"""The Pusey-Barrett-Rudolph (PBR) Theorem
=========================================

In this tutorial, we will explore the Pusey-Barrett-Rudolph (PBR) theorem
:footcite:`Pusey_2012_On`, a significant no-go theorem in the foundations
of quantum mechanics. We will describe the theorem's core argument and then
use :code:`|toqito⟩` to verify the central mathematical property that
the theorem relies on.

The PBR theorem addresses a fundamental question: Is the quantum state
(e.g., the wavefunction :math:`|\\psi\\rangle`) a real, objective property of a
single system (an *ontic* state), or does it merely represent our
incomplete knowledge or information about some deeper underlying reality
(an *epistemic* state)?
"""

# %%
# PBR Argument
# ----------------
#
# The PBR theorem :footcite:`Pusey_2012_On` argues against a broad class of epistemic models.
#
# 1.  **Epistemic Hypothesis**: An epistemic model assumes there is a
#     "real" physical state of the system, often denoted by :math:`\lambda`.
#     The quantum state :math:`|\psi\rangle` is then just a probability
#     distribution over the possible values of :math:`\lambda`. A key
#     implication is that the distributions for two different quantum states,
#     say :math:`|\psi_0\rangle` and :math:`|\psi_1\rangle`, could overlap. This
#     means that for some underlying physical states :math:`\lambda`, the system
#     could have been prepared in *either* :math:`|\psi_0\rangle` or
#     :math:`|\psi_1\rangle`.
#
#     We can visualize the overlap of these hypothetical probability distributions.
#     To do this, we will create a simple illustrative plot. We are not assuming
#     any specific physical model for :math:`\lambda`; the plot is purely a visual aid to
#     make the concept of overlapping probability distributions concrete.
#
#     For this illustration, we represent the space of possible ontic states :math:`\lambda`
#     on the x-axis. We then choose two simple, overlapping normal (Gaussian)
#     distributions to represent the hypothetical probability densities
#     :math:`p(\lambda | \psi_0)` and :math:`p(\lambda | \psi_1)`. The specific choice of Gaussian distributions
#     is arbitrary; any pair of distinct, overlapping distributions would
#     demonstrate the same essential feature.

import matplotlib.pyplot as plt
import numpy as np
from scipy.stats import norm

fig, ax = plt.subplots(figsize=(8, 4), dpi=100)
lambda_space = np.linspace(-4, 4, 1000)
dist_0 = norm(loc=-1, scale=1)
dist_1 = norm(loc=1, scale=1)
p_lambda_0 = dist_0.pdf(lambda_space)
p_lambda_1 = dist_1.pdf(lambda_space)
ax.plot(lambda_space, p_lambda_0, label=r"$p(\lambda | \psi_0)$")
ax.plot(lambda_space, p_lambda_1, label=r"$p(\lambda | \psi_1)$")
ax.fill_between(
    lambda_space,
    np.minimum(p_lambda_0, p_lambda_1),
    color="gray",
    alpha=0.5,
    label="Overlap Region (Δ)",
)
ax.set_xlabel(r"Ontic State Space ($\lambda$)", fontsize=12)
ax.set_ylabel(r"Probability Density", fontsize=12)
ax.set_yticks([])
ax.legend(fontsize=12)
ax.spines["top"].set_visible(False)
ax.spines["right"].set_visible(False)
ax.spines["left"].set_visible(False)
plt.tight_layout()

# %%
#  .. note::
#     The shaded region :math:`\Delta` represents the set of ontic states
#     :math:`\lambda` that are ambiguous—the system could have been prepared as
#     :math:`|\psi_0\rangle` or :math:`|\psi_1\rangle`. The PBR theorem shows that the
#     existence of any such overlap (for any pair of distinct states) leads to a
#     contradiction with quantum theory's predictions.
#     Figure adapted from the PBR paper :footcite:`Pusey_2012_On`.
#
# 2.  **Thought Experiment**: The PBR paper :footcite:`Pusey_2012_On` constructs a thought
#     experiment to show this leads to a contradiction. Consider two
#     non-orthogonal quantum states, for example:
#
#     .. math::
#        |0\rangle \quad \text{and } \quad |+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)
#
#     If their underlying reality distributions overlap, it's possible to
#     prepare a system where its true state :math:`\lambda` is consistent
#     with both preparations. Now, imagine we prepare two such systems
#     independently. There is a non-zero chance that the combined physical
#     state :math:`(\lambda_1, \lambda_2)` is compatible with *any* of the
#     four possible quantum preparations:
#
#     .. math::
#        |0\rangle \otimes |0\rangle, \quad |0\rangle \otimes |+\rangle, \text{ }
#        |+\rangle \otimes |0\rangle, \quad |+\rangle \otimes |+\rangle
#
# 3.  **Contradiction via Antidistinguishability**: The crux of the
#     theorem is to show that quantum mechanics allows for a special
#     entangled measurement on the two systems. This measurement has a
#     remarkable property: each of its possible outcomes is strictly
#     forbidden (has zero probability) for at least one of the four
#     product states.
#
#     This property is known as **antidistinguishability**. A set of states
#     :math:`\{|\Psi_i\rangle\}` is antidistinguishable if there exists a
#     measurement with outcomes :math:`\{M_i\}` such that
#     :math:`\langle \Psi_i | M_i | \Psi_i \rangle = 0` for all :math:`i`.
#
#     This leads to a contradiction:
#
#     *   The epistemic model predicts that sometimes the underlying
#         reality :math:`(\lambda_1, \lambda_2)` is ambiguous.
#     *   In these cases, the measurement (which only depends on
#         :math:`\lambda`) must produce *some* outcome, say outcome :math:`k`.
#     *   But what if the state was *actually* prepared as
#         :math:`|\Psi_k\rangle`? Quantum mechanics says outcome :math:`k`
#         is impossible for this state.
#
#     This contradiction implies that the initial assumption—that the
#     distributions for :math:`|0\rangle` and :math:`|+\rangle` overlap—must
#     be false. The PBR theorem generalizes this to any pair of distinct
#     quantum states. The conclusion is that, under the assumption of
#     preparation independence, the quantum state must be ontic.


# %%
# Verifying Antidistinguishability with `|toqito⟩`
# ------------------------------------------------
#
# We can now use :code:`|toqito⟩` to verify the key requirement for the PBR
# theorem that the set of four states constructed from :math:`|0\rangle`
# and :math:`|+\rangle` are indeed antidistinguishable.

import numpy as np

from toqito.matrices import standard_basis
from toqito.matrix_ops import tensor
from toqito.state_props import is_antidistinguishable

# Define the single-qubit states |0> and |+>.
e_0, e_1 = standard_basis(2)
state_0 = e_0
state_plus = (e_0 + e_1) / np.sqrt(2)

# Construct the four 2-qubit product states from the PBR paper's simple example.
psi_00 = tensor(state_0, state_0)
psi_0_plus = tensor(state_0, state_plus)
psi_plus_0 = tensor(state_plus, state_0)
psi_plus_plus = tensor(state_plus, state_plus)

pbr_states = [psi_00, psi_0_plus, psi_plus_0, psi_plus_plus]

# Check if this set of states is antidistinguishable.
is_ad = is_antidistinguishable(pbr_states)

print(f"Are the four PBR states antidistinguishable? {is_ad}")

# %%
# The result confirms that there exists a measurement
# that can perfectly exclude each of the four states, providing the
# necessary ingredient for the PBR no-go theorem's contradiction.
#
# This result, derived from a solvable semidefinite program within
# :code:`|toqito⟩`'s :py:func:`~toqito.state_opt.state_exclusion.state_exclusion` function supports the theorem's
# conclusion that the quantum state has a strong claim to being an
# objective feature of reality.

# %%
# General PBR States
# ------------------
# The theorem holds for any pair of non-orthogonal states. The `toqito`
# library provides a function to generate the states from the more general
# proof in the PBR paper :footcite:`Pusey_2012_On`, which are defined by an angle :math:`\theta`.
#
# .. math::
#    |\psi_0\rangle = \cos(\frac{\theta}{2})|0\rangle + \sin(\frac{\theta}{2})|1\rangle \quad \text{and }
#    |\psi_1\rangle = \cos(\frac{\theta}{2})|0\rangle - \sin(\frac{\theta}{2})|1\rangle
#
# For instance, we can generate a set of :math:`2^n` states for some :math:`n` and :math:`\theta`.

from toqito.states import pusey_barrett_rudolph

# Generate states for n=2 systems and theta = pi/3
general_pbr_states = pusey_barrett_rudolph(n=2, theta=np.pi / 3)

# %%
# The inner product of the two base states is :math:`\cos(\theta)`.
# For these to be antidistinguishable, we need to check the condition from the paper.
# The theorem states that if :math:`2^{1/n} - 1 < \tan(\theta/2)`, a contradiction is obtained.
# For :math:`n=2` and :math:`\theta=\pi/3`, we have :math:`\tan(\theta/2) = \tan(\pi/6) \approx 0.577`.
# The other side of the inequality is :math:`2^{1/2} - 1 \approx 0.414`.
# Since :math:`0.414 < 0.577`, the theorem applies and this set should be antidistinguishable.

is_ad_general = is_antidistinguishable(general_pbr_states)

print(f"\nAre the four general PBR states (n=2, theta=pi/3) antidistinguishable? {is_ad_general}")

# %%
#
#
# References
# ----------
#
# .. footbibliography::