Entanglement in Quantum Optics

Optical entanglement is a statement about specified modes and degrees of freedom. A source can emit one photon into each of two arms without entangling their polarization, and an entangled total state can have an unentangled polarization reduction if frequency or timing reveals the alternatives. Partial traces and mode decompositions make these distinctions calculable.

Helpful background. Mode decompositions identifies independent optical subsystems; Bell states describes polarization qubits; Gaussian states describes quadrature correlations. Quantum optics develops the preparation and detection methods.

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

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

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.

Projecting the model onto its one-signal-photon/one-idler-photon sector and normalizing gives the pure component

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

Here $\sum_{\mu\nu}|F_{\mu\nu}|^2=1$, and the signal and idler mode families are distinct orthogonal subsystems. 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.

Destructive detection of both photons selects coincidence statistics; it does not leave a usable pair behind. An unobserved mode or classical source fluctuation can also make the conditional component mixed, in which case the pure-state rank test no longer applies. Parametric down-conversion develops source spectra, phase matching, and heralding.

The vacuum and multipair terms matter. 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. For example, take

$$\lvert\Psi\rangle=\frac{\lvert HH\rangle\lvert f\rangle+e^{i\phi}\lvert VV\rangle\lvert g\rangle}{\sqrt2}, \qquad \gamma=\langle g|f\rangle.$$

The unobserved states $f,g$ are normalized. Tracing them out multiplies the $\lvert HH\rangle\langle VV\rvert$ coherence by $e^{-i\phi}\gamma/2$. The reduced polarization concurrence is $|\gamma|$, and its largest overlap with a phase-adjusted Bell state is $(1+|\gamma|)/2$. Identical unobserved states retain a Bell pair; orthogonal unobserved states leave only the separable $HH/VV$ mixture. In both cases each photon’s local polarization state remains $I/2$, so its local entropy alone cannot distinguish them.

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 $1/2$ and the number amplitudes above have positive $(\tanh r)^n$. A minus sign in those amplitudes instead squeezes $q_A+q_B$ and $p_A-p_B$ unless one also rotates a mode phase. Homodyne measurements can reveal these correlations directly in quadrature data.

Entanglement certification requires an inequality or reconstruction procedure appropriate to the state class and noise model. For instance, the calibrated variance sum $\operatorname{Var}(q_A-q_B)+\operatorname{Var}(p_A+p_B)<2$ certifies entanglement even without a Gaussian assumption. Phase drift, loss, excess noise, and uncertainty in the vacuum-noise calibration must be included in an experimental evaluation.

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. For independent pure single-photon wavepackets $f,g$ and an ideal balanced beam splitter, the probability of one photon in each output is

$$P_{\rm coin}=\frac{1-|\langle f|g\rangle|^2}{2}.$$

It ranges from zero for identical packets to $1/2$ for orthogonal packets. This interference is a probe of indistinguishability, not by itself a complete certification of entanglement in a specified output degree of freedom.

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.

References

• 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. 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.
• L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press (1995).
• C. Weedbrook, S. Pirandola, R. García-Patrón, 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. Two different overlap measurements. A polarization source has unobserved-state overlap $|\gamma|=0.6$. Separately, two pure photons entering a balanced beam splitter have wavepacket overlap $0.6$. Find the polarization concurrence and best Bell fidelity in the first experiment, and the coincidence probability in the second.

Solution

The polarization concurrence is $0.6$ and the phase-adjusted Bell fidelity is $(1+0.6)/2=0.8$. The separate Hong–Ou–Mandel experiment gives $P_{\rm coin}=(1-0.6^2)/2=0.32$. These overlaps refer to different state preparations and measurements; a coincidence dip is not numerically the concurrence of an arbitrary polarization state.