Quantum Entanglement: When Two Particles Become One!

Quantum Entanglement Explained

Quantum Entanglement is a phenomenon where two particles become linked such that the quantum properties of one immediately depend on the properties of the other, regardless of the distance separating them.

This phenomenon raises questions about the principles of Einstein's special relativity, which states that nothing can travel faster than light in a vacuum, and thus, it is not possible to send information instantaneously over large distances.

So, how can two particles be in a superposition of entangled states concerning different physical properties (energy, momentum, polarization, spin, etc.), and the measurement of one instantly define the state of the other, regardless of the distance between them?

Regarding the energy of two entangled photons, when a single high-energy photon \( E_0 \) is converted into two lower-energy photons \( E_1 \), \( E_2 \), energy conservation requires that: \( E_0 \)=\( E_1 \)+\( E_2 \). Thus, any measurement of one photon's energy immediately affects the other photon.

Example of Polarization of an Entangled Photon Pair

With a special crystal, it is possible to create two entangled photons.

A single photon (ω_p) enters the crystal and splits into two new entangled photons (ω_s and ω_c). These two entangled photons share a common quantum property: polarization. In other words, they maintain a quantum correlation relationship, and the sum of their respective frequencies corresponds to the frequency of the initial photon (ω_p = ω_s + ω_c).

The system can be described in one of the possible superposition states: \( |HH\rangle \) + \( |HV\rangle \) + \( |VH\rangle \) + \( |VV\rangle \)

If only two states are superimposed: \( |HV\rangle \) + \( |VH\rangle \), then the two particles are in an entangled state.

\[ |\psi\rangle = \frac{1}{\sqrt{2}} \left( |H\rangle_A |V\rangle_B + |V\rangle_A |H\rangle_B \right) \]

• A and B denote the two separated photons.
• \( |H\rangle \) represents a horizontally polarized photon.
• \( |V\rangle \) represents a vertically polarized photon.
• \( |\psi\rangle \) denotes the quantum state of the system (Dirac notation).
• \( |H\rangle_A |V\rangle_B \) represents the polarized states.

What Does This Formula Mean?

This means that the global state of the system cannot be simply described by the state of each particle individually but by a single wave function that includes both particles simultaneously.

In our example, the photons do not have a defined polarization before measurement, but their polarization is correlated: if one photon is measured and found to be horizontally polarized (𝐻), the other will be vertically polarized (𝑉), and vice versa.

The interaction between two entangled particles in the context of quantum entanglement cannot be viewed as "classical communication" in the sense of a signal or information traveling through space at a certain speed.

The properties of each particle are not defined before measurement; they only become "real" at the moment of measurement. The fact that the measurement results are instantaneously correlated, even at a distance, does not violate relativity because no information is actually sent. It is simply a phenomenon of correlation between the states of the two particles, which are interdependent through their shared entangled state.

N.B.: The wave function does not represent an objective physical reality but only our knowledge of the system. The quantum state has no defined properties before measurement.

Collapse of the Wave Function

When an observer measures particle A, the wave function collapses instantaneously, and particle B takes on a correlated state, regardless of the distance between them. This correlation is instantaneous but does not imply information transfer, as the result remains random.

Imagine two observers, Alice and Bob, sharing a pair of entangled particles. Alice measures her particle and gets a result (+1 or -1, for example). Bob, measuring his, will always get a result correlated with Alice's. But Alice cannot choose her result. She cannot encode a message by manipulating the state of her particle. The measurement results are fundamentally random, preventing any control.

N.B.: The collapse of the wave function is instantaneous but does not represent a real physical action, only an update of the observer's information.

Non-Local Correlation

Quantum entanglement is based on a fundamental phenomenon of quantum mechanics, non-local correlation.

Non-local correlation is an experimentally proven phenomenon; it does not allow for exploitable information transmission faster than light. It simply shows that quantum mechanics violates classical intuitions about locality and realism.

Local Variable: Causality and Speed Limits

A local variable is an internal property of a system influenced only by local interactions, i.e., by causes in its immediate vicinity that respect relativistic causality.

• Depends only on interactions in its immediate region.
• Influenced only by signals that respect the speed of light limit (c).
• Respects relativistic causality: no influence can propagate instantaneously.

Non-Local Variable: Beyond the Speed of Light

A non-local variable is a property of a system that can be instantly affected by an event occurring elsewhere, regardless of distance.

• Can be instantly influenced by a distant event.
• Seems to violate the speed of light limit.
• Implies that the state of a particle can change immediately based on a measurement made on another, even if they are separated by light-years.

N.B.: The speed of light plays a fundamental role in distinguishing between local and non-local variables, as it imposes a limit on the propagation of physical influences.