Revisiting Schrödinger's Cat

Did we really prepare a quantum cat—or did an algebraic test merely say PASS?

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The answer in one minute

This repository begins with a familiar story and ends with a stricter scientific question.

Question	Short answer
Is the unmeasured cat automatically in $	+\rangle=(
Can measuring the atom steer the cat into $	+\rangle$?
Does $A_{00}=A_{01}$ alone prove success?	No. The equality also accepts the empty branch $A_{00}=A_{01}=0$.
What makes the claim trustworthy?	A closed-form derivation, randomized implementation tests, and deterministic physical counterexamples—used together.

New to quantum computing? Start with the interactive zero-prerequisite OER, then use the longer concept guide (English + 中文速览).

The original notebook remains the PennyLane Challenge solution. The OER and this README ask the next question: does passing the challenge's amplitude-equality test guarantee a physically realizable conditional state?

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Q1 — Is Schrödinger's cat already “alive and dead at the same time”?

Not as a local state of the cat. After the atom and cat become maximally entangled, their joint state can be

$$ |\Phi^+\rangle_{AC} =\frac{|0\rangle_A|0\rangle_C+|1\rangle_A|1\rangle_C}{\sqrt2}. $$

The joint atom–cat system is in a coherent entangled superposition. But if we ignore the atom and examine only the cat, we trace the atom out:

$$ \rho_C=\mathrm{Tr}_A!\left(|\Phi^+\rangle\langle\Phi^+|\right) =\frac12|0\rangle\langle0|+\frac12|1\rangle\langle1| =\frac{I}{2}. $$

The diagonal entries give 50/50 probabilities; the off-diagonal entries—the cat's local coherence—are zero. So the cat alone sits at the center of the Bloch sphere, not on its surface at $|+\rangle$.

    

The GIF is a teaching interpolation between two state descriptions, not the literal continuous-time trajectory of one cat during measurement.

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Q2 — If the cat is mixed, how can it become a genuine superposition?

By quantum steering: choose a measurement basis for the atom, measure it, and keep a specified outcome. Conditioned on that outcome, the cat can land in

$$ |+\rangle_C=\frac{|0\rangle_C+|1\rangle_C}{\sqrt2}. $$

The logic has four steps:

1. prepare the atom and cat in $|00\rangle$;
2. entangle them with a two-qubit unitary $U$;
3. rotate the atom's measurement basis with $U3(\theta,\phi,\lambda)$;
4. measure the atom and post-select its $|0\rangle$ outcome.

Post-selection is conditional: it describes the retained subensemble. It neither guarantees that the selected outcome occurs nor enables faster-than-light signalling.

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Q3 — What exactly did the PennyLane challenge ask us to solve?

The original circuit is shown first; the generalized circuit is shown second.

For a fixed input $|00\rangle$, only the first column of $U$ matters:

$$ U|00\rangle=a|00\rangle+b|01\rangle+c|10\rangle+d|11\rangle. $$

After applying $U3(\theta,\phi,\lambda)$ to the atom, the two amplitudes in the atom-$|0\rangle$ branch are

$$ A_{00}=a\cos\frac{\theta}{2}-c,e^{i\lambda}\sin\frac{\theta}{2}, $$

$$ A_{01}=b\cos\frac{\theta}{2}-d,e^{i\lambda}\sin\frac{\theta}{2}. $$

The challenge validator asks for

$$ A_{00}=A_{01}. $$

If this branch is non-zero, equal amplitudes mean its normalized cat state is $|+\rangle$ up to a global phase. The phrase “if this branch is non-zero” is the crucial physical qualification.

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Q4 — Can the measurement basis be solved analytically?

Yes—no optimizer is required. Define

$$ \alpha=a-b,\qquad \beta=c-d. $$

Then the equality condition becomes

$$ \alpha\cos\frac{\theta}{2} =\beta e^{i\lambda}\sin\frac{\theta}{2}. $$

For the general case $|\alpha|>0$ and $|\beta|>0$,

$$ \lambda=\arg(\alpha)-\arg(\beta), \qquad \theta=2\arctan\frac{|\alpha|}{|\beta|}, \qquad \phi=0. $$

• $\lambda$ aligns the two complex phases.
• $\theta$ balances the two magnitudes.
• $\phi$ only changes the discarded atom-$|1\rangle$ branch, so it does not enter the equality constraint.

Case	Condition	One valid equality solution
General	$	\alpha
Degenerate $\alpha$	$	\alpha
Degenerate $\beta$	$	\beta
Both zero	$	\alpha

Why does the closed form work?

For non-zero $\alpha$ and $\beta$, the chosen $\lambda$ makes $\beta e^{i\lambda}$ point in the same complex direction as $\alpha$. The chosen $\theta$ gives

$$ \cos\frac{\theta}{2}=\frac{|\beta|}{\sqrt{|\alpha|^2+|\beta|^2}}, \qquad \sin\frac{\theta}{2}=\frac{|\alpha|}{\sqrt{|\alpha|^2+|\beta|^2}}. $$

Both sides therefore have the same phase and the same magnitude

$$ \frac{|\alpha||\beta|}{\sqrt{|\alpha|^2+|\beta|^2}}. $$

The degenerate rows force the remaining sine or cosine factor to zero. This proves the amplitude-equality formula for every input column $(a,b,c,d)$.

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Q5 — If the amplitudes are equal, have we prepared the cat?

Not necessarily. Equality is an algebraic condition; preparation is a physical claim.

First ask whether the selected branch can occur:

$$ p_0=|A_{00}|^2+|A_{01}|^2. $$

Only when $p_0>0$ does the conditional cat state exist:

$$ |\mathrm{cat}_0\rangle =\frac{A_{00}|0\rangle+A_{01}|1\rangle}{\sqrt{p_0}}. $$

Then ask whether that state is the target:

$$ F=|\langle+|\mathrm{cat}_0\rangle|^2. $$

So a complete success claim requires:

1. amplitude equality: $A_{00}\approx A_{01}$;
2. reachability: $p_0>0$;
3. conditional correctness: $F\approx1$.

If $p_0=0$, normalization divides by zero. No conditional state exists, so fidelity must be reported as N/A, not zero.

Case	Equality validator	$p_0$	$F$	Physical conclusion
Bell preparation: $(H\otimes I)$ then CNOT	PASS; $A_{00}=A_{01}=0.5$	$0.5$	$1$	Reachable and correct
CNOT on $	00\rangle$	PASS; $A_{00}=A_{01}=0$	$0$	N/A

This is the central loophole: $0=0$ is true, but it does not prepare a quantum state.

For a Schmidt-rank-2 state $U|00\rangle$, the cat's reduced state has full support, so a non-zero branch steering it to $|+\rangle$ exists. Rank-1 boundary cases require the separate reachability check above.

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Q6 — If we have a derivation, why run numerical simulations?

Because a correct formula can still be implemented incorrectly.

The figure uses 50 Haar-random $4\times4$ unitaries to visualize the solved angles and their equality errors. The script then uses 100 Haar-random unitaries as a larger stress test.

The third panel is a 3D scatter in $(\theta,\lambda,\text{error})$. Because $\lambda$ is periodic, the $(\theta,\lambda)$ parameter domain can be interpreted topologically as a cylinder; the plot itself is not a drawn cylinder.

The randomized tests check that:

• the code extracts the correct first column of $U$;
• phase alignment and magnitude balancing are implemented correctly;
• the returned angles make $A_{00}$ and $A_{01}$ equal to floating-point precision;
• the implementation works across many typical complex-valued inputs.

The maximum observed amplitude-equality error is $2.4\times10^{-16}$—machine precision.

But randomized agreement is not a proof of the formula, and it does not establish physical validity for every boundary case.

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Q7 — Why test a deterministic counterexample if 100/100 random tests pass?

Because exact zero-probability branches form a measure-zero boundary. Haar-random sampling almost surely produces a Schmidt-rank-2 state and almost surely misses that boundary, no matter how visually convincing a 100/100 pass rate looks.

A deliberately chosen case such as CNOT acting on $|00\rangle$ exposes the semantic gap immediately:

$$ \mathrm{CNOT}|00\rangle=|00\rangle. $$

For the returned degenerate equality solution, the retained atom-$|0\rangle$ branch has

$$ A_{00}=A_{01}=0, \qquad p_0=0. $$

The original equality assertion passes, yet the claimed conditional state is physically undefined. Identity, SWAP, and suitable phase-gate inputs reveal the same class of boundary failure.

A deterministic counterexample is therefore not competing with the random test. It asks a different question that random sampling is structurally unlikely to ask.

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Q8 — How can I explore and reproduce the project?

The fastest route is the Colab notebook. For a local run:

git clone https://github.com/sunshineluyao/schrodingers-cat.git
cd schrodingers-cat
pip install -r requirements.txt

# Run the NumPy-only solver and 100-unitary stress test
python scripts/quantum_sandbox.py

# Regenerate all static figures and GIFs
python scripts/generate_figures.py

# Explore the original challenge notebook
jupyter notebook "Revisiting_Schrodinger's_Cat.ipynb"

Show the copy-paste PennyLane solution

import pennylane as qp
import pennylane.numpy as np

dev = qp.device("default.qubit", wires=["atom", "cat"])

@qp.qnode(dev)
def evolve_atom_cat(unitary, params):
    qp.QubitUnitary(unitary, wires=["atom", "cat"])
    qp.U3(params[0], params[1], params[2], wires="atom")
    return qp.state()

def u3_parameters(unitary):
    """Closed-form U3 angles for the challenge equality condition."""
    a, b, c, d = unitary @ np.array([1, 0, 0, 0], dtype=complex)
    alpha = a - b
    beta = c - d
    abs_alpha = np.abs(alpha)
    abs_beta = np.abs(beta)
    phi = 0.0

    if np.isclose(abs_alpha, 0) and np.isclose(abs_beta, 0):
        theta, lam = 0.0, 0.0
    elif np.isclose(abs_alpha, 0):
        theta, lam = 0.0, 0.0
    elif np.isclose(abs_beta, 0):
        theta, lam = np.pi, 0.0
    else:
        lam = np.angle(alpha) - np.angle(beta)
        theta = 2 * np.arctan(abs_alpha / abs_beta)

    return np.array([theta, phi, lam])

H = np.array([[1, 1], [1, -1]]) / np.sqrt(2)
CNOT = np.array(
    [[1, 0, 0, 0],
     [0, 1, 0, 0],
     [0, 0, 0, 1],
     [0, 0, 1, 0]],
    dtype=complex,
)
U_bell = CNOT @ np.kron(H, np.eye(2))

params = u3_parameters(U_bell)
state = evolve_atom_cat(U_bell, params)
assert np.isclose(state[0], state[1], atol=5e-2)
print("PASS: equal-amplitude challenge condition")

This assertion reproduces the original challenge condition. For a physical preparation claim, also compute $p_0$ and conditional fidelity $F$ using the equations above. The interactive OER demonstrates both the reachable Bell case and the zero-probability CNOT counterexample.

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Q9 — Where is everything in the repository?

├── Revisiting_Schrodinger's_Cat.ipynb   # original challenge notebook
├── assets/
│   ├── hero/                            # banners
│   ├── figures/                         # static SVG and PNG figures
│   └── anim/                            # GitHub-safe GIF animations
├── scripts/
│   ├── quantum_sandbox.py               # NumPy solver + 100-unitary stress test
│   └── generate_figures.py              # reproducible figure/GIF generator
├── docs/
│   └── quantum-computing-101.md         # concept guide (EN + 中文速览)
├── oer/
│   ├── index.html                       # interactive physical-validity lesson
│   ├── README.md                        # Hugging Face Space configuration
│   └── assets/                          # self-contained deployment assets
├── certificates/                        # PennyLane and WISER records
├── Citation.cff                         # GitHub citation metadata
└── requirements.txt

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Q10 — What is the project's provenance?

This project was completed as part of the PennyLane “Revisiting Schrödinger's Cat” challenge and the WISER 2026 summer program.

  

All certificate files (PDF / PNG / SVG) are collected in certificates/.

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Q11 — What should I read or cite?

References

1. PennyLane: Revisiting Schrödinger's Cat challenge
2. PennyLane U3 gate documentation
3. Nielsen & Chuang, Quantum Computation and Quantum Information (Cambridge, 2010), ch. 2 & 4
4. Schrödinger, E. (1935), “Die gegenwärtige Situation in der Quantenmechanik,” Naturwissenschaften 23, 807–812
5. Wiseman & Milburn, Quantum Measurement and Control (Cambridge, 2009) — quantum steering and post-selection
6. Mezzadri, F. (2007), “How to generate random matrices from the classical compact groups,” Notices of the AMS 54(5), 592–604

This repository ships a Citation.cff file, which powers GitHub's Cite this repository button. If you use this work, please cite:

@misc{zhang2026schrodingerscat,
  author = {Zhang, Luyao (Sunshine)},
  title  = {Revisiting Schr\"{o}dinger's Cat: A Complete Guide
            (PennyLane Quantum Challenge)},
  year   = {2026},
  url    = {https://github.com/sunshineluyao/schrodingers-cat},
  note   = {Closed-form U3 equality solution, randomized verification,
            deterministic physical-validity tests, and an interactive OER}
}

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Q12 — What do the three forms of evidence establish together?

They answer three different scientific questions.

Evidence layer	Question it answers	What it establishes	What it cannot establish alone
Mathematical derivation	Is the equal-amplitude formula correct for the stated algebraic problem?	The closed form satisfies $A_{00}=A_{01}$, including degenerate cases.	Whether the code implements the formula correctly; whether the selected branch has non-zero probability.
Randomized numerical simulation	Did we implement the formula correctly on diverse, typical inputs?	100/100 Haar-random tests reach machine-precision amplitude equality.	A universal proof; reliable coverage of measure-zero boundaries; physical meaning of a PASS.
Deterministic counterexample	Does the validator's PASS always mean a realizable quantum state?	No: $A_{00}=A_{01}=0$ passes equality while $p_0=0$ and $F$ is undefined.	The general closed-form solution or broad implementation reliability.

The complete verification record is therefore:

Test	Equality result	Reachability result	Correct interpretation
Bell preparation	$A_{00}=A_{01}=0.5$	$p_0=0.5$	Reachable; $F=1$
One sampled random unitary	PASS	$p_0>0$ almost surely	Reachable for that sampled full-rank state
100 Haar-random unitaries	100/100 PASS	Exact zero is almost surely not sampled	Implementation stress test, not a boundary proof
Identity, SWAP, CNOT, or phase gate on $	00\rangle$	Can PASS with $A_{00}=A_{01}=0$	$p_0=0$

Final lesson: the mathematical derivation proves the equal-amplitude formula; randomized simulation checks its implementation; deterministic counterexamples test its physical meaning. All three are indispensable.

This is the broader trustworthy-computing principle behind the project: a syntactically satisfied assertion is not yet an operationally reachable outcome, and an operational outcome is not yet the intended physical state.

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_{PennyLane challenge solution · interactive physical-validity OER added August 2026 · all figures reproducible via scripts/generate_figures.py}