Quantum entanglement

Understanding Quantum Entanglement: A Phenomenon Beyond Imagination

Quantum entanglement is a mind-bending phenomenon where particles interact or share spatial proximity in such a way that their quantum state cannot be described independently, even at large distances. Imagine two dancers performing a dance so intricate and connected that no matter how far apart they are, one move by the first dancer instantly affects the second. That’s what entanglement is like.

The Birth of Entanglement

Albert Einstein introduced a thought experiment about a box emitting a photon in 1931, recognizing the phenomenon that would later be called entanglement. Erwin Schrödinger defined the equations of quantum entanglement in 1932 but set them aside. In 1935, Einstein, Podolsky, and Rosen published their EPR paradox paper, followed by Schrödinger’s discussion of entanglement as a characteristic trait of quantum mechanics.

From Theory to Reality

The concept was later studied by John Archibald Wheeler and Chien-Shiung Wu. Quantum entanglement has been demonstrated with various particles and is being researched for its use in communication and computation. The singlet state described above is the basis for one version of the EPR paradox, where a source emits particles that are sent in opposite directions, creating an entangled pair.

Entanglement: A Resource for Quantum Computing

Quantum systems can become entangled through various interactions, including parametric down-conversion. The mathematical definition of entanglement highlights the maximal knowledge about a system not implying maximal knowledge about individual parts. For some ways in which entanglement may be achieved for experimental purposes, see the section below on methods.

Breaking Entanglement

Entanglement is broken when the entangled particles decohere through interaction with the environment; for example, when a measurement is made. Mathematically, an entangled system can be defined to be one whose quantum state cannot be factored as a product of states of its local constituents; that is to say, they are not individual particles but are an inseparable whole.

Nonlocality and Entanglement

The distance and timing of the measurements can be chosen so as to make the interval between the two measurements spacelike, hence, any causal effect connecting the events would have to travel faster than light. According to the principles of special relativity, it is not possible for any information to travel between two such measuring events.

Experiments and Applications

A number of experiments have shown in practice that Bell’s inequality is not satisfied. When measurements of the entangled particles are made in moving relativistic reference frames, in which each measurement (in its own relativistic time frame) occurs before the other, the measurement results remain correlated. It has been proven mathematically that compatible measurements cannot show Bell-inequality-violating correlations, and thus entanglement is a fundamentally non-classical phenomenon.

Quantum Entanglement in Practice

The text provides a comprehensive overview of quantum mechanics and its applications in physics. Here’s a summary of the main points:

• Hilbert space: A vector space used to describe quantum states.
• Superposition: A particle can exist in multiple states simultaneously.
• Entanglement: Two or more particles become correlated, even when separated by large distances.
• Measurement: The state collapses to one of the possible outcomes upon measurement.
• Quantum operators: Mathematical objects that describe physical quantities such as energy and momentum.
• Reduced density matrices: A way to represent a quantum state on a smaller subsystem, by tracing out the degrees of freedom of the larger system.

Classification Schemes for Entanglement

States can be classified into different categories based on their properties. Local unitary (LU) classes: two states can be transformed into each other by a local unitary operation. Stochastic LOCC (SLOCC) classes: two states can be transformed into each other by local operations including measurements with probability larger than 0. Multi-copy transformation classes: the possibility of transforming an arbitrarily large number of copies of a state into at least one pure entangled state.

Quantum Entanglement in Quantum Information Theory

In quantum information theory, entangled states are considered a ‘resource’, i.e., something costly to produce that allows implementing valuable transformations. Entanglement swapping is a variant of teleportation that allows two parties that have never interacted to share an entangled state.

Applications and Experiments

Quantum states describing systems made of more than two pieces can also be entangled, such as the GHZ state or W state. Tracing out one qubit from the GHZ state turns it into a separable state, while tracing out any of the three qubits in the W state keeps it entangled.

Quantum Entanglement and Time

A fundamental conflict between quantum mechanics and general relativity regarding time. The Wheeler-DeWitt equation predicts the state of the universe is timeless or static, contrary to ordinary experience. Work by Don Page and William Wootters suggests that entanglement can give rise to the illusion of time for observers on the inside.

Conclusion

The integration of quantum mechanics and general relativity faces many problems. The text provides a detailed exploration of how quantum entanglement works, its applications in various fields, and the ongoing research that continues to push the boundaries of our understanding. Entanglement is not just a theoretical concept; it’s a practical tool with real-world implications for communication, computation, and even our perception of time itself.