Coherent States

Definition

In quantum optics and QFT, a coherent state is an eigenstate of the annihilation operator:

$a|\alpha ⟩=\alpha |\alpha ⟩,\alpha \in \mathbb{C}.$

Coherent states are the minimum-uncertainty states: they saturate the Heisenberg uncertainty relation $\Delta x\Delta p=\hslash /2$ with equal uncertainties in both quadratures (unlike squeezed states).

In the Weyl operator language, the coherent state is the displaced vacuum:

$|\alpha ⟩=W(\alpha )|0⟩=D(\alpha )|0⟩,$

where $D(\alpha )=\exp (\alpha a^{†}-{\alpha }^{\ast }a)$ is the displacement operator. In QFT with a smeared field, the coherent state associated with test function $f$ is

$|{\psi }_f⟩=W(f)|0⟩,$

corresponding to a classical field configuration $⟨\phi (x){⟩}_f=⟨{\psi }_f|\phi (x)|{\psi }_f⟩\ne 0$.

The expectation value of Weyl operators in coherent states is:

$⟨{\psi }_f|W(g)|{\psi }_f⟩=e^{i\text{Re}⟨f,g{⟩}_{+}}{e}^{-\frac{1}{2}\Vert g{\Vert }_{+}^2},$

where $⟨f,g{⟩}_{+}$ is the positive-frequency inner product. This Gaussian form makes all correlators exactly computable.

The Araki-Uhlmann relative entropy between a coherent state ${\omega }_f$ and the vacuum ${\omega }_0$, both restricted to the algebra of a wedge, is

$S({\omega }_f\Vert {\omega }_0)=\Vert f{\Vert }_{+}^2,$

the one-particle norm squared of the displacement function. On the global algebra the two states are pure and the relative entropy diverges, so the formula is a local statement.

Role in Research

Coherent states appear in the quantum information program as the simplest excited states for which Bell-CHSH violations and relative entropy can be computed explicitly. They form a baseline against which squeezed and cat states are compared.

Key computations:

• Bell-CHSH violation for coherent states: The CHSH correlator is computed explicitly using the Gaussian expectation value formula. Coherent states produce Bell violations that depend on the displacement $f$ and the test functions used for the Weyl operators.

• Relative entropy: The formula $S({\omega }_f\Vert {\omega }_0)=\Vert f{\Vert }_{+}^2$ gives a simple geometric interpretation: relative entropy is the squared distance in one-particle Hilbert space between the coherent displacement and the vacuum.

• Comparison class for cat states: The 2026 paper (Phys. Rev. D 113, 065008) computes Bell violations for cat states $|\alpha ⟩+|-\alpha ⟩$, using coherent states as the building blocks and highlighting how quantum interference modifies the Bell correlator relative to the incoherent mixture of $|\alpha ⟩$ and $|-\alpha ⟩$ separately.

Relations

• weyl-operators — coherent states are defined by $W(f)|0⟩$; the Weyl algebra structure makes all expectation values exactly computable via Gaussian integrals
• araki-uhlmann-relative-entropy — $S({\omega }_f\Vert {\omega }_0)=\Vert f{\Vert }_{+}^2$ is the prototype computable relative entropy, serving as the baseline for more complex states
• squeezed-states — squeezed states reduce uncertainty in one quadrature; they are Bogoliubov transformations of the vacuum rather than displacements; relative entropy involves Bogoliubov coefficients
• cat-states — cat states are superpositions of coherent states $|\alpha ⟩+|-\alpha ⟩$; their Bell violations and relative entropy differ from those of individual coherent states due to interference
• bell-chsh-inequality — explicit CHSH correlators for coherent states are computed in the quantum information program, providing benchmarks for optimization
• type-iii-von-neumann-algebras — even coherent states are mixed when restricted to local algebras; the type III₁ structure means there are no pure normal local states

Papers

See bell-inequalities-qft and relative-entropy-qft for full paper lists.

Notes

• Coherent states form an overcomplete basis: $\frac{1}{\pi }\int |\alpha ⟩⟨\alpha |d^2\alpha =1$. This overcomplete resolution of identity (with the phase space measure $d^2\alpha$) is the basis of the Glauber-Sudarshan P-representation.
• Coherent states are not orthogonal: $|⟨\alpha |\beta ⟩{|}^2=e^{-|\alpha -\beta {|}^2}$. This overlap is what makes cat states ($|\alpha ⟩\pm |-\alpha ⟩$) physically interesting — for large $|\alpha |$ the two components are nearly orthogonal, approximating a genuine superposition of macroscopically distinct states.
• In QFT, the coherent state $W(f)|0⟩$ is a Fock-space state only when $f$ has finite one-particle norm. For general $f$ (e.g., in the thermodynamic limit), it may be a unitarily inequivalent representation — an important subtlety for infinite-volume QFT.