Do Individual Photons Gain Entropy as They Travel?

A photon does not carry a fixed amount of entropy just because it is moving. In quantum physics, entropy describes a state—often the state of a system such as a photon, a group of photons, or radiation in a volume. Whether that entropy changes depends on how the state is prepared, what you count as the system, and whether the photon interacts with anything on its way.

For an ideal photon in a pure state traveling through empty space, its von Neumann entropy does not increase. Entropy assigned to the photon alone can change if it becomes mixed or entangled with its surroundings.

What “entropy of a photon” means

For a quantum state represented by a density operator $\rho$, the von Neumann entropy is

$$S(\rho)=-k_{\mathrm B}\operatorname{Tr}(\rho\ln\rho),$$

where $k_{\mathrm B}$ is Boltzmann’s constant. A pure state has one definite quantum state description, represented by a density operator with one eigenvalue equal to 1 and the others 0. Its von Neumann entropy is zero. A mixed state has multiple nonzero eigenvalues and positive entropy.

This is not the same as saying a pure photon has no uncertainty in every sense. A photon can have a spread-out wave packet, and measurements of its position or momentum can have uncertain outcomes. That measurement uncertainty is not, by itself, a positive von Neumann entropy for a pure state.

What happens during ideal travel?

In empty space, a photon’s state evolves according to quantum mechanics. For an isolated system, this evolution is unitary. Unitary evolution can change the state’s phase and its distribution over positions or frequencies, but it preserves the state’s von Neumann entropy. In particular, a pure state remains pure.

So a wave packet spreading as it propagates does not automatically mean that entropy is being produced. The photon’s position may become less localized, but the full quantum state still contains the information needed to describe that spread.

A worked example: polarization

Suppose a photon is prepared with horizontal polarization, $|H\rangle$. Its polarization state is pure, so its polarization entropy is zero. If it travels through ideal empty space without interacting with anything, that remains true.

Now compare this with a source that prepares horizontal and vertical polarization at random, with equal probabilities. The state is a mixture:

$$\rho=\tfrac12|H\rangle\langle H|+\tfrac12|V\rangle\langle V|.$$

Its two nonzero eigenvalues are each $1/2$, giving

$$S=-k_{\mathrm B}\left(\tfrac12\ln\tfrac12+\tfrac12\ln\tfrac12\right)=k_{\mathrm B}\ln 2.$$

Ideal travel through empty space does not change this entropy either. The contrast is in the preparation: the first photon has a pure polarization state, while the second is described by a mixed polarization state. Being “one photon” does not specify its entropy.

A coherent superposition, such as $(|H\rangle+|V\rangle)/\sqrt{2}$, is also pure, and therefore has zero von Neumann entropy. It is different from the random mixture above, even though a measurement of polarization gives the same probabilities for the two outcomes.

When entropy assigned to a photon can change

A photon can interact with matter, a detector, or other radiation. Such interactions can correlate its properties with its surroundings. For example, if a photon’s polarization becomes entangled with a detector’s state, the combined photon-plus-detector state may still be pure, while the photon considered by itself is mixed. Its reduced entropy can then be positive.

This does not mean the total entropy of the isolated combined system necessarily rose. Under unitary evolution, the total von Neumann entropy is conserved. Entropy assigned to one part can change as correlations form; the entropy of the other part and the correlations matter too. If you ignore environmental details or use a coarse-grained thermodynamic description, entropy can appear to increase because information about those details is no longer tracked.

Absorption is another case: after a photon is absorbed, it is no longer a freely traveling photon. Its energy is transferred to matter, and the entropy of the larger system depends on the interaction and subsequent evolution.

Common mistakes

• Treating entropy as a property of motion: Travel alone does not determine entropy. The quantum state and system boundary do.
• Equating uncertainty with entropy: A pure wave packet can have uncertain measurement outcomes while having zero von Neumann entropy.
• Confusing a mixture with a superposition: Randomly prepared horizontal or vertical polarization is mixed; a coherent superposition of those polarizations is pure.
• Assuming local entropy tells the whole story: A photon’s reduced state can become mixed through entanglement even while the joint system remains pure.
• Assuming energy loss is entropy production: A photon’s energy may change in an interaction or due to a changing background, but energy change alone does not establish that entropy increased.

The useful question is therefore not whether a photon gains entropy simply by traveling, but whether the photon’s state becomes mixed, whether it interacts with other systems, and which parts of the overall system are included in the entropy calculation.