Continuous-Variable Systems

A continuous-variable system is a quantum system whose useful degrees of freedom are described by continuous spectra or infinite-dimensional Hilbert spaces. Particle position, momentum, oscillator quadratures, optical modes, and field modes are the standard examples.

The word “continuous” does not mean classical. A continuous-variable system is still a quantum system: states are rays or density operators, observables are operators, and composite systems are described by tensor products. The difference is that bases such as $\{\lvert x\rangle\}$ are usually generalized bases, while normalizable states live in spaces such as $L^2(\mathbb R)$ or in Fock spaces with infinitely many possible occupation numbers.

This page is the entry point for continuous-variable composition. The detailed mathematics of $L^2$ spaces belongs to the Mathematical Toolkit, while wavefunction probability densities belong to Wave Mechanics. Here the focus is how continuous-variable degrees of freedom form composite systems and support entanglement.

Continuous-Variable Quantum Computation uses these foundations to define mode registers, finite-energy input families, Gaussian and non-Gaussian computational operations, continuous-output decoding, and model-level resource accounting. This page retains the Hilbert-space, composition, generalized-basis, and normalizability foundations.

Continuous-Variable Platforms uses this structure to compare optical and microwave sources, Gaussian control, non-Gaussian resources, detectors, noise, and scaling. Those engineering questions remain there rather than being duplicated below.

Infinite-Dimensional Tensor Products

For two distinguishable systems, the composition rule is still

$$\mathcal H_{AB} = \mathcal H_A\otimes\mathcal H_B.$$

The new feature is that $\mathcal H_A$ and $\mathcal H_B$ may be infinite-dimensional. For one particle moving on a line,

$$\mathcal H = L^2(\mathbb R).$$

For two distinguishable particles on a line,

$$\mathcal H_{12} = L^2(\mathbb R)\otimes L^2(\mathbb R) \cong L^2(\mathbb R^2).$$

The isomorphism says that a two-particle state can be represented by a square-integrable wavefunction $\Psi(x_1,x_2)$, with normalization

$$\int_{-\infty}^{\infty}dx_1 \int_{-\infty}^{\infty}dx_2\, |\Psi(x_1,x_2)|^2 = 1.$$

This is the continuous-variable version of a coefficient array $\psi_{ij}$ for two finite-dimensional systems. The indices $i,j$ have become continuous coordinates $x_1,x_2$, and sums have become integrals.

For finitely many subsystems, this tensor-product rule is conceptually close to the finite-dimensional case. Infinite tensor products over infinitely many modes, spatial points, or field degrees of freedom are more delicate. In nonrelativistic many-body physics one often works with a finite or countable mode basis and then takes limits carefully. In relativistic QFT, local algebras and operator-valued distributions require additional structure beyond this introductory page.

Two Particles on a Line

A pure product state of two distinguishable particles on a line has the form

$$\Psi(x_1,x_2) = \psi(x_1)\phi(x_2).$$

Here $\psi$ is a normalized state of particle 1 and $\phi$ is a normalized state of particle 2. The probability density factorizes:

$$|\Psi(x_1,x_2)|^2 = |\psi(x_1)|^2|\phi(x_2)|^2.$$

An entangled pure state is a normalized $\Psi(x_1,x_2)$ that cannot be written as one product wavefunction. A common structural form is a Schmidt expansion,

$$\Psi(x_1,x_2) = \sum_n \sqrt{p_n}\, u_n(x_1)v_n(x_2),$$

where $\{u_n\}$ and $\{v_n\}$ are orthonormal sets and $p_n\ge0$ with $\sum_n p_n=1$. The state is product exactly when only one nonzero Schmidt coefficient appears.

The reduced density operator of particle 1 has position-space kernel

$$\rho_1(x,x') = \int_{-\infty}^{\infty}dy\, \Psi(x,y)\Psi^*(x',y).$$

This is the partial trace written in a continuous position representation. The formula is useful, but it should not obscure the operator meaning: $\rho_1$ is the unique operator that reproduces all expectation values of observables acting only on particle 1.

For a product state $\Psi(x,y)=\psi(x)\phi(y)$, the reduced kernel becomes

$$\rho_1(x,x') = \psi(x)\psi^*(x') \int_{-\infty}^{\infty}dy\,|\phi(y)|^2 = \psi(x)\psi^*(x'),$$

so particle 1 remains pure. For an entangled state, $\rho_1$ is mixed, and the subsystem entropy can be nonzero.

The detailed position-space treatment, including phase correlations, reduced kernels, and center-of-mass coordinates, is developed in Position-Space Two-Particle States.

Two Modes of an Oscillator Field

Continuous-variable systems are not limited to particle positions. A harmonic oscillator mode is also an infinite-dimensional quantum system. Its Hilbert space is spanned by number states,

$$\mathcal H_{\rm mode} = \operatorname{span} \{\lvert0\rangle,\lvert1\rangle,\lvert2\rangle,\ldots\}.$$

For two modes $A$ and $B$,

$$\mathcal H_{AB} = \mathcal H_A\otimes\mathcal H_B, \qquad \lvert n_A,n_B\rangle = \lvert n_A\rangle_A\otimes\lvert n_B\rangle_B.$$

The state

$$\lvert\Psi\rangle = \frac{1}{\sqrt2} \left( \lvert1_A,0_B\rangle + \lvert0_A,1_B\rangle \right)$$

is entangled with respect to the mode decomposition into subsystem $A$ and subsystem $B$. It is not a product state of mode $A$ and mode $B$. This example is finite in excitation number, but each mode Hilbert space remains infinite-dimensional.

Mode entanglement is decomposition-dependent. If one changes the mode basis, for example by a beam-splitter-like transformation, a state that looks entangled in one mode decomposition may look simple in another. This is not a contradiction; it is the continuous-variable version of the general rule that entanglement is defined relative to a specified tensor-product structure.

Oscillator modes also have quadrature operators. In dimensionless units,

$$q = \frac{a+a^\dagger}{\sqrt2}, \qquad p = \frac{a-a^\dagger}{i\sqrt2},$$

with

$$[q,p]=i.$$

Quadratures behave mathematically like position and momentum variables for a mode. They are the natural language of quantum optics, continuous-variable quantum information, and Gaussian-state methods.

Normalizability Caveats

Many useful continuous-variable expressions are distributions, not normalizable states. Position eigenkets obey

$$\langle x\vert x'\rangle = \delta(x-x'),$$

so $\lvert x\rangle$ is not a vector in $L^2(\mathbb R)$. Momentum eigenkets and plane waves have the same status. They are generalized eigenstates used inside integrals, not physical normalizable states by themselves.

The same warning applies to idealized EPR-like expressions such as

$$\int_{-\infty}^{\infty}dx\, \lvert x\rangle_1\lvert x\rangle_2.$$

This expression captures perfect position correlation, but it is not normalizable. Its formal norm contains an infinite volume factor. Physical states must use wavepackets or Gaussian approximations with finite variance.

The EPR State Preview develops this warning into the standard pair of commuting collective observables, $X_1-X_2$ and $P_1+P_2$.

Normalizability matters because entanglement measures, reduced density operators, and expectation values can fail to be well-defined for ideal distributions. A delta-correlated expression may be an excellent limiting model, but the limit should not be confused with an ordinary Hilbert-space vector.

Gaussian-State Preview

Gaussian states are the most important controlled family of continuous-variable states. They are states whose Wigner functions or characteristic functions are Gaussian in phase-space variables. For $N$ modes, collect the quadratures into

$$R = (q_1,p_1,\ldots,q_N,p_N)^T.$$

A Gaussian state is determined by its first moments

$$d_i = \langle R_i\rangle$$

and covariance matrix

$$V_{ij} = \frac12 \langle \Delta R_i\Delta R_j+\Delta R_j\Delta R_i \rangle, \qquad \Delta R_i=R_i-\langle R_i\rangle.$$

This finite set of moments can encode an infinite-dimensional state because Gaussianity is a strong restriction. Coherent states, squeezed states, thermal oscillator states, and two-mode squeezed states are central examples.

The preview should be read with two cautions:

• not every continuous-variable state is Gaussian;
• Gaussian entanglement has efficient covariance-matrix criteria, but those criteria do not replace the general definition of entanglement.

The Gaussian States Preview develops this language more carefully. For now, the key point is that continuous-variable entanglement often becomes tractable when the state is Gaussian and the relevant subsystems are modes.

Physical Interpretation

Continuous-variable composition appears in several guises:

• two or more particles with spatial wavefunctions;
• center-of-mass and relative coordinates for composite motion;
• optical modes, microwave modes, phonon modes, and trap modes;
• collective variables in many-body systems;
• field modes that prepare the transition to QFT.

The conceptual question is always the same: what are the subsystems? For particles, the tensor factors may be particle Hilbert spaces. For fields and optics, they may be modes. For spatial regions, the factorization question becomes more subtle, especially in QFT and gauge theories.

Common Mistakes

• Treating $\lvert x\rangle$ and $\lvert p\rangle$ as normalizable states instead of generalized eigenstates.
• Forgetting the integration measure when normalizing wavefunctions or tracing out a coordinate.
• Assuming every two-variable wavefunction $\Psi(x_1,x_2)$ is entangled; product wavefunctions are not.
• Confusing particle entanglement with mode entanglement.
• Changing the mode basis without reconsidering the tensor-product structure.
• Applying finite-dimensional entropy intuition without checking convergence and trace-class conditions.
• Treating ideal EPR correlations as physical states rather than limiting models.

Cross-Links

• Tensor Products of Hilbert Spaces
• Product States
• Entangled States
• Entanglement Depends on a Decomposition
• Partial Trace
• Subsystem Entropy
• Mode Decompositions
• Two-Mode Entanglement
• EPR State Preview
• Gaussian States Preview
• Squeezed States as Entangled Modes
• Position-Space Two-Particle States
• Mode Occupations
• Mode Expansions
• Field Operators
• Entanglement in Quantum Optics
• Formula Sheet
• L2 Spaces
• Position and Momentum Representations
• Wavefunctions and Probability Density
• Gaussian Wave Packets

References

• L. E. Ballentine, Quantum Mechanics: A Modern Development, 2nd ed., World Scientific, 2014.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013.
• S. L. Braunstein and P. van Loock, “Quantum information with continuous variables”, Reviews of Modern Physics 77, 513-577, 2005.
• C. Weedbrook, S. Pirandola, R. Garcia-Patron, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information”, Reviews of Modern Physics 84, 621-669, 2012.
• A. Serafini, Quantum Continuous Variables: A Primer of Theoretical Methods, CRC Press, 2017.
• D. F. Walls and G. J. Milburn, Quantum Optics, 2nd ed., Springer, 2008.

Exercises

1. Product wavefunction. Let $\Psi(x_1,x_2)=\psi(x_1)\phi(x_2)$ with $\psi$ and $\phi$ normalized. Show that $\Psi$ is normalized.

Solution

Compute

$$\int dx_1\,dx_2\, |\Psi(x_1,x_2)|^2 = \int dx_1\,|\psi(x_1)|^2 \int dx_2\,|\phi(x_2)|^2.$$

Each factor equals $1$, so the product equals $1$.

2. Reduced kernel for a product state. For $\Psi(x,y)=\psi(x)\phi(y)$, compute $\rho_1(x,x')$.

Solution

By definition,

$$\rho_1(x,x') = \int dy\, \Psi(x,y)\Psi^*(x',y).$$

Substituting the product form gives

$$\rho_1(x,x') = \psi(x)\psi^*(x') \int dy\,|\phi(y)|^2 = \psi(x)\psi^*(x'),$$

because $\phi$ is normalized.

3. Mode entanglement. For

$$\lvert\Psi\rangle = \frac{1}{\sqrt2} \left( \lvert1_A,0_B\rangle + \lvert0_A,1_B\rangle \right),$$

trace out mode $B$ and find the reduced state of mode $A$.

Solution

The density operator is $\rho=\lvert\Psi\rangle\langle\Psi\rvert$. Orthogonality of $\lvert0_B\rangle$ and $\lvert1_B\rangle$ removes the cross terms under the trace over $B$, giving

$$\rho_A = \frac12 \lvert1_A\rangle\langle1_A\rvert + \frac12 \lvert0_A\rangle\langle0_A\rvert.$$

The reduced state is mixed, so the two modes are entangled.

4. Ideal EPR normalization. Explain why $\int dx\,\lvert x\rangle_1\lvert x\rangle_2$ is not normalizable.

Solution

Regulate the expression by putting the system in a box of length $L$ and considering

$$\int_{-L/2}^{L/2}dx\, \lvert x\rangle_1\lvert x\rangle_2.$$

Its formal norm contains

$$\int dx\,dx'\, \delta(x-x'), = L,$$

which diverges as $L\to\infty$. Equivalently, the state has perfect position correlation over an infinite range with no normalizable envelope. Physical approximations replace the ideal expression by wavepackets with finite widths and finite total norm.

5. Quadrature commutator. Using $[a,a^\dagger]=1$, show that $q=(a+a^\dagger)/\sqrt2$ and $p=(a-a^\dagger)/(i\sqrt2)$ obey $[q,p]=i$.

Solution

Compute

$$\begin{aligned} [q,p] &= \frac{1}{2i} [a+a^\dagger,a-a^\dagger] \\ &= \frac{1}{2i} \left( -[a,a^\dagger]+[a^\dagger,a] \right) \\ &= \frac{1}{2i} \left( -1-1 \right) = i. \end{aligned}$$