Cat States

Definition

A cat state (or Schrödinger cat state) is a superposition of two macroscopically distinct coherent states:

$|{\text{cat}}_{\pm }⟩\propto |\alpha ⟩\pm |-\alpha ⟩,\alpha \in \mathbb{C},$

with normalization ${\mathcal{N}}_{\pm }=\sqrt{2(1\pm e^{-2|\alpha {|}^2})}$. The name refers to Schrödinger’s thought experiment: for $|\alpha |\gg 1$ the states $|\alpha ⟩$ and $|-\alpha ⟩$ are nearly orthogonal ($|⟨\alpha |-\alpha ⟩|=e^{-2|\alpha {|}^2}\approx 0$) and represent macroscopically distinguishable field amplitudes — “dead” and “alive” superposed.

In the Wigner function representation, cat states have oscillatory interference fringes between the two coherent-state peaks — a hallmark of quantum superposition with no classical analogue.

The expectation value of a Weyl operator $W(g)$ in the cat state $|{\text{cat}}_{+}⟩$ is

$⟨{\text{cat}}_{+}|W(g)|{\text{cat}}_{+}⟩=\frac{2\cos \Delta (f,g)e^{-\frac{1}{2}\Vert g{\Vert }^2}+e^{-\frac{1}{2}\Vert g-2f{\Vert }^2}+e^{-\frac{1}{2}\Vert g+2f{\Vert }^2}}{2(1+e^{-2\Vert f{\Vert }^2})},$

where $\Delta (f,g)$ is the smeared Pauli–Jordan form that appears in the Weyl relation and $\Vert \cdot \Vert$ is the one-particle norm. The first term is what the incoherent mixture $\frac{1}{2}(|\alpha ⟩⟨\alpha |+|-\alpha ⟩⟨-\alpha |)$ gives on its own; the two Gaussians are the cross terms $⟨\alpha |W(g)|-\alpha ⟩$, and since $\Vert g\mp 2f{\Vert }^2=\Vert g{\Vert }^2\mp 4⟨f,g⟩+4\Vert f{\Vert }^2$ they carry the overlap suppression $e^{-2\Vert f{\Vert }^2}$. At $g=0$ the four terms and the normalization combine to $1$, which is the check the expression has to pass.

In QFT, the cat state is constructed as $W(f)|0⟩+W(-f)|0⟩$ for a test function $f$ with compact support, providing a well-localized field excitation.

Role in Research

Cat states are the focus of the 2026 paper Phys. Rev. D 113, 065008, which computes Bell-CHSH violations for cat states in QFT vacuum. The key findings:

1. Interference-driven Bell violation. The CHSH correlator for cat states carries the two cross terms above, which the incoherent mixture does not have. They can enhance or suppress the violation depending on the test functions, and the overlap $e^{-2\Vert f{\Vert }^2}$ controls their size: the interference is a small-amplitude effect, and at large amplitude the cat correlator approaches the mixture’s.

2. New qualitative features. For a coherent state the correlator depends on the test functions through $\Vert g\Vert$ and the symplectic phase alone; for a cat state it depends on the inner product $⟨f,g⟩$ as well, through the cross terms, which opens parameter space for optimization.

3. Large-$|\alpha |$ limit. For large amplitude $|\alpha |\gg 1$, the cat state approaches a superposition of macroscopically distinct field configurations. Bell violation in this regime tests quantum superposition of classically distinguishable states — connecting quantum information to the measurement problem.

4. Connection to quantum interference. The enhancement of Bell violation by quantum interference (compared to the incoherent mixture) is a concrete quantitative demonstration that quantum coherence — not merely entanglement — plays a role in Bell inequality violations in QFT.

Relations

• coherent-states — cat states are superpositions of coherent states $|\alpha ⟩\pm |-\alpha ⟩$; Bell correlators for cat states contain the coherent-state correlators plus interference cross-terms
• weyl-operators — cat state expectation values of Weyl operators are computable in closed form as sums of Gaussian terms with interference; the Weyl algebra structure is essential
• bell-chsh-inequality — the 2026 paper explicitly computes the CHSH correlator for cat states, finding qualitatively new features from interference
• squeezed-states — complementary family of non-classical states; together with cat states they explore the space of non-Gaussian vs Gaussian-but-squeezed excitations
• araki-uhlmann-relative-entropy — relative entropy $S({\omega }_{\text{cat}}\Vert {\omega }_0)$ can be computed for cat states; involves both the coherent-state contribution and interference corrections
• type-iii-von-neumann-algebras — in the type III₁ local algebra context, cat states (as global Fock-space states) restrict to mixed states when localized; their Bell properties are computed using the algebraic machinery

Papers

See bell-inequalities-qft for full paper list. The primary cat state paper is Phys. Rev. D 113, 065008 (2026).

Notes

• The normalization factor $e^{-2|\alpha {|}^2}$ in the denominator matters for small $|\alpha |$; for large $|\alpha |$ the even and odd cat states $|{\text{cat}}_{\pm }⟩$ become equally normalized and nearly orthogonal.
• Cat states are experimentally realized in microwave cavity QED, superconducting qubits, and (approximately) in optical systems. They are highly sensitive to decoherence: even a single photon loss destroys the coherence between $|\alpha ⟩$ and $|-\alpha ⟩$.
• Common confusion: “cat state” in quantum information can sometimes refer to GHZ-type states (multi-qubit analogs). In the QFT context here, it specifically means $|\alpha ⟩\pm |-\alpha ⟩$ in terms of field coherent states.
• The Wigner function of a cat state is non-positive (it has regions where the quasi-probability distribution is negative), confirming its non-classical nature — in contrast to coherent and squeezed states whose Wigner functions are non-negative Gaussians.