.$$

(5)

The same basic equation tells us that the integrated effect information of B’s state at time $t_{0}$, $\phi_{e}(b_{t_{0}}=0)$, also equals 1.

The integrated cause information for A is calculated in a slightly different manner and illustrates a time asymmetry in the equations of IIT. As in the effect case, the past state of A contains no information about the present state of A, and likewise for B. We only consider the information B’s past state has on A’s current state and the information A’s past state has on B’s current state. Specifically, given a current state of (1,0), equation 42 in [4] gives

$$\phi_{c}(a_{t_{0}}=1)=p(b_{t_{-1}}=1|a_{t_{0}}=1)\log_{2}\left[\frac{p(a_{t_{0}}=1|b_{t_{-1}}=1)}{p^{\theta}(a_{t_{0}}=1|b_{t_{-1}}=\textrm{noise})}\right]$$

(6)

where $p(b_{t_{-1}}=1|a_{t_{0}}=1)$ is calculated according to Bayes’ rule as follows:

$$p(b_{t_{-1}}=1|a_{t_{0}}=1)=\frac{p(a_{t_{0}}=1|b_{t_{-1}}=1)\cdot p(b_{t_{-1}}=1)}{p(a_{t_{0}}=1)}$$

(7)

where $p(a_{t_{0}}=1)$ and $p(b_{t_{-1}}=1)$ are unconstrained probabilities (see equations 6-8 in [4]) and are both equal to 0.5 since, at any given time step and with no knowledge of past or future states, the probability that we will find either channel in a given state is 0.5 because there are only two states. Here we also have that $p(a_{t_{0}}=1|b_{t_{-1}}=1)$ is the probability that A’s current state is 1 if B’s past state is 1 and $p(b_{t_{-1}}=1|a_{t_{0}}=1)$ is the probability that B’s past state was 1 given that A’s state is currently 1. As before, $p^{\theta}(a_{t_{0}}=1|b_{t_{-1}}=\textrm{noise})$ noises the system and is equal to 0.5. Since $p(a_{t_{0}}=1|b_{t_{-1}}=1)=1$, Bayes’ rule given by equation (7) tells us that $p(b_{t_{-1}}=1|a_{t_{0}}=1)=1$. As before, then, we find that $\phi_{c}(a_{t_{0}}=1)=1$. Likewise, the same process tells us that $\phi_{c}(b_{t_{0}}=0)$ also equals 1.

Equation 45 in [4] then tells us that the integrated information of a part is the minimum of its integrated effect and integrated cause information, i.e.

	$\displaystyle\phi(a_{t_{0}}=1)$	$\displaystyle=\min\left[\phi_{c}(a_{t_{0}}=1),\phi_{e}(a_{t_{0}}=1)\right]$		(8)
	$\displaystyle\phi(b_{t_{0}}=0)$	$\displaystyle=\min\left[\phi_{c}(b_{t_{0}}=0),\phi_{e}(b_{t_{0}}=0)\right]$		(9)

respectively, which are both trivially 1.

The amount of consciousness ($\Phi$) in the state of the whole system is then simply a sum of the integrated information of the smaller subsystems as calculated above. The state of the dyad at the time $t_{0}$ therefore has

	$\displaystyle\Phi(t_{0})$	$\displaystyle=\phi(a_{t_{0}}=1)+\phi(b_{t_{0}}=0)$		(10)
		$\displaystyle=1+1=2$

units of consciousness.

No matter which of its four possible states the dyad is in, all of the above reasoning applies, and we find that it always has two units of consciousness. It is therefore not possible to put the dyad into a superposition of $\Phi$-values. What we can do, however, is put the system into a superposition of different states of consciousness.

To understand this distinction intuitively, compare experiencing a green screen with experiencing a blue screen. It might be that these two experiences do not correspond to any difference in $\Phi$ (why would changing only the color change the amount of consciousness?). Now imagine that we put a subject into a superposition of experiencing a blue screen and experiencing a green screen. By assumption this is not a $\Phi$ superposition, but it is clearly a superposition of distinct conscious experiences. One might doubt that distinct human states of consciousness could ever have identical $\Phi$ [26], but IIT allows for this in AI, and IIT3.0 and IIT4.0 predict that this is indeed the case for our simple dyad, as we now explain.

If two qualitatively distinct states of consciousness are quantitatively identical (i.e. they have identical $\Phi$), then their distinctness must come down to the different ways in which each state generates that $\Phi$-value. This difference is what is captured in a Q-shape.¹¹1This has come under various labels in the literature. In [40] it is primarily referred to as a “maximally irreducible conceptual structure (MICS)”. But it is also referred to as a “shape in qualia space”, and so we adopt the simpler terminology, “Q-shape”. In [4] it is referred to as a “$\Phi$-structure”. In this section we define Q-shapes for all four states of the dyad. We show that these Q-shapes are distinct. It follows that IIT (as presently formulated) must treat these states as corresponding to qualitatively distinct states of consciousness. Finally, we discuss some differences in how Q-shapes are understood in IIT3.0 versus IIT4.0, which will be relevant to the collapse model proposed in the next section.

The dyad states $(1,0)$ and $(0,0)$ each have $\Phi=2$, but for different reasons. This can be seen by partitioning the dyad, replacing some of the parts by noise, as defined above, and then noting that $(1,0)$ and $(0,0)$ induce different forward and backward probability distributions. These different distributions lead to different Q-shapes. In the general case of more complex systems, we also have to weigh the parts according to their individual values of $\phi$. The simple structure of the dyad allows us to bypass this (since $\phi(A)=\phi(B)=1$), but we will return to the more general case in the next section.

We begin with part A, when the dyad is in state $(1,0)$. The prescription of partitioning means that we replace the complement of A (that is, B) by noise, i.e. an equiprobable distribution of $0$ and $1$, while keeping $A$ in state $1$. Evolving this forward in time, we obtain a probability distribution $(0,\frac{1}{2},0,\frac{1}{2})$, where we have labelled the four states in lexicographical order: $(0,0),(0,1),(1,0),(1,1)$. Evolving it backwards in time, i.e. retrodicting the dyad’s state at one time step earlier, we obtain exactly the same probability distribution. This gives us the first two rows in the Q-shape matrix

$$Q(1,0)=\left(\begin{array}[]{cccc}0&\frac{1}{2}&0&\frac{1}{2}\\ 0&\frac{1}{2}&0&\frac{1}{2}\\ \frac{1}{2}&\frac{1}{2}&0&0\\ \frac{1}{2}&\frac{1}{2}&0&0\end{array}\right).$$

(11)

The third and fourth row are the forward (effect) and backward (cause) probability distributions that we obtain if we consider the subsystem B instead, keeping it in state $0$ and replacing A by noise as above. Thus, the Q-shape of a given state (such as $(1,0)$) is a collection of four probability distributions over the four dyad states, represented by the four rows in our representation matrix.

Performing the calculation for the other dyad states (which each have $\Phi=2$ due to the two parts always having $\phi$=1), we obtain

$$Q(0,0)=\left(\begin{array}[]{cccc}\frac{1}{2}&0&\frac{1}{2}&0\\ \frac{1}{2}&0&\frac{1}{2}&0\\ \frac{1}{2}&\frac{1}{2}&0&0\\ \frac{1}{2}&\frac{1}{2}&0&0\end{array}\right),\enspace Q(0,1)=\left(\begin{array}[]{cccc}\frac{1}{2}&0&\frac{1}{2}&0\\ \frac{1}{2}&0&\frac{1}{2}&0\\ 0&0&\frac{1}{2}&\frac{1}{2}\\ 0&0&\frac{1}{2}&\frac{1}{2}\end{array}\right),\enspace Q(1,1)=\left(\begin{array}[]{cccc}0&\frac{1}{2}&0&\frac{1}{2}\\ 0&\frac{1}{2}&0&\frac{1}{2}\\ 0&0&\frac{1}{2}&\frac{1}{2}\\ 0&0&\frac{1}{2}&\frac{1}{2}\end{array}\right).$$

(12)

Our Q-shapes are not really “shapes”; they are just matrices of probability distributions. But we can turn them into shapes by following the IIT3.0 prescription described in [40] (see especially Figures 10-12). To obtain such visualizations for our dyad states, we simply interpret two probability distributions over the dyad’s state space (which have four real entries each) as an element of the eight-dimensional vector space $\mathbb{R}^{8}$. This is the phase space of the dyad. Let us use this to build the “shape” corresponding to $Q(1,0)$ from equation 11 above. Consider the first two rows of that matrix. They determine the location of part A. The last two rows determine the location of part B. This gives us two points in the eight-dimensional space. Since $\phi=1$ in all our cases, it does not help us to distinguish Q-shapes, so we have ignored it.

IIT3.0 therefore predicts that the dyad is (minimally) conscious, and can be in one of four qualitatively distinct conscious states. It is natural therefore to wonder what it is like to be the dyad, and what these qualitative differences actually consist of. This is a question that IIT actually aims to answer. That is, IIT wants to be able to say something about what the experience of any given conscious system is like, especially when the system is incapable of verbal reports. The general idea is to extrapolate from features of our own Q-shapes.

For example, consider what it is like to be an echolocating bat. In [39] it was famously argued that this question is intractable. However, more recently in [49] it was argued that IIT makes it tractable. The idea is to consider the general properties of human visual experience Q-shapes and human auditory experience Q-shapes. Then, we compare them with bat experience Q-shapes. If bat experience Q-shapes are “more similar” to, say, human auditory experience Q-shapes, then we can say something about what it is like to be a bat (it is more like human auditory experience than human visual experience). Of course for both human and bat experience, deriving exact Q-shapes is far too complicated. Consequently, there is also no straightforward way to compare the dyad Q-shapes with (aspects of) our Q-shapes.

Nonetheless, there is a curious discussion about this in [40] (see Figure 19), that considers a system that is only slightly more complex than our dyad, which they call a “photodiode”. It also involves two parts, labelled ‘D’ and ‘P’, that specify each other’s states at each time step. (The main difference is that D receives two external inputs and has a threshold $\geq$ 2. All connections have weight 1. Meanwhile P serves as a memory for the previous state of D and its feedback to D serves as a predictor of the next external input by effectively decreasing the threshold of D.) Despite these differences, its Q-shapes are very similar to our dyad’s Q-shapes. They also involve two points in an 8D space. About its experience, they say the following:

“It is instructive to consider the quality of experience specified by such a minimally conscious photodiode. […] D says something about P’s past and future, and P about D’s, and that is all. Accordingly, the shape in qualia space is a constellation having just two [points], and is thus minimally specific. […] Moreover, the symmetry of the [Q-shape] implies that the quality of the experience would be the same regardless of the system’s state: the photodiode in state DP=00, 01, or 10, receiving one external input, generates exactly the same [Q-shape] as DP=11. In all the above cases, the experience might be described roughly as “it is like this rather than not like this”, with no further qualifications. The photodiode’s experience is thus both quantitatively and qualitatively minimal.”

If all four states of the photodiode have the same Q-shape, then they must all correspond to the same probability distributions. For as they say (in the IIT3.0 jargon), the probability distributions (or “cause effect repertoires”) for each part (or each “concept”) specifies what each part “contributes to the quality of the experience”. (Meanwhile, the $\phi$ of each part is said to be “how much” the part is present in experience.) But as we have seen, our feedback dyad does not yield this result: the four possible states correspond to distinct Q-shapes. It is therefore not possible to simply describe each of the four possible conscious states of the dyad as “it is like this rather than not like this”. What could the differences in our four dyad Q-shapes possibly translate to in experience? These are difficult questions for IIT.

We have mostly followed the IIT3.0 rather than the IIT4.0 prescription for building Q-shapes. In IIT4.0, they are somewhat simpler, in that they replace probability distributions with states (see equation (56) in [4]). In particular, the IIT4.0 Q-shape of any dyad state is given by the $\phi$-values of A and B as well as the states that these $\phi$-values were maximized over. So in the case of $Q(1,0)$, A and B both have $\phi=1$; for A this was maximized over B being in state 1 (i.e. $b=1$), while for B this was maximized over A being in state 0 (i.e. $a=0$). For $Q(0,0)$, A and B both have $\phi=1$; for A this was maximized over B being in state 0 (i.e. $b=0$), while for B this was maximized over A being in state 1 (i.e. $a=1$). The four states of the dyad therefore correspond to distinct Q-shapes in IIT4.0, consistently with IIT3.0.

The choice of how to represent Q-shapes here seems somewhat arbitrary, as both options satisfy the constraint of identifying differences in how the parts contributed to an overall $\Phi$-value for the system. However, as we explain in the next section, the IIT3.0 choice is much better suited for a certain application of IIT: defining a fully general consciousness-collapse model.

In [17] a dynamical collapse model is proposed in which Q-shape superpositions are unstable and tend to collapse. The following general form for continuous collapse models ([10, p.27]) is used:

$$d\psi_{t}=[-i\hat{H}_{0}dt+\sqrt{\lambda}(\hat{A}-\langle\hat{A}\rangle_{t})dW_{t}-\frac{\lambda}{2}(\hat{A}-\langle\hat{A}\rangle_{t})^{2}dt]\psi_{t}.$$

(13)

The first term on the right-hand side of the equation represents Schrödinger evolution, while the remaining two terms represent the collapse evolution. Here, $\hat{H}_{0}$ is the Hamiltonian of the system, $\lambda$ is a real-valued parameter governing the collapse rate, $\hat{A}$ is a collapse operator whose eigenstates the system collapses towards, $\langle\hat{A}\rangle_{t}$ is its expected value at time $t$, and $W_{t}$ is a noise process which ensures that collapse happens stochastically at a rate determined by a measure of difference between the superposed $\hat{A}$ eigenstates.

The pure state $\rho_{t}^{W}:=|\psi_{t}\rangle\langle\psi_{t}|$ therefore evolves stochastically. All statistical predictions that we can extract from this state are linear in $\rho_{t}^{W}$ due to the Born rule. Hence, given a single realization of the process (13), all statistical predictions (say, about outcomes of any measurement that we might decide to perform at some point while the process unfolds) can be computed from $\rho_{t}:=\mathbb{E}[\rho_{t}^{W}]$ [10]. As a consequence of (13), this resulting state evolves according to the Lindblad equation

$$\frac{d}{dt}\rho_{t}=-i[\hat{H}_{0},\rho_{t}]-\frac{\lambda}{2}[\hat{A},[\hat{A},\rho_{t}]]$$

(14)

(for the derivation see e.g. [10]). Hence, the system can evolve via Schrödinger dynamics, via collapse, or via some combination of the two. To understand the collapse term we can ignore the Schrödinger dynamics term by setting its Hamiltonian to zero, $\hat{H}_{0}=0$. The collapse term only has an effect when the system is in a superposition of eigenstates of $\hat{A}$. In this situation, the double commutator will be non-zero and the state will evolve. The “speed” at which it evolves is a function of the eigenvalues $a_{i}$ of $\hat{A}$. This is because the $(i,k)$th matrix entry of the double commutator in $\hat{A}$’s eigenbasis is

$$[\hat{A},[\hat{A},\rho]]_{ik}=\rho_{ik}(a_{i}-a_{k})^{2}.$$

(15)

The dampening of the off-diagonal elements of $\rho$ occurs at a rate that grows with $(a_{i}-a_{k})^{2}$ where if $a_{i}\neq a_{k}$ the system is in a superposition. We see that the eigenbasis of $\hat{A}$ determines the collapse basis, i.e. the basis in which the state becomes “classical”, while its eigenvalues tell us which superpositions of pairs of such states are removed more quickly (namely, those with large $(a_{i}-a_{k})^{2}$).

Let us now use this prescription to construct the simplest possible consciousness collapse model for the dyad. Subsequently, we will compare this with the more general, but more involved approach in [17]. For the moment, let us only mention that our simple model contains only a single collapse operator, whereas the one in [17] involves several such operators, generalizing Eq. (13). We will say more about the similarities and differences below.

The four states of the dyad are mutually distinct states of consciousness, spanning the total Hilbert space. Therefore, we expect a consciousness-collapse model to lead to a state for large times $t$ that is diagonal in that basis. Therefore, our collapse operator $\hat{Q}$ will have the form

$$\hat{Q}=\lambda_{00}|00\rangle\langle 00|+\lambda_{01}|01\rangle\langle 01|+\lambda_{10}|10\rangle\langle 10|+\lambda_{11}|11\rangle\langle 11|,$$

(16)

with four eigenvalues $\lambda_{ij}$. Any consciousness-collapse model should arguably imply the following principle for the choice of those eigenvalues:

If two states of the dyad (say, $ij$ and $kl$) are qualitatively very different states of consciousness, then superpositions of these states should vanish very quickly, i.e. $|\lambda_{ij}-\lambda_{kl}|$ should be very large.

That is, it is natural to allow superpositions of “qualitatively similar” states to persist for longer, while qualitatively different states must decohere quickly.

For a quantitative application of this prescription, we need a way to compare states of consciousness, i.e. a distance measure on Q-shapes. Since Q-shapes are collections of probability distributions, it is natural to define their distance in terms of distance measures on probability distributions, which is a classical and well-studied topic in information theory. The preferred distance measures on probability distributions in IIT have changed in almost every successive version. IIT2.0 used the well-known Kullback-Leibler divergence. IIT3.0 used Earth Mover’s distance [42]. IIT4.0 uses the intrinsic difference measure from section 3. IIT3.0’s measure was explicitly turned into a generalized distance measure for Q-shapes. IIT4.0’s measure is not so well suited for this task, a point we will return to at the end of the section.

A natural choice is to define the distance of two Q-shapes $Q=(q_{1},q_{2},q_{3},q_{4})^{\top}$ (i.e. with rows $q_{1},\ldots,q_{4}$) and $\tilde{Q}=(\tilde{q}_{1},\tilde{q}_{2},\tilde{q}_{3},\tilde{q}_{4})^{\top}$ as

$$\mathcal{D}(Q,\tilde{Q}):=\sum_{i=1}^{4}\mathcal{D}(q_{i},\tilde{q}_{i}),$$

(17)

where $\mathcal{D}$ is some choice of distance measure on the set of probability distributions. That is, the distance of two Q-shapes is the sum of the distances of their probability distributions. (This is precisely the form of IIT3.0’s extended Earth mover’s distance measure.)

Now we have a large choice of possible distance measures $\mathcal{D}$ at our disposal. However, note that the four Q-shapes of the dyad (Eqs. (11) and (12)) consist of a small variety of very simple probability distributions only: all entries are $0$ or $\frac{1}{2}$, and any two rows are either equal, or they differ in all $4$ entries. Two identical rows must have distance zero. Furthermore, it is natural to demand that every two probability distributions arising as rows in these Q-shapes that differ in all four places all have the same distance, which we can set to unity by a choice of scaling factor. For example,

$$\textstyle\mathcal{D}\left((0,\frac{1}{2},0,\frac{1}{2}),(\frac{1}{2},0,\frac{1}{2},0)\right)=1.$$

We can then determine the distances between all pairs of Q-shapes of the dyad and obtain the following values, writing $\mathcal{D}(Q,\tilde{Q})$ as the $Q\tilde{Q}$-entry of a table:

	Q(0,0)	Q(0,1)	Q(1,0)	Q(1,1)
Q(0,0)	$0$	$2$	$2$	$2$
Q(0,1)	$2$	$0$	$4$	$2$
Q(1,0)	$2$	$4$	$0$	$2$
Q(1,1)	$2$	$2$	$2$	$0$

Let us now return to our consciousness-collapse principle. Formulating it in terms of this distance measure, it reads: If the distance $\mathcal{D}$ between two Q-shapes $Q(i,j)$ and $Q(k,l)$ is large, then the distance between the eigenvalues $\lambda_{ij}$ and $\lambda_{kl}$ of the collapse operator must also be large.

This desideratum could always be satisfied by the arbitrary prescription to make all eigenvalues extremely large and distant from each other. However, this would typically induce almost-instantaneous collapse, a behavior that we do not expect for simple systems such as the dyad. Thus, we are searching for a choice of eigenvalues that is as tame as possible while still satisfying the above postulate.

This leads us to define the eigenvalues in terms of an optimization problem:

Minimize $\lambda_{00}+\lambda_{01}+\lambda_{10}+\lambda_{11}$ subject to $\lambda_{ij}\geq 0$, $|\lambda_{ij}-\lambda_{kl}|\geq\mathcal{D}(Q(ij),Q(kl))$.

This prescription keeps the collapse behavior “tame” by demanding that the eigenvalues are not arbitrarily large, but only as large as they need to be (in their total sum) to satisfy our principle for all pairs of Q-shapes. Note that the total time scale of the collapse is not determined by $\hat{Q}$ and its eigenvalues, which do not have any physical units. Instead, it is determined by the noise term of (13), i.e. the parameter $\lambda$ in (14). This will remain a parameter of the collapse model that needs to be determined experimentally. The above considerations tell us only the relative speed at which superpositions between distinct Q-shapes are suppressed, whereas the total speed would depend on $\lambda$ and hence on further considerations as to which states of consciousness are implausible to remain in superposition for significant amounts of time because of, say, human experience.

As we show in appendix C, this optimization problem has twelve solutions: one of them is

$$\lambda_{00}=2,\enspace\lambda_{01}=0,\enspace\lambda_{10}=4,\enspace\lambda_{11}=6,$$

and the other solutions are permutations of this one ($(2,0,4,6)$), such as $(6,4,0,2)$ — indeed, all permutations of these four numbers such that $|\lambda_{01}-\lambda_{10}|\geq 4$. This degeneracy can be understood as a consequence of the symmetry of the problem: for example, the table of pairwise distances does not change if we exchange $Q(0,0)$ and $Q(1,1)$. Indeed, these solutions do not only minimize the sum of the $\lambda_{ij}$, but they also minimize the expression

$$\frac{1}{2}\sum_{i,j,k,l}|\lambda_{ij}-\lambda_{kl}|=|\lambda_{00}-\lambda_{01}|+|\lambda_{00}-\lambda_{10}|+|\lambda_{00}-\lambda_{11}|+|\lambda_{01}-\lambda_{10}|+|\lambda_{01}-\lambda_{10}|+|\lambda_{10}-\lambda_{11}|,$$

i.e. the total sum of the pairwise collapse rates, under the assumption (that we can always make) that one of the $\lambda_{ij}$ is zero.

We can simply pick one of the six solutions and use it to define our collapse operator. For the sake of the argument, let us pick the above, but the choice does not matter for the following discussion.

Let us interpret the result by looking at some example collapse rates. We have $\mathcal{D}(Q(00),Q(01))=2$ which is small, and $|\lambda_{00}-\lambda_{01}|=2$ is also small (and, indeed, identical). Superpositions of the two dyad states $00$ and $01$ can thus remain stable for a relatively long time. On the other hand, $\mathcal{D}(Q(01),Q(10))=4$ is large, and so is $|\lambda_{01}-\lambda_{10}|=4$. Hence, superpositions between the dyad states $01$ and $10$ will be killed off more quickly.

However, consider the two dyad states $01$ and $11$. Their distance is small, $\mathcal{D}(Q(01),Q(11))=2$, and our principle demands that the corresponding difference of eigenvalues (i.e. the associated collapse rate) is at least as large as that. However, it is actually $|\lambda_{01}-\lambda_{11}|=6$, which is much larger than required. Thus, any superposition of these two dyad states would fall off much faster than what would be expected by considering the difference between their Q-shapes alone.

We can understand this behavior by noting that the $n=4$ dyad states lead to $n(n-1)/2=6$ distance values (the table above), from which $n=4$ eigenvalues of the collapse operator have to be determined. Thus, every value of $|\lambda_{ij}-\lambda_{kl}|$ must depend on more than just the number $\mathcal{D}(Q(ij),Q(kl))$. If our principle is satisfied, then a large value of the latter implies a large value of the former, but the converse is not in general true. The quantum limitation of only having $n$ eigenvalues introduces additional constraints.

It seems that this must be a general phenomenon: if we have $n$ distinct Q-shapes, but $m\ll n/2$ collapse operators, then the $m\cdot n$ eigenvalues are smaller in number than the $n(n+1)/2$ distance values. We must hence have pairs of Q-shapes whose superposition must collapse more quickly than what their mere qualitative distance as states of consciousness would suggest. Superposition resistance hence cannot only resemble the structure of conscious experience, but is also additionally constrained by the general structure of quantum mechanics.

We can now see how the elements of the construction above are realized in greater generality in [17]. Here, there is not a single Q-shape operator whose eigenstates are all the classical Q-shapes. Rather, a Q-shape is associated with an ensemble of orthogonal self-adjoint collapse operators. The eigenvalue of each operator does not pick out a Q-shape, but an element of a Q-shape, which is either an entry in a probability matrix or a $\phi$-value. This solves the above problem and allows for superposition resistance to resemble the structure of conscious experience more closely. However it does so at the cost of having a very complex model.

To illustrate this complexity, consider how many collapse operators we need to capture all the details of a Q-shape of a classical system. A system of n elements will have $2^{n}-1$ subsystems. So if we have two elements, as with our dyad, we have three subsystems (A, B, and AB). We were able to ignore AB in our simple case but we cannot do that in general. Every element has d possible states giving a total of $d^{n}$ possible states for the system. Each subsystem is associated with two probability distributions, as we saw in the previous section, and one $\phi$-value. To capture all of this, the number of collapse operators we need is $(2^{n}-1)\times(2\times d^{n}+1)$. So for our classical dyad, where $n=d=2$, we need 27 collapse operators. It would be extraordinary if Nature were to operate at such a high level of complexity for such a simple system.

But this still is not sufficient when dealing with quantum systems, since qubit elements do not have $d=2$ possible states, but have infinitely many possible pure states. It is for this reason that the model in [17] formulates everything in terms of the QIIT found in ([53],[27]). Here each subsystem is associated with two appropriate density matrices instead of two appropriate classical probability distributions. The density matrices for the quantum dyad have more entries than the classical probability distributions associated with the classical dyad, so we need more collapse operators. In particular, we now need $(2^{n}-1)\times(2\times d^{2n}+1)$ collapse operators in general, and so 99 collapse operators for our dyad.

The use of QIIT raises a further complication. In QIIT, every quantum system is assigned a well-defined Q-shape (which in many cases may be the null Q-shape) whether or not the system is in a superposition of classical Q-shapes. That is, for QIIT, distinct states of consciousness do not always correspond to mutually orthogonal quantum states, and this makes it in general impossible to have physical processes whose observable behavior depends on all the properties of those states of consciousness (because non-orthogonal states cannot be perfectly distinguished). In particular, this excludes collapse models where the rate of collapse is proportional to the “size” of the superposition, e.g. to the qualitative difference of the superposed conscious states. The ensemble of collapse operators defined in equation (2) of [17] therefore adds an additional constraint that restricts these operators to just those states that are associated with classical Q-shapes.

Thus, in an attempt to be completely general, the collapse model in [17] became very complex. But as has been demonstrated here, if one just wants a collapse model for some simple system whose physical properties are known, then much of those complexities can be bypassed, as we can instead define a single collapse operator as we have done here.

Finally, we note that specific predictions of one’s collapse model may vary with the use of IIT formalism, which is constantly being updated. The choice of distance measure in particular has undergone significant revision.

IIT2.0 used the Kullback-Leibler divergence to measure the distance between probability distributions. But that was rejected in part because it is not symmetric.

IIT3.0 adopted Earth Mover’s distance (EMD). This is symmetric and, as shown in [40], yields different results than the IIT2.0 measure, even for the dyad. The EMD can easily be generalized to become a measure of distance between Q-shapes, as in equation 17. This Q-shape distance measure was essential to IIT3.0 because it was used to calculate $\Phi$ (by measuring the distance between the Q-shape of a system’s state and the Q-shape of that state partitioned). But the EMD has recently been abandoned, in part because of problems raised in [7]. Alternative measures can be found in [45].

IIT4.0 adopts the intrinsic difference measure, described in IIT4.0 and in section 3 above, as its preferred distance measure. The intrinsic difference is infinite if the denominator is zero. This is avoided when the denominator involves some partition and therefore white noise. But measuring the distance between two Q-shapes doesn’t involve any partitioning. So, the intrinsic difference is not suitable for measuring Q-shape distance. Consequently, IIT4.0 does not supply such a distance measure. This led to a simpler calculation of $\Phi$ in IIT4.0: the system $\Phi$ is a sum of subsystem $\phi$-values. QIIT in its most recent version is consistent with IIT4.0 B.

Consider the following simple implementation of the dyad as depicted in Figure 3: channels A and B are optical cables and the dyad does nothing more than cross those cables, without contact. The outputs are then fed back into the inputs, creating a kind of feedback cycle. We have two photons in the cables, and each of them can carry one of two perfectly distinguishable “classical” states, corresponding to horizontal $|0\rangle$ or vertical polarization $|1\rangle$. What horizontal or vertical means is determined by an external reference frame; for what follows, the exact choice of reference is unimportant, except that the physical situation must tell us what we mean by both photons carrying identical or orthogonal polarization directions (e.g. $|00\rangle$ in the first case, and $|01\rangle$ in the second). It is clear that we are not restricted to preparing the photons in the classical basis states, but we can prepare them in arbitrary superpositions, such as that of Equation 3. This is a necessary condition to implement “Schrödinger’s dyad” as introduced in the previous sections.

However, if we identify our basic units as the photons that traverse the cables, then it may not seem like IIT applies here, since IIT requires causal relationships. In particular, to be a basic unit of IIT, something should have the power to “take a difference” (be affected by something) and “make a difference” (produce effects on something) [4]. The concern with this implementation is that the photons do not take or make a difference, because they never change their polarization states. Indeed, under this interpretation of the physical setup, we would not even have implemented the dyad, but another system (two bits and an identity gate).

On the other hand, this depends on counting the photons as our basic causal units. If we instead identify our basic units as the polarization qubits at the physical locations $A$ and $B$ in space, we get a different result. In particular, we may say that the photon polarization state at A_t0 causes the state at B_t+1 and was caused by the state at B_t-1. Under this way of identifying our basic units, the system has non-zero $\Phi$ and is the simplest conscious system according to IIT. This may even fit well with interpretations of modern physics in which the basic causal objects are spacetime points [43].

IIT does not want the $\Phi$ or Q-shape of a system to depend on some arbitrary choice: these are meant to be objective properties. So at least one of the above two causal interpretations of the system must be ruled out. IIT does not give clear criteria for what to do in such a situation. However, one option is suggested by the IIT4.0 principle of maximal existence, which states that “what exists is what exists the most”. This ontological principle is used to motivate the exclusion principle, which effectively states that if two overlapping sets of units have non-zero $\Phi$, then only the system with maximal $\Phi$ is conscious. Thus, if there are multiple interpretations of what the causal units are in the first place, we might similarly only consider the interpretation that yields the greater $\Phi$. In that case, we have found a very simple implementation of the dyad by identifying the qubits with locations in space.

In this paper we have described some simple predictions of IIT that have enabled us to make a number of observations.

First, we showed that either IIT is false, or a simple system like the feedback dyad is conscious and can easily be put into a quantum superposition of conscious states (i.e. a superposition of Q-shapes). This result was shown to be robust across successive IIT formalisms.

Second, we identified the simplest consciousness-collapse (or Q-shape-collapse) model. It involves a single Q-shape collapse operator, whose eigenstates are the four possible states of the dyad. For the model to do what is needed (make the collapse rate proportional to a measure of difference between the superposed Q-shapes), we found that the four eigenvalues must depend on six distance values.

In such models, the rate of collapse of a superposition of two states of consciousness must therefore depend on more than the relation between the two states. More complex models may avoid this by defining an ensemble of orthogonal Q-shape collapse operators. However, this can get very complicated, so for practical purposes (like testing Q-shape-collapse models), the prescription that we have provided here may be more useful.

Finally, we have made several observations about the general structure of IIT. For example, we argued that while treating gates as having states is permissible if gates are neurons, this does not work in general, and especially not for computers, where gates operate on systems that possess states. This is especially clear for quantum computers, where qubits, and not gates, are superposed. We have also noted that to apply IIT to a physical system, we need a specification of the basic causal units in the system. Insofar as physics does not specify such things, IIT is not fully applicable to physical systems.

In further research, it would be interesting to investigate whether any existing quantum computers (or other quantum systems) can maintain states like the $t_{0}$ state of our quantum dyad, and for how long. Such systems may place bounds on the fundamental parameters of IIT-based consciousness-collapse models.

We are grateful to Thomas D. Galley for discussions, and to Larissa Albantakis for helpful feedback on an earlier draft. This research was supported by grant number FQXi-RFP-CPW-2015 from the Foundational Questions Institute and Fetzer Franklin Fund, a donor advised fund of Silicon Valley Community Foundation. Moreover, this research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science, and Economic Development, and by the Province of Ontario through the Ministry of Colleges and Universities.