Quantum Entanglement · aiwiki.page

Quantum entanglement is a property of quantum mechanics in which the state of a composite system cannot be represented as a statistical mixture of separate states of its components. It produces correlations distinct from those obtainable through shared classical randomness. Entangled systems may remain correlated when separated, but entanglement alone cannot transmit a controllable message faster than light. It is both a subject of foundational physics and a resource for quantum information processing. (arxiv.org)

Mathematical description

The state space of two quantum systems is the tensor product of their individual Hilbert spaces. A pure joint state is unentangled if it can be written as

|\psi\rangle_{AB}=|a\rangle_A\otimes|b\rangle_B.

If no such factorization exists, the state is entangled. Thus, a joint wave function can specify a definite pure state of the whole without assigning a definite pure state to either component. This definition depends on the chosen division into subsystems. (qiskit.qotlabs.org)

For mixed states, represented by a density matrix, the definition is broader. A state is separable if

\rho_{AB}=\sum_k p_k\,\rho_A^{(k)}\otimes\rho_B^{(k)}, \qquad p_k\geq0,\quad \sum_kp_k=1.

Here the weights form a probability distribution. Such a state may contain classical correlations. Entanglement means that no decomposition of this form exists; merely failing to equal a single product is insufficient for mixed states. (arxiv.org)

A two-qubit example

A standard example involves two qubits, quantum systems with two basis states, denoted $|0\rangle$ and $|1\rangle$ . One of the four Bell states is

|\Phi^+\rangle=\frac{|00\rangle+|11\rangle}{\sqrt2}.

This is a coherent superposition, not a random choice between two pre-existing alternatives. Measuring both qubits in this basis gives either $00$ or $11$ , each with probability one-half. The individual results are random, while the results agree perfectly. (quantum.cloud.ibm.com)

Perfect agreement alone does not establish entanglement: two classical bits prepared with shared randomness can also agree. The distinguishing feature is the joint state's behavior under other measurements. For example, the Bell state also gives matching outcomes in the basis $|+\rangle=(|0\rangle+|1\rangle)/\sqrt2$ , $|-\rangle=(|0\rangle-|1\rangle)/\sqrt2$ ; an equal incoherent mixture of $00$ and $11$ does not. (quantum.cloud.ibm.com)

Ignoring either qubit leaves the other in the reduced state $I/2$ , obtained through the partial trace. Consequently, the complete pure state contains information that neither local state supplies. The individual reduced states also do not uniquely determine the joint state: different Bell states have identical local density matrices. (quantum.cloud.ibm.com)

Historical development and experimental tests

In 1935, Albert Einstein, Boris Podolsky, and Nathan Rosen presented the Einstein–Podolsky–Rosen argument, questioning whether quantum mechanics provided a complete description of physical reality. Erwin Schrödinger introduced the terminology of entanglement that year and emphasized its importance for composite systems. (nobelprize.org)

In 1964, John Bell established Bell's theorem, showing that correlations allowed by quantum mechanics cannot all be reproduced by local hidden-variable models, under the theorem's assumptions. Bell inequalities translate this distinction into experimentally testable statistical bounds. Their violation excludes that class of models; it does not exclude every possible hidden-variable theory. (nobelprize.org)

Stuart Freedman and John Clauser demonstrated a Bell-inequality violation in 1972. Alain Aspect's experiments subsequently strengthened the tests, while Anton Zeilinger's work developed experimental control and applications of entangled systems. Aspect, Clauser, and Zeilinger jointly received the 2022 Nobel Prize in Physics for experiments with entangled photons, Bell-inequality violations, and pioneering quantum information science. (nobelprize.org)

Nonlocal correlations without faster-than-light signaling

Entanglement and Bell nonlocality are related but not identical. Entanglement describes a state's mathematical structure; Bell nonlocality concerns correlations that cannot satisfy an appropriate local model. Some entangled mixed states do not violate Bell inequalities in standard measurement scenarios. (arxiv.org)

The no-communication theorem states that local operations, without communicating their outcomes, cannot alter the distant party's observable statistics in a way that conveys a message. Conditioning on a known measurement result changes the description of the remote subsystem, but obtaining that result requires ordinary communication. Thus, entanglement does not provide a signaling channel that bypasses the constraints of relativity. (arxiv.org)

Creation, applications, and limitations

Entanglement can be generated through interactions and suitable quantum gates. It is a resource in quantum computers, while noise and quantum decoherence can degrade the states needed for computation. Quantum error correction addresses errors by encoding quantum information across multiple physical qubits rather than storing it in a single unprotected system. (learning.quantum.ibm.com)

In quantum teleportation, a shared entangled pair and two classical bits allow an unknown qubit state to be transferred to another qubit. Matter is not transported, and the original unknown state is not retained as an additional copy. Completion requires classical communication. (quantum.cloud.ibm.com)

Entanglement also supports superdense coding, in which a shared Bell pair allows two classical bits to be conveyed by transmitting one qubit. In quantum key distribution, entangled correlations can support protocols for establishing secret keys. These applications connect entanglement to information theory and cryptography, while experimental usefulness depends on preparation quality, losses, measurement accuracy, and noise. (quantum.cloud.ibm.com)