The 1935 Thought Experiment

Erwin Schrödinger introduced the cat in a paper titled “Die gegenwärtige Situation in der Quantenmechanik” (Naturwissenschaften, 1935). The paper is a critique of the Copenhagen interpretation of quantum mechanics, not an endorsement of macroscopic superposition.

The setup is familiar: a cat is placed in a sealed chamber with a radioactive atom, a Geiger counter, a hammer, and a vial of poison. If the atom decays in one hour, the counter fires, the hammer falls, the vial breaks, and the cat dies. If the atom does not decay, the cat lives. The atom is a quantum system; after one hour it is in a superposition of decayed and undecayed states.

Quantum mechanics — specifically, the Schrödinger equation, applied without any special rule for measurement — says the entire system (atom + counter + hammer + vial + cat) evolves into a superposition:

$$|\Psi\rangle = \frac{1}{\sqrt{2}}\bigl(|\text{decayed}\rangle|\text{cat dead}\rangle

• |\text{undecayed}\rangle|\text{cat alive}\rangle\bigr).$$

Schrödinger’s point was that this is absurd: the cat is either dead or alive, not a superposition of both, and any interpretation of quantum mechanics that predicts otherwise is failing at the level of macroscopic physical reality. He intended the cat as a reductio ad absurdum — a demonstration that taking the wave function literally at macroscopic scales leads to nonsense.

He was not proposing that cats are literally in superposition. He was proposing that the theory was incomplete.

What Actually Resolves the Cat

The resolution that modern physics offers is decoherence — the process by which a quantum superposition is destroyed through entanglement with the environment.

A macroscopic object — a cat, a hammer, a Geiger counter — is coupled to an enormous number of environmental degrees of freedom: air molecules, photons, phonons in its own structure. Each of these interactions entangles the macroscopic system with the environment, and the entanglement effectively destroys the coherence between branches of the superposition. What starts as

$$|\Psi\rangle = \frac{1}{\sqrt{2}}(|\text{decayed}\rangle|\text{dead}\rangle

• |\text{undecayed}\rangle|\text{alive}\rangle)$$

rapidly becomes, after environmental entanglement (tracing over environmental degrees of freedom $|E\rangle$):

$$\rho = \frac{1}{2}|\text{decayed}\rangle\langle\text{decayed}| \otimes |\text{dead}\rangle\langle\text{dead}|

• \frac{1}{2}|\text{undecayed}\rangle\langle\text{undecayed}| \otimes |\text{alive}\rangle\langle\text{alive}|.$$

This is a mixed state, not a superposition. The off-diagonal terms (the interference terms that distinguish a superposition from a classical mixture) vanish on a timescale

where $E_\mathrm{int}$ is the interaction energy with each environmental degree of freedom and $N$ is the number of such degrees of freedom. For a macroscopic object at room temperature, $\tau_\mathrm{decoherence}$ is of order $10^{-20}$–$10^{-30}$ seconds — unmeasurably short. The cat is never in a superposition for any observable duration. The superposition collapses before any measurement can resolve it.

This is not a philosophical solution to the measurement problem — it does not explain why a particular measurement outcome is obtained, only why we never observe interference between macroscopic branches — but it does explain why Schrödinger’s setup does not produce an observable macroscopic superposition. The cat’s entanglement with its own environment (the box, the air, its own thermal photons) destroys the coherence long before any observation.

What a Cat State Actually Is

In quantum optics, a cat state is not a cat in a superposition. It is a specific quantum state of a harmonic oscillator (typically a mode of the electromagnetic field) that was named in honour of Schrödinger’s thought experiment.

A coherent state $|\alpha\rangle$ is the quantum state that most closely resembles a classical oscillating electromagnetic field with amplitude $\alpha \in \mathbb{C}$. Coherent states are eigenstates of the annihilation operator: $\hat{a}|\alpha\rangle = \alpha|\alpha\rangle$. The mean photon number is $\bar{n} = |\alpha|^2$.

A cat state is a superposition of two coherent states with opposite phases:

where $\mathcal{N}_\pm = 1/\sqrt{2(1 \pm e^{-2|\alpha|^2})}$ is the normalisation constant. For large $|\alpha|$, the two coherent states are nearly orthogonal: $\langle -\alpha | \alpha \rangle = e^{-2|\alpha|^2} \approx 0$.

The Wigner quasi-probability distribution of a cat state is revealing. The Wigner function of a coherent state $|\alpha\rangle$ is a Gaussian peaked at $(x, p) = (\sqrt{2}\,\mathrm{Re}\,\alpha, \sqrt{2}\,\mathrm{Im}\,\alpha)$. The cat state Wigner function is:

$$W_{\mathrm{cat}+}(x,p) = \mathcal{N}+^2\bigl[W_{|\alpha\rangle}(x,p) + W_{|-\alpha\rangle}(x,p)

• 2W_\mathrm{int}(x,p)\bigr],$$

where the interference term $W_\mathrm{int}$ has negative values in the region between the two Gaussian peaks. Negative regions of the Wigner function are a signature of non-classical states; they cannot arise from any classical probability distribution. The cat state is quantum mechanical in a way that coherent states are not.

Haroche and the Nobel Prize

Serge Haroche (ENS Paris) spent two decades developing techniques to create, control, and observe cat states of the electromagnetic field in real time. His experiment used a superconducting microwave cavity — a polished copper box cooled to near absolute zero — in which single microwave photons could be trapped for hundreds of milliseconds, and a beam of single Rydberg atoms to probe the field non-destructively.

Haroche created cat states of cavity photons and, crucially, watched their decoherence in real time: as the quantum coherence between the two branches $|\alpha\rangle$ and $|-\alpha\rangle$ was progressively destroyed by coupling to the environment, the Wigner function’s negative region (the interference fringe) smoothed out and disappeared, leaving a classical mixture. The decoherence rate was proportional to $|\alpha|^2$ — the mean photon number, which measures how “macroscopic” the cat state is:

where $\kappa$ is the photon loss rate of the cavity. A larger cat (larger $|\alpha|^2$) decoheres faster, as Schrödinger’s argument implicitly requires.

Haroche shared the 2012 Nobel Prize in Physics with David Wineland “for ground-breaking experimental methods that enable measuring and manipulation of individual quantum systems.”

Cat Qubits: From Paradox to Engineering

The step from fundamental physics to quantum computing was taken when researchers noted that the two coherent states $|\alpha\rangle$ and $|-\alpha\rangle$ can serve as the two computational basis states of a qubit:

The cat qubit encodes a logical qubit in this pair of coherent states. Its remarkable property is an intrinsic asymmetry between error types.

Bit-Flip Suppression

A bit-flip error ($|0\rangle_L \leftrightarrow |1\rangle_L$, i.e., $|\alpha\rangle \leftrightarrow |-\alpha\rangle$) requires flipping the amplitude of the oscillator from $+\alpha$ to $-\alpha$. For a stabilised cat qubit (confined to the cat-state manifold by a parametric drive), this requires overcoming an energy barrier proportional to $|\alpha|^2$. The bit-flip time scales exponentially:

where $T_1$ is the single-photon loss time. For modest values of $|\alpha|^2$ (mean photon numbers of 5–10), the bit-flip time can exceed minutes.

A phase-flip error (the other error type) is not suppressed — the cat qubit is still vulnerable to dephasing at a rate proportional to $|\alpha|^2$. This creates a strongly biased noise channel: only one of the two error types is relevant.

The Engineering Consequence

Biased noise is useful because it allows the error-correcting code to focus its resources on only one error type. A repetition code (a string of cat qubits where phase errors are corrected by majority vote) can suppress the phase-flip error arbitrarily while the exponential bit-flip suppression handles the other. The hardware overhead for fault tolerance — the ratio of physical qubits to logical qubits — is dramatically reduced compared to codes that must handle both error types equally.

In 2023 and 2024, several groups demonstrated cat qubits with bit-flip times of seconds to minutes:

• Grimm et al. (2020, Nature 584, 205): Kerr cat qubit with exponential bit-flip suppression demonstrated in a superconducting circuit.
• Berdou et al. (2023, PRX Quantum 4, 020350): Cat qubit with $T_X$ exceeding $100$ seconds.
• Reglade et al. (2024, Nature 629, 778–783): Cat qubits from Alice & Bob demonstrating exponential scaling $T_\mathrm{bit-flip} \propto e^{2|\alpha|^2}$ with mean photon numbers up to $|\alpha|^2 \approx 10$, pushing bit-flip times beyond $10$ seconds in the laboratory and, in subsequent chip demonstrations, beyond several minutes.

This is the state of the art as of early 2025: the cat qubit is no longer a curiosity but a competitive architecture for fault-tolerant quantum computing, with bit-flip coherence times exceeding the best alternative approaches.

The Wigner Function and Quantum Non-Classicality

The Wigner quasi-probability distribution provides the most informative picture of a quantum state’s non-classicality. For a state with density matrix $\rho$, the Wigner function is:

For the cat state $|\mathrm{cat}_+\rangle$ with $|\alpha|^2 = 4$ (four mean photons in each coherent component), the Wigner function has two positive Gaussian peaks at $(x, p) = (\pm\sqrt{2}|\alpha|, 0)$ and an oscillating interference fringe between them with negative regions of amplitude $\sim -2/\pi$. The negativity of the Wigner function is a necessary condition for the state to exhibit quantum features that no classical mixture can reproduce.

As decoherence proceeds (e.g., through photon loss in a cavity), the negative regions shrink and eventually vanish — the Wigner function becomes everywhere non-negative, and the state becomes classically describable as a mixture of coherent states. This is the quantum-to-classical transition, made visible in phase space.

Haroche’s team measured this process directly, frame by frame, in real time. It is one of the most dramatic experimental visualisations of decoherence ever achieved.

What Schrödinger Would Make of This

Schrödinger was a physicist, not a philosopher of language. If told in 1935 that ninety years later, the superposition of two distinguishable states of a harmonic oscillator — named after his cat, with the same formal structure as his thought experiment — would be the leading candidate for the basic unit of a fault-tolerant quantum computer, he would have had two questions.

The first: how do you maintain the superposition against decoherence? The answer is that you work at millikelvin temperatures in superconducting circuits, and you use an active parametric drive to confine the state to the cat-state manifold.

The second, I think, would have been: does this resolve the measurement problem? And the honest answer remains: no, not fully. Decoherence explains why macroscopic superpositions are unobservable, but it does not explain why any particular measurement outcome occurs. That question is as open as it was in 1935.

What has changed is the practical relationship between quantum theory and technology. The uncertainty Schrödinger was pointing at — the strangeness of superposition, the fragility of coherence, the role of the environment — is now a resource to be engineered, not a conceptual embarrassment to be resolved. The cat qubit works precisely because the decoherence is asymmetric: bit flips are exponentially suppressed while phase flips are correctable. The asymmetry is exploited, not apologised for.

My two cats, meanwhile, are in definite classical states. One is on the radiator. The other is on the keyboard.

References

• Grimm, A., Frattini, N.E., Puri, S., Mundhada, S.O., Touzard, S., Mirrahimi, M., Girvin, S.M., Shankar, S., & Devoret, M.H. (2020). Stabilization and operation of a Kerr-cat qubit. Nature, 584, 205–209. https://doi.org/10.1038/s41586-020-2587-z

• Haroche, S., & Raimond, J.-M. (2006). Exploring the Quantum: Atoms, Cavities, and Photons. Oxford University Press.

• Reglade, U., Bocquet, A., Gautier, R., et al. (2024). Quantum control of a cat qubit with bit-flip times exceeding ten seconds. Nature, 629, 778–783. https://doi.org/10.1038/s41586-024-07294-3

• Mirrahimi, M., Leghtas, Z., Albert, V.V., Touzard, S., Schoelkopf, R.J., Jiang, L., & Devoret, M.H. (2014). Dynamically protected cat-qubits: A new paradigm for universal quantum computation. New Journal of Physics, 16, 045014. https://doi.org/10.1088/1367-2630/16/4/045014

• Schrödinger, E. (1935). Die gegenwärtige Situation in der Quantenmechanik. Naturwissenschaften, 23(48), 807–812; 23(49), 823–828; 23(50), 844–849. https://doi.org/10.1007/BF01491891

• Walls, D.F., & Milburn, G.J. (2008). Quantum Optics (2nd ed.). Springer.

• Zurek, W.H. (2003). Decoherence, einselection, and the quantum origins of the classical. Reviews of Modern Physics, 75(3), 715–775. https://doi.org/10.1103/RevModPhys.75.715

Changelog

• 2026-02-17: Updated “bit-flip times exceeding seven minutes” in the summary to “exceeding minutes,” aligning with the sourced figures: the body text reports “beyond several minutes” and Reglade et al. (2024) report “exceeding ten seconds.”

A three-year-old cannot put her shoes on before her socks. Not because she lacks motor skills — because the operations do not commute.

The same structural constraint, dressed in the language of operators on a Hilbert space, is why Heisenberg’s uncertainty principle holds. This post is about that connection: the accidental algebra lesson built into getting dressed, and why the physicists of 1925 had to abandon one of arithmetic’s most taken-for-granted assumptions.

Getting Dressed Is a Non-Abelian Problem

Start with the mundane. Your morning routine imposes a strict partial order on operations: underwear before trousers, socks before shoes, cap before chin-strap if you cycle. Try reversing any pair and the sequence fails — physically, not just socially. You cannot pull a sock over a shoe.

The operation “put on socks” followed by “put on shoes” produces a wearable human; the reverse produces neither, and no amount of wishing commutativity into existence will help.

In the language of abstract algebra, two operations \(A\) and \(B\) commute if \(AB = BA\) — if doing them in either order yields the same result. Everyday life is full of operations that do not commute: rotate a book 90° around its vertical axis then 90° around its horizontal axis; now reverse the order. The final orientations differ. Turn right then turn left while driving; left then right. Different positions.

The intuition is not hard to build. What is surprising is how rarely we note it, and what it costs us when we finally hit a domain — quantum mechanics — where non-commutativity is not an inconvenient edge case but the central fact.

Piaget Said Seven; Toddlers Disagreed

Jean Piaget argued that children do not acquire operational thinking — the ability to mentally perform and reverse sequences of actions — until the concrete operational stage, roughly ages seven to eleven (Inhelder & Piaget, 1958). Before that, he claimed, children lack the understanding that an operation can be undone or reordered.

Post-Piagetian research pushed back hard. Patricia Bauer and Jean Mandler tested infants aged sixteen and twenty months on novel, multi-step action sequences (Bauer & Mandler, 1989). For causally structured sequences — where step A physically enables step B — infants reproduced the correct order after a two-week delay. They were not told the order was important. They had no language to encode it. They just knew, implicitly, that the operations had a necessary direction.

A 2020 study by Klemfuss and colleagues tested 100 children aged roughly two-and-a-half to five on temporal ordering questions (Klemfuss et al., 2020). Children answered “what happened first?” questions correctly 82% of the time. The errors that did appear followed an encoding-order bias — children defaulted to reporting the next event in the sequence as originally experienced, regardless of what was asked. The ordering knowledge was intact. What children lack, for Piaget’s full seven years, is the formal recursive conception of reversibility. The procedural knowledge — that some sequences must be done in the right order and cannot be freely rearranged — is there from the second year of life.

Which means: learning that \(AB \neq BA\) is not learning something exotic. It is articulating something the nervous system already knows.

The Mathematician’s Commutator

Abstract algebra formalized this intuition in the nineteenth century. A group is abelian (commutative) if every pair of elements satisfies \(ab = ba\). Integers under addition: abelian. Rotations in three dimensions: not.

Arthur Cayley’s 1858 memoir established matrix algebra as a formal theory (Cayley, 1858). Multiply two \(2 \times 2\) matrices:

\(AB \neq BA\). Non-commutativity is not a curiosity; it is the generic condition for matrix products. Commutativity is the special case — and requiring justification.

William Rowan Hamilton had already gone further. On 16 October 1843, walking along the Royal Canal in Dublin, he discovered the quaternions and carved their multiplication rule into the stone of Broom Bridge:

From this it follows immediately that \(ij = k\) but \(ji = -k\). Hamilton’s four-dimensional number system — the first algebraic structure beyond the complex numbers — was non-commutative by construction. He did not apologize for it. He celebrated it.

The Lie algebra structure underlying these commutator relations is the same skeleton that governs Messiaen’s modes of limited transposition, which I traced in a previous post on group theory and music — a very different physical domain, but identical algebraic machinery.

Born, Jordan, and the Physicist’s Shock

Classical mechanics treats position \(x\) and momentum \(p\) as ordinary real numbers. Real numbers commute: \(xp = px\). The Poisson bracket \(\{x, p\} = 1\) encodes a classical relationship, but the underlying quantities are scalars, and scalars commute.

In July 1925, Werner Heisenberg published a paper that could not quite bring itself to say what it was doing (Heisenberg, 1925). He replaced classical dynamical variables with arrays of numbers — what we would now call matrices — and found, uncomfortably, that the resulting quantum condition required order to matter.

While Heisenberg was on vacation, Max Born and Pascual Jordan finished the translation into matrix language (Born & Jordan, 1925). They wrote the commutation relation explicitly, recognized it as the fundamental law, and showed that it reproduced the known quantum results:

Non-commutativity of position and momentum was not a mathematical accident. It was the theory.

The uncertainty principle followed four years later as a theorem, not an additional postulate. Howard Robertson proved in 1929 that for any two observables \(\hat{A}\) and \(\hat{B}\), the Cauchy–Schwarz inequality on Hilbert space yields (Robertson, 1929):

Substituting \(\hat{A} = \hat{x}\), \(\hat{B} = \hat{p}\), \([\hat{x}, \hat{p}] = i\hbar\):

This is the uncertainty principle. It does not say nature is fuzzy or that measurement disturbs systems in some vague intuitive sense. It says: position and momentum are operators that do not commute, and the Robertson inequality then constrains their joint variance. Non-commutativity is the uncertainty principle. Put the shoes on before the socks and the state is not defined.

The same logic applies to angular momentum. The three components satisfy:

This is the Lie algebra \(\mathfrak{su}(2)\). You cannot simultaneously determine two components of angular momentum to arbitrary precision — not because the measurement apparatus is noisy, but because the operations of measuring them do not commute.

The fiber bundle language that underlies these rotation groups also appears, in different physical dress, in the problem of the falling cat and geometric phases — another case where the order of rotations has non-trivial physical consequences (see that post).

Connes and Non-Commutative Space

Alain Connes asked what happens if we allow the coordinates of space itself to be non-commutative. In ordinary geometry, the algebra of coordinate functions on a manifold is commutative: \(f(x) \cdot g(x) = g(x) \cdot f(x)\). Connes’ non-commutative geometry replaces this with a spectral triple \((\mathcal{A}, \mathcal{H}, D)\): an algebra \(\mathcal{A}\) of operators (possibly non-commutative) acting on a Hilbert space \(\mathcal{H}\), with a generalized Dirac operator \(D\) encoding the geometry (Connes, 1994).

The payoff was remarkable. With Ali Chamseddine, Connes showed that if \(\mathcal{A}\) is chosen as a specific non-commutative product of the real numbers, complex numbers, quaternions, and matrix algebras, the spectral action principle reproduces the full Lagrangian of the Standard Model coupled to general relativity from a single geometric principle (Chamseddine & Connes, 1996). The Higgs field, the gauge bosons, the graviton: all from the geometry of a non-commutative space.

Classical geometry is the special case where the coordinate algebra is commutative. Drop that assumption and you open up a vastly richer landscape. Quantum mechanics lives in that landscape. Possibly, so does the structure of spacetime at the Planck scale.

The Lesson Pre-Schoolers Already Know

There is an irony here that I cannot quite leave alone. Students learning linear algebra for the first time consistently make the same mistake. Anna Sierpinska documented it carefully: they assume \(AB = BA\) for matrices because they have spent years in arithmetic and scalar algebra where multiplication commutes (Sierpinska, 2000). The commutativity of ordinary multiplication is so deeply internalized that abandoning it feels like breaking a rule.

But the pre-schooler in the sock-and-shoe scenario never had that problem. Her procedural memory, documented in infants as young as sixteen months by Bauer and Mandler, encoded the correct asymmetry directly. The order of operations is the first thing a developing mind learns about actions in the world, before the arithmetic of school teaches it the convenient fiction that order is irrelevant.

Arithmetic is the outlier. \(3 + 5 = 5 + 3\) because counting does not depend on where you start. But putting on clothes, multiplying matrices, rotating rigid bodies, measuring quantum observables: these operations carry memory of order, and they repay the attention a child already brings to them before she can name a number.

The universe is non-abelian. We are born knowing it. School briefly convinces us otherwise. Physics eventually agrees with the pre-schooler.

References

• Inhelder, B., & Piaget, J. (1958). The Growth of Logical Thinking from Childhood to Adolescence. Basic Books.
• Bauer, P. J., & Mandler, J. M. (1989). One thing follows another: Effects of temporal structure on 1- to 2-year-olds’ recall of events. Developmental Psychology, 25, 197–206.
• Klemfuss, J. Z., McWilliams, K., Henderson, H. M., Olaguez, A. P., & Lyon, T. D. (2020). Order of encoding predicts young children’s responses to sequencing questions. Cognitive Development, 55, 100927. DOI: 10.1016/j.cogdev.2020.100927
• Cayley, A. (1858). A memoir on the theory of matrices. Philosophical Transactions of the Royal Society of London, 148, 17–37. DOI: 10.1098/rstl.1858.0002
• Heisenberg, W. (1925). Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen. Zeitschrift für Physik, 33, 879–893.
• Born, M., & Jordan, P. (1925). Zur Quantenmechanik. Zeitschrift für Physik, 34, 858–888.
• Robertson, H. P. (1929). The uncertainty principle. Physical Review, 34, 163–164. DOI: 10.1103/PhysRev.34.163
• Connes, A. (1994). Noncommutative Geometry. Academic Press. ISBN 0-12-185860-X.
• Chamseddine, A. H., & Connes, A. (1996). Universal formula for noncommutative geometry actions: Unification of gravity and the standard model. Physical Review Letters, 77, 4868–4871. DOI: 10.1103/PhysRevLett.77.4868
• Sierpinska, A. (2000). On some aspects of students’ thinking in linear algebra. In J.-L. Dorier (Ed.), On the Teaching of Linear Algebra (pp. 209–246). Kluwer Academic Publishers. DOI: 10.1007/0-306-47224-4_8

Changelog

• 2026-02-03: Corrected the age range for the Klemfuss et al. (2020) study from “two to four” to “roughly two-and-a-half to five” — the actual participants were aged 30–61 months.
• 2026-02-03: Updated the characterisation of Klemfuss et al. (2020) findings to reflect the paper’s central result: errors follow an encoding-order bias (children default to the next event in encoding sequence). The paper’s title — “Order of encoding predicts young children’s responses” — names the mechanism.