Entanglement in Quantum Optics

Quantum optics is one of the cleanest experimental homes of entanglement. Optical modes can be prepared, interfered, delayed, filtered, phase shifted, counted, and measured by homodyne detection with high precision. The same laboratory flexibility also makes the conceptual bookkeeping demanding: entanglement claims must specify the modes, degrees of freedom, and measurement operations.

This page bridges the composite-systems language to quantum-optical examples. The detailed definitions of mode decompositions, two-mode entanglement, squeezed states as entangled modes, Gaussian states, and Bell states live elsewhere in this volume. Full source engineering, nonlinear optics, and detector modeling continue in Quantum Optics, with biphoton source design canonical in Parametric Down-Conversion; security proofs and optical network protocols belong in the quantum-information treatment.

The guiding distinction is:

$$\text{photon pair} \quad\neq\quad \text{entangled pair} \quad\neq\quad \text{usable optical resource}.$$

A source may produce photon pairs without producing high-quality entanglement in the intended degrees of freedom. Loss, distinguishability, spectral correlations, phase drift, detector dark counts, multipair emission, and mode mismatch can all matter.

Optical Modes

An optical mode is not just “a photon.” A complete mode label may include:

• spatial path or transverse profile;
• frequency or temporal wavepacket;
• polarization;
• propagation direction;
• orbital angular momentum or other transverse structure;
• cavity or waveguide mode index.

For a complete mode $\mu$, the creation operator $a_\mu^\dagger$ creates one excitation in that mode:

$$\lvert1_\mu\rangle = a_\mu^\dagger\lvert0\rangle.$$

Two modes $\mu,\nu$ define a mode tensor product when their creation operators refer to independent orthogonal degrees of freedom. Entanglement can then live between path modes, polarization modes, frequency bins, time bins, spatial modes, or collections of such labels.

The phrase “polarization-entangled photons” is therefore shorthand. It usually means that, after controlling or tracing over spatial and spectral degrees of freedom, the polarization state of two selected optical modes is entangled. If hidden spectral or path information reveals which alternative occurred, the reduced polarization entanglement can be degraded.

Photon Pairs from Down-Conversion

Spontaneous parametric down-conversion is a standard source of photon pairs. In a low-gain regime, the output state has the schematic form

The detailed pump, phase-matching, joint-spectrum, polarization/time-bin source, and heralding treatment belongs to Parametric Down-Conversion. Here the pair amplitude is used only to expose the subsystem and entanglement bookkeeping.

$$\lvert\Psi\rangle \approx \lvert0\rangle + \epsilon \sum_{\mu,\nu} f_{\mu\nu}\, a_\mu^\dagger b_\nu^\dagger \lvert0\rangle + O(\epsilon^2).$$

Here $a_\mu^\dagger$ and $b_\nu^\dagger$ create excitations in signal and idler mode families, while $f_{\mu\nu}$ is a joint mode amplitude determined by the pump, phase matching, filtering, and collection optics. Energy and momentum constraints appear schematically as

$$\omega_p \approx \omega_s+\omega_i, \qquad \mathbf k_p \approx \mathbf k_s+\mathbf k_i$$

up to medium, dispersion, and quasi-phase-matching details.

Conditioned on detecting or selecting one pair, the normalized two-photon component is

$$\lvert\psi_2\rangle = \sum_{\mu,\nu} F_{\mu\nu}\, \lvert1_\mu\rangle_s \lvert1_\nu\rangle_i.$$

This pair is entangled across signal and idler if the coefficient matrix $F_{\mu\nu}$ is not factorizable as $u_\mu v_\nu$. The entanglement might be in polarization, frequency, time, transverse momentum, path, or several degrees at once.

The vacuum and multipair terms are not cosmetic. In experiments they affect heralding, coincidence rates, accidental counts, and visibility. A source description should state whether it discusses the full optical state, the postselected one-pair state, or a reduced state after loss and filtering.

Polarization Entanglement

If signal and idler occupy well-defined spatial-spectral modes and each has a two-dimensional polarization degree of freedom, polarization can act as a qubit. A standard polarization-entangled state is

$$\lvert\Phi_\phi\rangle_{\mathrm{pol}} = \frac{1}{\sqrt2} \left( \lvert H\rangle_s\lvert H\rangle_i + e^{i\phi} \lvert V\rangle_s\lvert V\rangle_i \right).$$

For $\phi=0$, this is the plus Bell state in the polarization basis. The one-photon polarization reduced state is maximally mixed:

$$\rho_s = \frac12 \left( \lvert H\rangle\langle H\rvert + \lvert V\rangle\langle V\rvert \right),$$

and the polarization entanglement entropy is one bit.

In a real optical setup, the total state may include frequency, time, path, and transverse-mode labels:

$$\lvert\Psi\rangle \in \mathcal H_{\mathrm{path}} \otimes \mathcal H_{\mathrm{freq}} \otimes \mathcal H_{\mathrm{pol}} \otimes \cdots .$$

If the $H,H$ and $V,V$ alternatives leave distinguishable traces in those other degrees of freedom, tracing them out turns the polarization state into a mixed state and reduces its coherence. Compensation, filtering, interferometric stability, and mode matching are therefore part of the entanglement source, not merely engineering decoration.

Path Entanglement

Path entanglement uses spatial modes as the subsystems. A single-photon path state after a balanced beam splitter can have the form

$$\lvert\psi\rangle = \frac{ \lvert1\rangle_A\lvert0\rangle_B + e^{i\phi} \lvert0\rangle_A\lvert1\rangle_B }{\sqrt2}.$$

This is mode entanglement across the $A\vert B$ path split. It is not entanglement between two photons; there is only one photon. Its operational use depends on available phase references and local operations, but the mode-reduced density operator is mixed:

$$\rho_A = \frac12 \lvert1\rangle\langle1\rvert + \frac12 \lvert0\rangle\langle0\rvert.$$

Two-photon path-entangled states include NOON-like states,

$$\lvert\mathrm{NOON}_2\rangle = \frac{ \lvert2\rangle_A\lvert0\rangle_B + e^{i\phi} \lvert0\rangle_A\lvert2\rangle_B }{\sqrt2}.$$

Such states are sensitive to phase shifts but also sensitive to loss. The word “path” should always refer to well-defined optical modes, not merely informal directions in a diagram.

Two-Mode Squeezing

Parametric interactions also generate continuous-variable entanglement. In a common phase convention, a two-mode squeezed vacuum is

$$\lvert\mathrm{TMSV}(r)\rangle = \frac{1}{\cosh r} \sum_{n=0}^{\infty} (-\tanh r)^n \lvert n\rangle_A \lvert n\rangle_B.$$

This is already a Schmidt decomposition across modes $A$ and $B$. For $r=0$, it is the product vacuum. For $r>0$, the two modes are entangled, and the photon numbers are perfectly correlated in the ideal state.

At low gain,

$$\lvert\mathrm{TMSV}(r)\rangle \approx \lvert0,0\rangle - r\lvert1,1\rangle + O(r^2).$$

This is why weak down-conversion is often described as pair creation. At higher gain, multipair terms are intrinsic rather than rare errors. Whether they are useful or harmful depends on the task and measurement scheme.

The mean photon number in each mode is

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

Finite squeezing gives finite energy and imperfect EPR correlations. The ideal EPR state is a limiting model, not a normalizable laboratory state.

Homodyne Correlations

Balanced homodyne detection measures optical quadratures by interfering a signal with a strong phase-controlled local oscillator. For one mode,

$$q_\theta = \frac{ a e^{-i\theta} + a^\dagger e^{i\theta} }{\sqrt2}.$$

The special choices $\theta=0$ and $\theta=\pi/2$ give quadratures often denoted $q$ and $p$:

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

For the two-mode squeezed vacuum, the EPR-like collective variances are

$$\operatorname{Var}(q_A-q_B) = e^{-2r}, \qquad \operatorname{Var}(p_A+p_B) = e^{-2r}$$

in the convention where each vacuum quadrature has variance $1/2$. Homodyne measurements can therefore reveal continuous-variable correlations directly in quadrature data.

Entanglement certification requires an inequality or reconstruction procedure appropriate to the state class and noise model. Strong correlations in two measured quadratures are suggestive, but loss, excess noise, phase drift, and mixedness must be accounted for.

Interference and Indistinguishability

Linear optics transforms modes. A beam splitter, for example, mixes input creation operators into output creation operators by a unitary transformation. Entanglement can appear or disappear relative to different input-output mode decompositions.

Hong–Ou–Mandel interference is the standard two-photon reminder that indistinguishability matters. Two identical photons entering opposite ports of a balanced beam splitter can bunch into the same output port, suppressing coincidences. This interference is not automatically the same as a useful entangled resource; it is a mode-transformation effect whose interpretation depends on the input state, output modes, and measurement.

The practical lesson is simple: optical entanglement experiments live or die by mode matching. Polarization, path, spectrum, temporal wavepacket, and spatial profile cannot be treated as invisible if they carry which-alternative information.

What This Page Owns

This page owns the conceptual bridge from composite-system entanglement to quantum-optical language:

• optical modes as physical subsystems;
• photon-pair amplitudes and postselected biphoton states;
• polarization entanglement as Bell-state physics in optical modes;
• path entanglement as occupation-mode entanglement;
• two-mode squeezed states as continuous-variable entanglement;
• homodyne correlations as quadrature measurements of entangled modes;
• mode mismatch, loss, and distinguishability as entanglement degraders.

It does not own:

• nonlinear-optical phase-matching theory;
• detailed spontaneous parametric down-conversion source design;
• detector tomography and calibration;
• Bell-test loophole analysis;
• quantum-key-distribution security proofs;
• optical cluster-state construction;
• continuous-variable error correction and fault tolerance.

Those subjects should link here for the subsystem and reduced-state language, while keeping their own canonical technical development.

Common Mistakes

• Saying “the photons are entangled” without naming the degree of freedom and complete modes.
• Treating a photon pair as automatically entangled.
• Ignoring spectral or temporal distinguishability when discussing polarization entanglement.
• Calling a single-photon path state two-particle entanglement.
• Treating finite squeezing as an exact EPR state.
• Forgetting that homodyne correlations require phase conventions and local-oscillator stability.
• Ignoring multipair emission in low-gain pair-source descriptions.
• Treating mode transformations as if they preserve every entanglement claim unchanged.

Cross-Links

• Mode Decompositions
• Two-Mode Entanglement
• Squeezed States as Entangled Modes
• Gaussian States Preview
• EPR State Preview
• Continuous-Variable Systems
• Bell States
• Number States
• Mode Occupations
• Creation and Annihilation Operators
• Field Operators
• Entanglement in Quantum Information
• Formula Sheet

References

• L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press, 1995.
• D. F. Walls and G. J. Milburn, Quantum Optics, 2nd ed., Springer, 2008.
• M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge University Press, 1997.
• S. E. Harris, M. K. Oshman, and R. L. Byer, “Observation of tunable optical parametric fluorescence”, Physical Review Letters 18, 732-734, 1967, doi:10.1103/PhysRevLett.18.732.
• D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in parametric production of optical photon pairs”, Physical Review Letters 25, 84-87, 1970, doi:10.1103/PhysRevLett.25.84.
• C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference”, Physical Review Letters 59, 2044-2046, 1987, doi:10.1103/PhysRevLett.59.2044.
• P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs”, Physical Review Letters 75, 4337-4341, 1995, doi:10.1103/PhysRevLett.75.4337.
• M. D. Reid, “Demonstration of the Einstein-Podolsky-Rosen paradox using nondegenerate parametric amplification”, Physical Review A 40, 913-923, 1989, doi:10.1103/PhysRevA.40.913.
• Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein-Podolsky-Rosen paradox for continuous variables”, Physical Review Letters 68, 3663-3666, 1992, doi:10.1103/PhysRevLett.68.3663.
• S. L. Braunstein and P. van Loock, “Quantum information with continuous variables”, Reviews of Modern Physics 77, 513-577, 2005, doi:10.1103/RevModPhys.77.513.
• 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, doi:10.1103/RevModPhys.84.621.

Exercises

1. Polarization Bell pair. For

$$\lvert\Phi^+\rangle_{\mathrm{pol}} = \frac{ \lvert H\rangle_s\lvert H\rangle_i + \lvert V\rangle_s\lvert V\rangle_i }{\sqrt2},$$

compute the reduced polarization state of the signal photon and its entropy in bits.

Solution

Tracing over the idler polarization gives

$$\rho_s = \frac12 \lvert H\rangle\langle H\rvert + \frac12 \lvert V\rangle\langle V\rvert.$$

The eigenvalues are $1/2$ and $1/2$, so

$$S(\rho_s) = 1$$

bit.

2. Factorable pair amplitude. Suppose a postselected biphoton state has coefficients $F_{\mu\nu}=u_\mu v_\nu$. Show that it is not entangled across signal and idler.

Solution

The state is

$$\lvert\psi_2\rangle = \sum_{\mu,\nu} u_\mu v_\nu \lvert1_\mu\rangle_s \lvert1_\nu\rangle_i.$$

This factors as

$$\left( \sum_\mu u_\mu\lvert1_\mu\rangle_s \right) \otimes \left( \sum_\nu v_\nu\lvert1_\nu\rangle_i \right),$$

so it is a product state across the signal-idler split.

3. Single-photon path entropy. For

$$\lvert\psi\rangle = \frac{ \lvert1\rangle_A\lvert0\rangle_B + \lvert0\rangle_A\lvert1\rangle_B }{\sqrt2},$$

compute the entropy of mode $A$ in bits.

Solution

Tracing out mode $B$ gives

$$\rho_A = \frac12 \lvert1\rangle\langle1\rvert + \frac12 \lvert0\rangle\langle0\rvert.$$

The entropy is therefore

$$S_A=1$$

bit. This is mode entanglement across the path split, not entanglement between two photons.

4. Low-gain pair probability. In the approximation

$$\lvert\mathrm{TMSV}(r)\rangle \approx \lvert0,0\rangle - r\lvert1,1\rangle$$

with $r\ll1$, what is the leading one-pair probability?

Solution

The one-pair amplitude is approximately $-r$, so the leading one-pair probability is

$$P_1 \approx r^2.$$

The omitted normalization and higher-pair terms change this only at higher orders in $r$.

5. Homodyne EPR variance. In the convention used above, what happens to $\operatorname{Var}(q_A-q_B)$ as $r\to\infty$ for the two-mode squeezed vacuum, and why is that limit not a physical normalizable state?

Solution

The variance is

$$\operatorname{Var}(q_A-q_B) = e^{-2r},$$

so it tends to zero as $r\to\infty$. But the mean photon number in each mode is

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

which diverges in the same limit. Exact EPR correlations in this family require infinite energy, so the ideal limit is not a normalizable laboratory state.