Continuous Variables and Modes

Continuous-variable systems have observables with continuous spectra or modes with unbounded occupation. Their Hilbert spaces are typically infinite-dimensional, but the organizing questions remain the same as for qubits: what is the subsystem decomposition, which states factorize, what information survives a partial trace, and which correlations are genuinely quantum?

The extra care comes from three sources:

• position and momentum eigenvectors are generalized states rather than normalizable vectors;
• different particle, mode, and spatial-region decompositions answer different physical questions;
• covariance matrices completely characterize Gaussian states but not arbitrary non-Gaussian states.

This chapter connects wave mechanics, Fock space, and entanglement without making quantum optics or continuous-variable quantum information duplicate canonical homes.

Chapter Map

Task	Canonical page	Main output
identify the infinite-dimensional setting	Continuous-Variable Systems	tensor products, normalizability, modes, and Gaussian preview
decide which modes define subsystems	Mode Decompositions	spatial, momentum, frequency, polarization, and wavepacket modes
analyze elementary mode entanglement	Two-Mode Entanglement	single-excitation, number-correlated, squeezed, and beam-splitter states
interpret ideal EPR correlations safely	EPR State Preview	commuting collective observables and normalizable approximations
use first and second moments	Gaussian States Preview	quadratures, covariance matrices, Wigner functions, and Gaussian criteria
connect squeezing to entanglement	Squeezed States as Entangled Modes	two-mode squeezing, reduced thermal states, and EPR-like noise suppression
translate wave mechanics into composite-state language	Position-Space Two-Particle States	product kernels, entangled wavefunctions, reductions, and exchange symmetry

What Makes a System Continuous-Variable?

A canonical degree of freedom has operators $q$ and $p$ satisfying

$$[q,p]=i\hbar I.$$

Neither operator has a finite spectrum, and the associated Hilbert space cannot be finite-dimensional. The standard position representation is

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

whose vectors are square-integrable wavefunctions modulo equality almost everywhere. The position kets $\lvert x\rangle$ are useful generalized eigenvectors, but

$$\langle x|x'\rangle = \delta(x-x')$$

shows that they are not normalized Hilbert-space vectors.

An oscillator mode is also continuous-variable even though its number basis is discrete:

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

The unbounded occupation and continuous quadrature spectra, not the notation used for the basis, make it a continuous-variable system.

Two Common Composite Structures

Particle-coordinate factorization

For two distinguishable particles on a line,

$$\begin{aligned} \mathcal H_{12} &= L^2(\mathbb R,dx_1) \otimes L^2(\mathbb R,dx_2) \\ &\cong L^2(\mathbb R^2,dx_1dx_2). \end{aligned}$$

A pure state is represented by $\Psi(x_1,x_2)$. It is normalized when

$$\int_{\mathbb R^2} dx_1dx_2\, \lvert\Psi(x_1,x_2)\rvert^2 =1.$$

It is a product state exactly when, up to sets of measure zero,

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

For identical particles, the formal coordinate slots do not automatically define operational subsystems. The wavefunction must also obey the appropriate exchange symmetry, and mode or region partitions may be more physical.

Mode factorization

Suppose the one-particle space decomposes into orthogonal mode subspaces,

$$\mathcal h = \mathcal h_A\oplus\mathcal h_B.$$

Bosonic or fermionic Fock space then factorizes, after an ordering convention is fixed in the fermionic case, as

$$\mathcal F_{\pm}(\mathcal h) \cong \mathcal F_{\pm}(\mathcal h_A) \otimes \mathcal F_{\pm}(\mathcal h_B).$$

This factorization makes modes $A$ and $B$ candidate subsystems. A state can therefore be separable in one mode decomposition and entangled in another. That is not a contradiction: entanglement is a property of a state together with a specified tensor-product structure and observable access.

Reduced States in Position Space

For a normalized two-particle wavefunction, the pure-state density kernel is

$$\rho(x_1,x_2;x_1',x_2') = \Psi(x_1,x_2) \Psi^*(x_1',x_2').$$

Tracing over particle or coordinate slot 2 gives

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

The kernel defines an operator through

$$(\rho_1 f)(x) = \int_{\mathbb R}dx'\, \rho_1(x,x')f(x').$$

For a product wavefunction, $\rho_1$ has rank one. For a pure entangled state it is mixed. The Position-Space Two-Particle States page owns the full wavefunction analysis and the cautions associated with center-of-mass coordinates and identical-particle symmetry.

Modes Are Chosen, Not Discovered Once and for All

Let $a_i$ annihilate an excitation in orthonormal mode $i$. A unitary one-particle basis change defines

$$b_\alpha = \sum_i U_{\alpha i}^*a_i, \qquad b_\alpha^\dagger = \sum_i U_{\alpha i}a_i^\dagger.$$

For bosons,

$$[b_\alpha,b_\beta^\dagger] = \delta_{\alpha\beta},$$

and the analogous fermionic anticommutator is preserved. Spatial, frequency, momentum, polarization, and wavepacket modes are therefore different bases of the same one-particle space when the transformation is unitary and complete.

The physically useful decomposition depends on preparation, dynamics, and measurement. A beam splitter changes the relation between input and output modes; a detector with finite bandwidth selects wavepacket modes; a lattice model privileges local orbitals. Mode Decompositions owns these choices and their operational caveats.

Quadrature Language

For one bosonic mode, set $\hbar=1$ and define

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

Then $[q,p]=i$. For $m$ modes, collect the quadratures into

$$R = (q_1,p_1,\ldots,q_m,p_m)^{\mathsf T}.$$

Their commutators are encoded by the symplectic form:

$$[R_j,R_k] = i\Omega_{jk}, \qquad \Omega = \bigoplus_{r=1}^{m} \begin{pmatrix} 0&1\\ -1&0 \end{pmatrix}.$$

For a state $\rho$, define the displacement vector and covariance matrix by

$$d_j = \operatorname{Tr}(\rho R_j),$$

and

$$\Gamma_{jk} = \frac12 \operatorname{Tr} \left[ \rho\{\Delta R_j,\Delta R_k\} \right],$$

where $\Delta R_j=R_j-d_j$. Every physical covariance matrix obeys

$$\Gamma + \frac{i}{2}\Omega \succeq 0.$$

With these conventions the vacuum covariance is $I/2$. Convention changes are common, so factors of two must be checked before importing an uncertainty or entanglement criterion.

What Gaussian Means

A Gaussian state has a Gaussian characteristic function or, equivalently, a Gaussian Wigner function. It is completely determined by $d$ and $\Gamma$. This turns many infinite-dimensional questions into finite-dimensional linear algebra.

For non-Gaussian states, the same moments remain useful but are not complete. Distinct states can share the same first and second moments, and a covariance-based test may fail to detect entanglement that is visible in higher moments or in the full density operator.

The Gaussian States Preview page develops purity, symplectic eigenvalues, Wigner functions, and the Gaussian partial-transpose criterion. This chapter uses only the structural lesson:

$$\text{Gaussian state} \Longrightarrow (d,\Gamma)\text{ is complete},$$

whereas

$$(d,\Gamma) \nRightarrow \rho.$$

Two-Mode Entanglement

One excitation shared by two modes

The state

$$\begin{aligned} \lvert\Psi_1\rangle &= \alpha\lvert1,0\rangle + \beta\lvert0,1\rangle, \\ \lvert\alpha\rvert^2+\lvert\beta\rvert^2 &=1. \end{aligned}$$

is entangled across the mode split when both coefficients are nonzero. Tracing out mode $B$ gives

$$\rho_A = \lvert\alpha\rvert^2\lvert1\rangle\langle1\rvert + \lvert\beta\rvert^2\lvert0\rangle\langle0\rvert.$$

This is mode entanglement, not entanglement between two persistent particle identities. Its operational use can depend on phase references, local number constraints, and which mode operations are available.

Two-mode squeezed vacuum

A central Gaussian example is

$$\begin{aligned} \lvert\mathrm{TMSV}(r)\rangle &= \sqrt{1-\lambda^2} \\ &\quad\times \sum_{n=0}^{\infty} \lambda^n\lvert n,n\rangle, \\ \lambda &=\tanh r. \end{aligned}$$

This is already a Schmidt decomposition. Either mode alone has a geometric number distribution,

$$\rho_A = (1-\lambda^2) \sum_{n=0}^{\infty} \lambda^{2n} \lvert n\rangle\langle n\rvert,$$

with mean occupation

$$\bar n = \sinh^2 r.$$

The reduced state is thermal in form, even though the joint two-mode state is pure. Its entanglement entropy is

$$S = (\bar n+1)\ln(\bar n+1) - \bar n\ln\bar n.$$

The Squeezed States as Entangled Modes page owns the squeezing transformation and its quantum-optical interpretation.

EPR Correlations and Their Limit

For a suitable two-mode squeezing phase, the collective quadratures obey

$$\begin{aligned} \operatorname{Var}(q_1-q_2) &=e^{-2r}, \\ \operatorname{Var}(p_1+p_2) &=e^{-2r}. \end{aligned}$$

The two collective observables commute:

$$[q_1-q_2,p_1+p_2]=0.$$

As $r$ increases, their fluctuations become small while conjugate combinations become noisy. The formal $r\to\infty$ limit has perfect EPR correlations, infinite energy, and no normalizable state vector. Physical experiments and calculations use finite-squeezing states or other normalizable wavepackets.

The EPR State Preview owns this limiting construction. The historical EPR argument and Bell inequalities have separate canonical homes in foundations; strong quadrature correlation alone should not be called a Bell-inequality violation.

Choosing an Entanglement Diagnostic

State and task	Appropriate first tool	Limitation to record
pure bipartite vector	Schmidt decomposition or reduced-state entropy	spectra may be infinite and entropy may diverge
mixed finite-mode state	separability criteria, witnesses, or negativity	no single scalar captures all operational resources
Gaussian bipartite state	covariance matrix and Gaussian partial transpose	convention and bipartition must be fixed
non-Gaussian state	full density operator, characteristic function, or tailored witness	second moments are generally incomplete
field or spatial-region problem	regulated algebra or mode decomposition	continuum factorization and ultraviolet limits require care

No diagnostic should be applied before the subsystem decomposition is stated. For infinite-dimensional systems, domains, convergence, energy constraints, and regularization may be part of the physical question rather than technical afterthoughts.

Analysis Workflow

1. Specify the physical degrees of freedom

State whether the objects are particles, oscillator modes, field modes, spatial regions, or effective collective coordinates.

2. Declare the tensor-product structure

Write the factorization or direct-sum decomposition explicitly. For identical particles, distinguish formal particle slots from accessible modes or regions.

3. Fix conventions

Record the value of $\hbar$, quadrature normalization, Fourier-transform convention, covariance ordering, and mode basis.

4. Check state legitimacy

Verify normalization, trace class, finite moments required by the calculation, and any energy or ultraviolet cutoff. Treat delta-correlated states as idealizations.

5. Choose a representation

Use wavefunction kernels for coordinate questions, number states for occupation structure, and phase-space or covariance methods for Gaussian states.

6. Reduce or transform consistently

Trace over the stated subsystem, or transform both states and observables when changing mode bases. Do not change the factorization silently.

7. Match the diagnostic to the state class

Use pure-state spectra, Gaussian criteria, or non-Gaussian witnesses only within their stated assumptions.

8. Interpret operationally

Identify the measurements, reference frames, and local controls that make the proposed subsystem correlations accessible.

Canonical Boundaries

This chapter owns the composite-system structure shared by continuous coordinates and modes. Detailed homes remain elsewhere:

• Fock Space and Occupation Number owns sector structure and number states.
• Creation, Annihilation, and Second Quantization owns mode operators and many-particle Hamiltonians.
• Bipartite Entanglement owns Schmidt decomposition, entropy, negativity, and witnesses in general form.
• Entanglement in Quantum Optics owns the bridge to optical preparation and measurement.
• Entanglement in QFT Preview owns the cautious bridge to vacuum and spatial-region entanglement.

Common Mistakes

• Calling every infinite-dimensional system continuous-variable. State which observables or unbounded mode occupations supply the continuous-variable structure.
• Treating position eigenstates as normalized vectors. They are distributions used inside a rigged-Hilbert-space framework.
• Assuming a mode decomposition is unique. Modes depend on basis, apparatus, and accessible observable algebra.
• Confusing particle entanglement with mode entanglement. Name the tensor factors explicitly.
• Ignoring exchange symmetry. A two-coordinate wavefunction for identical particles must lie in the physical symmetry sector.
• Using covariance data as a complete state description without Gaussianity. Higher moments can contain essential information.
• Mixing quadrature conventions. Vacuum variances and uncertainty inequalities change by factors of two.
• Calling the ideal EPR state physical. Perfect correlations require a non-normalizable, infinite-energy limit.
• Inferring Bell nonlocality from entanglement or squeezing alone. Bell violation requires a specified inequality and measurement scenario.
• Letting a basis change silently redefine locality. Transformations that mix the chosen subsystems can change the entanglement being discussed.

Reading Paths

Wave-mechanics route: Continuous-Variable Systems → Position-Space Two-Particle States → Reduced States and Partial Trace.

Mode route: Mode Decompositions → Fock Space and Occupation Number → Two-Mode Entanglement.

Gaussian route: Gaussian States Preview → Squeezed States as Entangled Modes → EPR State Preview.

Field-theory bridge: Mode Expansions → Field Operators → Entanglement in QFT Preview.

Across-fields bridge: Entanglement Across Fields → Entanglement in Quantum Optics or Entanglement in QFT Preview.

Practice bridge: Reference, Problems, and Notebooks → Entanglement Diagnostic Table → Computational Notebooks.

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.
• D. F. Walls and G. J. Milburn, Quantum Optics, 2nd ed., Springer, 2008.
• S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Reviews of Modern Physics 77, 513–577, 2005.
• C. Weedbrook et al., “Gaussian quantum information,” Reviews of Modern Physics 84, 621–669, 2012.
• A. Serafini, Quantum Continuous Variables: A Primer of Theoretical Methods, CRC Press, 2017.
• R. Simon, “Peres–Horodecki separability criterion for continuous variable systems,” Physical Review Letters 84, 2726–2729, 2000.
• S. D. Bartlett, T. Rudolph, and R. W. Spekkens, “Reference frames, superselection rules, and quantum information,” Reviews of Modern Physics 79, 555–609, 2007.

Exercises

Exercise 1: Product kernel and purity

Let $\Psi(x,y)=\psi(x)\phi(y)$, where both factors are normalized. Compute the reduced kernel $\rho_1(x,x')$ and show that the reduced state is pure.

Solution

Substitution into the partial trace gives

$$\begin{aligned} \rho_1(x,x') &= \int dy\, \psi(x)\phi(y) \psi^*(x')\phi^*(y) \\ &= \psi(x)\psi^*(x'), \end{aligned}$$

because $\int dy\,\lvert\phi(y)\rvert^2=1$. Thus $\rho_1=\lvert\psi\rangle\langle\psi\rvert$, so

$$\rho_1^2=\rho_1, \qquad \operatorname{Tr}(\rho_1^2)=1.$$

Exercise 2: Mode algebra under a basis change

Let $b_\alpha=\sum_iU_{\alpha i}^*a_i$, where $U$ is unitary and $[a_i,a_j^\dagger]=\delta_{ij}$. Verify the canonical commutator for the $b$ modes.

Solution

Using linearity,

$$\begin{aligned} [b_\alpha,b_\beta^\dagger] &= \sum_{i,j} U_{\alpha i}^*U_{\beta j} [a_i,a_j^\dagger] \\ &= \sum_i U_{\alpha i}^*U_{\beta i} \\ &= \delta_{\alpha\beta}. \end{aligned}$$

The last equality is unitarity. A unitary one-particle basis change therefore preserves the bosonic mode algebra.

Exercise 3: Reduced state of one shared excitation

For $\lvert\Psi_1\rangle=\alpha\lvert1,0\rangle+\beta\lvert0,1\rangle$, find the eigenvalues of $\rho_A$ and state when the mode state is entangled.

Solution

The two terms contain orthogonal states of mode $B$, so their cross terms vanish under the partial trace:

$$\rho_A = \lvert\alpha\rvert^2 \lvert1\rangle\langle1\rvert + \lvert\beta\rvert^2 \lvert0\rangle\langle0\rvert.$$

Its nonzero eigenvalues are $\lvert\alpha\rvert^2$ and $\lvert\beta\rvert^2$. The pure joint state is entangled across the $A|B$ mode split exactly when both eigenvalues are nonzero.

Exercise 4: Vacuum covariance and uncertainty

With $q=(a+a^\dagger)/\sqrt2$ and $p=(a-a^\dagger)/(i\sqrt2)$, show that the vacuum covariance is $I_2/2$ and saturates the uncertainty relation.

Solution

In the vacuum, $\langle q\rangle=\langle p\rangle=0$. Using $a\lvert0\rangle=0$ and $[a,a^\dagger]=1$ gives

$$\langle q^2\rangle = \langle p^2\rangle = \frac12,$$

while the symmetrized cross covariance vanishes. Hence

$$\Gamma_0 = \frac12 \begin{pmatrix} 1&0\\ 0&1 \end{pmatrix}.$$

Therefore

$$\operatorname{Var}(q) \operatorname{Var}(p) = \frac14,$$

which saturates the Robertson uncertainty bound for $[q,p]=i$.

Exercise 5: Marginal of a two-mode squeezed state

Starting from the Schmidt coefficients of $\lvert\mathrm{TMSV}(r)\rangle$, verify normalization and compute the mean occupation of either reduced mode.

Solution

With $\lambda=\tanh r$, normalization follows from the geometric series:

$$(1-\lambda^2) \sum_{n=0}^{\infty}\lambda^{2n} =1.$$

The reduced probabilities are $p_n=(1-\lambda^2)\lambda^{2n}$. Thus

$$\begin{aligned} \bar n &= (1-\lambda^2) \sum_{n=0}^{\infty} n\lambda^{2n} \\ &= \frac{\lambda^2}{1-\lambda^2} \\ &= \sinh^2 r. \end{aligned}$$

The geometric marginal is thermal in form even though the joint state is pure.