Quantum-like families#
Each family states the quantum-like model and the classical baselines it is compared with, in the form the code computes. All of them are fitted and compared as on the core page.
Question-order effects#
Design. Two yes/no questions \(A\), \(B\) asked in either order. Outcomes per order are \((yy, yn, ny, nn)\), the first letter being the first answer.
3D quantum-like model (QuantumOrderModel). State \(\psi = e_1 \in \mathbb{R}^3\), question vectors
A rank-1 “yes” is the ray \(P_q = u_q u_q^T\); a rank-2 “yes” is the plane \(P_q = I - u_q u_q^T\). Then
and the same with \(A\) and \(B\) exchanged for the order \(BA\). Three parameters \((a, b, g)\); the ranks are a structural choice fitted over \(\{1, 2\}^2\).
4D model that nests Bayes (QuantumOrderModel4D). Basis \(e_1..e_4\) = \((y,y), (y,n), (n,y), (n,n)\) of a
classical \(2\times 2\) table; \(\psi(t_1, t_2, t_3) \in \mathbb{R}^4\) from hyperspherical angles;
\(P_A = \operatorname{span}(e_1, e_2)\) and
where \(U(\varphi)\) rotates by \(\varphi\) in the planes \((e_1, e_4)\) and \((e_2, e_3)\). For \(\varphi = 0\) the projectors commute and \(p(x) = \psi_x^2\) is an ordinary joint table, i.e. exactly the Bayesian model; for \(\varphi \neq 0\) the order matters.
QQ equality. For any state and any projectors,
qq_test uses \(z = \hat q / \widehat{\mathrm{se}}(\hat q)\) with the binomial standard errors of the two
“different answers” proportions, \(\widehat{\mathrm{se}}^2 = \hat s_{AB}(1 - \hat s_{AB})/N_{AB} + \hat s_{BA}(1 - \hat s_{BA})/N_{BA}\),
and a two-sided normal \(p\)-value.
Baselines.
Bayes (
BayesOrderModel): marginals \(p_A, p_B\) and a correlation \(\rho \in (-1, 1)\) scaled between the Fréchet bounds; the same joint table in both orders.Anchoring (
AnchoringOrderModel): the first answer has its first-position rate \(p_1\); for weight \(w \ge 0\), \(P(\text{2nd yes} \mid \text{1st yes}) = p_2 + w(1 - p_2)\) and \(P(\text{2nd yes} \mid \text{1st no}) = p_2(1 - w)\) (mirror image for \(w < 0\)), with a weight per question.Saturated (
SaturatedOrderModel): a free softmax distribution per order (6 parameters), the best possible fit.
Quantum-like Bayesian networks#
A Bayesian network gives each full configuration \(x\) the probability \(P(x) = \prod_i P(x_i \mid \mathrm{pa}_i)\).
The quantum-like network gives it the amplitude \(\sqrt{P(x)}\,e^{i\theta(x)}\) and sums amplitudes over
the hidden configurations \(h\) (QLBNModel):
The phases are the parameters; the classical network (ClassicalBNModel) has none.
Belief dynamics#
Walks over \(N\) ordered confidence states \(x_j = j/(N - 1)\), grouped into \(K\) response categories; the initial distribution is a Gaussian bump around \(0.5\).
Markov walk: \(\dot\phi = K\phi\) with tridiagonal \(K\), up rate \(\alpha = \mu\gamma\), down rate \(\beta = (1 - \mu)\gamma\), reflecting ends; \(\phi(t) = e^{Kt}\phi(0)\).
Quantum walk: \(\psi(t) = e^{-iHt}\psi(0)\), \(\psi(0) = \sqrt{\phi(0)}\), with \(H_{jj} = \mu\,x_j\) and \(H_{j,j+1} = H_{j+1,j} = \sigma\); \(P(\text{state } j) = \lvert\psi_j(t)\rvert^2\).
Open-system walk: the Lindblad equation with the same \(H\) and dephasing jumps \(L_j = \lvert j\rangle\langle j\rvert\) at rate \(\lambda\); \(\lambda = 0\) is the quantum walk, large \(\lambda\) behaves like a Markov walk.
A judgement at \(t_1\) collapses the state onto the chosen category \(C\): \(\psi \mapsto P_C\psi/\lVert P_C\psi\rVert\) (or \(\rho \mapsto P_C\rho P_C/\operatorname{tr}(P_C\rho)\)). This is how an intermediate judgement changes a later one; in the Markov walk the marginal at \(t_2\) is unchanged on average.
Trust (or any belief) updated by events and queried as yes/no.
Open-system belief (OpenSystemBelief): a real qubit with \(\lvert 0\rangle\) = “yes/trust”,
\(\rho_0 = vv^T\), \(v = (\cos\tfrac{\phi_0}{2}, \sin\tfrac{\phi_0}{2})\). Each event applies
\(R_y(\alpha) = \begin{pmatrix}\cos\frac\alpha2 & -\sin\frac\alpha2 \\ \sin\frac\alpha2 & \cos\frac\alpha2\end{pmatrix}\) with
\(\alpha = -a_+\) after a success and \(\alpha = a_-\) after a failure, then dephases:
A query answers “yes” with probability \(\rho_{00}\) and collapses \(\rho\) to \(\lvert 0\rangle\langle 0\rvert\) or \(\lvert 1\rangle\langle 1\rvert\). The model predicts the full distribution over answer sequences.
Markov belief (MarkovBelief): \(p \mapsto p + (1 - p)\,u\) after a success and \(p \mapsto p\,(1 - d)\) after
a failure; a query reads \(p\) without changing it.
The question effect is \(P(\text{yes at } t \mid \text{asked at } t' < t) - P(\text{yes at } t \mid \text{not asked})\), marginalised over the intermediate answer; it is zero for the Markov belief.
Bistable perception (quantum Zeno)#
The percept is a qubit whose basis states are the two interpretations. Between observations, every \(\Delta t\), it rotates by \(U = e^{-ig\Delta t\,\sigma_x}\); each observation collapses it. The switch probability per observation and the dwell-time distribution are
Halving \(\Delta t\) doubles the mean dwell time. Baselines: Markov switching at rate \(r\), \(P(T > t) = e^{-rt}\), independent of \(\Delta t\); gamma renewal, \(T \sim \mathrm{Gamma}(k, \theta)\). The predictions are binned into dwell-time histograms (with a final open bin) and fitted as multinomials.
References: Wang & Busemeyer (2013); Busemeyer et al. (2011); Pothos & Busemeyer (2009); Moreira & Wichert (2016); Busemeyer, Kvam & Pleskac (2019); Yukalov & Sornette (2011); Kujala, Dzhafarov & Larsson (2015); Pothos, Busemeyer & Trueblood (2013); Eisert, Wilkens & Lewenstein (1999); Brainerd, Wang & Reyna (2013); Aerts (2009); Atmanspacher, Filk & Römer (2004).