Gaussian StateEdit

Gaussian states occupy a foundational place in the theory of quantum information with continuous variables. They describe the quantum states of bosonic modes—such as modes of the electromagnetic field or a quantum harmonic oscillator—that have Gaussian statistics in phase space. Their defining feature is that the Wigner function, a quasi-probability distribution used to represent quantum states in phase space, is Gaussian. As a result, all properties of a Gaussian state are completely determined by just two ingredients: the first moments (the mean values of the quadratures) and the second moments (the variances and covariances of the quadratures) of the state.

Because of this mathematical simplicity, Gaussian states are both conceptually transparent and experimentally accessible. They arise naturally in many optical experiments and serve as the workhorse for a wide range of protocols in quantum communication, sensing, and computation. The theory links closely to the phase-space language of classical optics, while preserving the essential quantum features that distinguish nonclassical states from classical light. See Continuous-variable quantum information and Quantum optics for broader context.

In the following sections, we summarize the formal framework, typical examples, operational tools, and the key places where Gaussian states intersect with current research and applications. Along the way, we emphasize the ways Gaussian states are worked with in practice, and how their limitations motivate the use of non-Gaussian resources for tasks such as universal quantum computation.

Definition and mathematical framework

Gaussian states are states of one or more bosonic modes whose Wigner function is Gaussian in the n-dimensional phase space of quadrature variables. For each mode, define the quadrature operators x_j and p_j (j = 1,...,n), which satisfy the canonical commutation relations [x_j, p_k] = i δ_jk (in natural units). Collect these into a phase-space vector R = (x_1, p_1, ..., x_n, p_n)^T. The Wigner function W(R) of a Gaussian state has the form W(R) = (1 / ((2π)^n sqrt(det V))) exp[-1/2 (R − d)^T V^(-1) (R − d)], where d is the first-moment (displacement) vector with components d_j = ⟨R_j⟩, and V is the 2n×2n covariance matrix with elements V_jk = 1/2 ⟨R_j R_k + R_k R_j⟩ − ⟨R_j⟩⟨R_k⟩. The covariance matrix V is a real, symmetric matrix that encodes all second-moment statistics and is constrained by the quantum uncertainty principle: V + i Ω ≥ 0, where Ω is the symplectic form, Ω = ⊕_j [ [0, 1], [−1, 0] ].

Key results connect V to the state’s purity and entanglement. Williamson’s theorem guarantees that V can be diagonalized by a symplectic transformation: V = S^T D S, with D = ⊕_j diag(ν_j, ν_j) and ν_j ≥ 1/2 (in units with ħ = 1). The ν_j are the symplectic eigenvalues and completely characterize the state up to symplectic (Gaussian) transformations.

For a single mode, a Gaussian state is completely specified by its displacement d = (⟨x⟩, ⟨p⟩) and its variances (⟨Δx^2⟩, ⟨Δp^2⟩), subject to the uncertainty relation ⟨Δx^2⟩⟨Δp^2⟩ ≥ (1/4). For multiple modes, cross-covariances ⟨Δx_j Δx_k⟩, ⟨Δx_j Δp_k⟩, etc., become part of V, encoding correlations that can be used to generate entanglement.

This framework links naturally to several operations. Quadratic Hamiltonians generate Gaussian unitary evolutions, which act linearly on the quadratures and transform V by a symplectic congruence: V → S V S^T, d → S d. Common Gaussian unitaries include single-mode phase shifts, squeezing, and mode-mixing devices like beam splitters. Displacements shift d but leave V unchanged. The phase-space picture also makes clear how Gaussian channels—such as loss, amplification, and thermal noise—affect states, often by adding classical noise to V and shifting d.

Typical Gaussian states and their properties

Vacuum state: The ground state of the quantum harmonic oscillator has d = 0 and V = (1/2) I, representing equal, minimum-uncertainty fluctuations in x and p. It is the purest Gaussian state and serves as the reference point for many experiments. See vacuum state.
Coherent states: Displaced vacuum states with d ≠ 0 and V = (1/2) I. They behave like classical oscillators in many respects and are eigenstates of the annihilation operator. See coherent state.
Squeezed states: States with reduced variance in one quadrature at the expense of increased variance in the conjugate quadrature, while preserving the uncertainty bound. They are described by a V with unequal diagonal elements after an appropriate basis change. See squeezed state.
Thermal states: Mixtures of number states that, in phase space, appear as isotropic Gaussian clouds with variance set by the mean photon number. They remain Gaussian and are central in modeling loss and noise in experiments. See thermal state.
Multi-mode Gaussian states: When several modes are involved, correlations encoded in V can produce entanglement. A canonical example is the two-mode squeezed vacuum, which is entangled and is fundamental to CV quantum teleportation and CV quantum communication. See two-mode squeezed vacuum.
Nonclassical features within Gaussian states: Even though their Wigner function is nonnegative, Gaussian states can exhibit nonclassical properties such as squeezing, which has no classical counterpart. See Wigner function and nonclassical light.

Gaussian operations, channels, and transformations

Gaussian unitary operations are generated by Hamiltonians that are at most quadratic in the quadratures. They include: - Displacements: shift d without changing V. - Phase shifts and rotations: rotate the quadrature axes in phase space. - Squeezing: change the aspect ratio of the Wigner function, altering V in a controlled way. - Beamsplitters and mode-mixing: entangle or distribute correlations between modes.

Under Gaussian channels, a state's covariance matrix transforms as V → T V T^T + N, where T represents added linear mixing (e.g., loss) and N encodes added noise. These channels model realistic processes like attenuation, amplifier gain with added noise, and coupling to environmental modes. See Gaussian channel; beam splitter.

Entanglement in Gaussian states is often analyzed via the covariance matrix and the PPT (positive partial transpose) criterion, which reduces to a simple condition on the symplectic spectrum of the partially transposed V. A convenient measure of entanglement for Gaussian states is logarithmic negativity. However, it is known that Gaussian operations alone cannot distill Gaussian entanglement, which motivates the use of non-Gaussian resources for certain tasks. See PPT criterion; logarithmic negativity; Gaussian channel; non-Gaussian.

Entanglement, mixedness, and informational aspects

Two-mode Gaussian states can be entangled when their covariances exhibit appropriate correlations. The PPT criterion provides a practical test: after partially transposing the state's covariance matrix, the resulting symplectic eigenvalues must satisfy certain bounds for separability. Entanglement measures derived from these eigenvalues, such as the logarithmic negativity, quantify the strength of nonlocal correlations that can be exploited for tasks like CV quantum teleportation and CV quantum key distribution (CV-QKD). See entanglement; logarithmic negativity; quantum teleportation; continuous-variable quantum key distribution.

Purity of a Gaussian state is determined by det(V); pure Gaussian states satisfy det(V) = (1/2)^{2n} (in ħ = 1 units). Mixedness arises from environmental interactions or intentional thermal noise and is a central consideration in real-world protocols. The theory also connects to quantum state tomography: Gaussian states are uniquely determined by a finite set of measurements of quadratures, often implemented via homodyne detection. See purity (quantum mechanics); homodyne detection.

Applications and practical relevance

Quantum communication: Gaussian states enable continuous-variable quantum key distribution (CV-QKD), where security proofs leverage Gaussian statistics and the Gaussian-optical channel model. See continuous-variable quantum key distribution.
Quantum teleportation and networking: The standard CV teleportation protocol uses Gaussian resources such as two-mode squeezed states and homodyne measurements to transmit coherent states. See quantum teleportation.
Quantum computation and metrology: Measurement-based CV quantum computing often relies on Gaussian cluster states; universality requires non-Gaussian resources or measurements. Gaussian states also underpin high-precision sensing and metrology in certain regimes, with squeezing improving sensitivity beyond the standard quantum limit for specific observables. See continuous-variable quantum information; cluster state; non-Gaussian.
Experimental platforms: Real-world generation of Gaussian states typically employs optical parametric amplifiers and beamsplitters, with state characterization via quantum tomography and Wigner-function reconstruction. See optical parametric amplifier; phase-insensitive amplifier; quantum state tomography.