Entanglement

Entanglement is a property of a composite quantum state relative to a specified subsystem decomposition. A bipartite pure state is entangled when it cannot be written as one product state. A bipartite mixed state is entangled when it cannot be written as a probabilistic mixture of product states.

In Symbols

Let

$$\mathcal H_{AB} =\mathcal H_A\otimes\mathcal H_B.$$

Pure States

A pure state is a product state if there exist normalized subsystem states such that

$$\lvert\psi\rangle_{AB} =\lvert\alpha\rangle_A \otimes \lvert\beta\rangle_B.$$

If no such factorization exists, $\lvert\psi\rangle_{AB}$ is entangled across $A\mathbin{:}B$.

Every finite-dimensional bipartite pure state has a Schmidt decomposition

$$\lvert\psi\rangle_{AB} =\sum_{j=1}^{r} \sqrt{\lambda_j}\, \lvert u_j\rangle_A \lvert v_j\rangle_B,$$

where

$$\lambda_j>0, \qquad \sum_{j=1}^{r}\lambda_j=1,$$

and both Schmidt-vector sets are orthonormal. The state is entangled exactly when its Schmidt rank satisfies

$$r>1.$$

Equivalently, the reduced state

$$\rho_A =\operatorname{Tr}_B \left( \lvert\psi\rangle\langle\psi\rvert \right) =\sum_j\lambda_j \lvert u_j\rangle\langle u_j\rvert$$

is mixed. For a bipartite pure state, the following conditions are equivalent:

• the joint state is a product;
• the Schmidt rank is one;
• $\rho_A$ is pure;
• $\rho_B$ is pure;
• the entanglement entropy is zero.

The pure-state entanglement entropy is

$$S(\rho_A) =-\operatorname{Tr} \left( \rho_A\log\rho_A \right) =-\sum_j\lambda_j\log\lambda_j.$$

The logarithm base fixes the unit: base two gives bits, while the natural logarithm gives nats.

Mixed States

A density operator is separable across $A\mathbin{:}B$ if it can be written

$$\rho_{AB} =\sum_k p_k\, \rho_A^{(k)} \otimes \rho_B^{(k)},$$

with

$$p_k\ge0, \qquad \sum_kp_k=1.$$

If no such decomposition exists, $\rho_{AB}$ is entangled. A mixed state can therefore be non-product yet separable. Non-factorization alone is not a valid mixed-state entanglement test.

The finite-sum form is sufficient in finite dimensions. In infinite-dimensional settings, the separable set is defined with the appropriate trace-norm closure, or equivalently with suitable probability measures over product states; this analytic qualification prevents finite decompositions from being treated as universal.

The decomposition above need not be unique. Entanglement asks whether at least one separable decomposition exists, not whether a displayed decomposition happens to contain entangled vectors.

Canonical Home

See Entangled States for the canonical full treatment, including mixed-state separability and operational criteria.

The Core Formalism introduction gives the pure-state first encounter. Schmidt Decomposition Overview and Reduced States provide the main diagnostics.

Subsystem Split

Entanglement is defined only after a tensor-product structure or an appropriate subsystem algebra has been declared. The same abstract vector space can admit different factorizations, and a state may be product with respect to one decomposition but entangled with respect to another.

A local change of basis,

$$\lvert\psi\rangle \longmapsto (U_A\otimes U_B)\lvert\psi\rangle,$$

cannot change whether a pure state is entangled. It preserves the Schmidt coefficients. A general global unitary can create or remove entanglement because it need not factor into local operations.

For identical particles, gauge constraints, and quantum fields, subsystem definitions can be subtler than assigning a labeled tensor factor to each particle. Mode entanglement, particle entanglement, spatial-region entanglement, and algebraic entanglement are not automatically interchangeable.

Bell-State Example

The Bell state

$$\lvert\Phi^+\rangle =\frac{ \lvert0\rangle_A\lvert0\rangle_B +\lvert1\rangle_A\lvert1\rangle_B }{\sqrt2}$$

has Schmidt coefficients $\lambda_1=\lambda_2=1/2$. It cannot be factored, so it is entangled. The joint state is pure:

$$\operatorname{Tr}(\rho_{AB}^2)=1.$$

Each reduced state is maximally mixed:

$$\rho_A=\rho_B=\frac{I}{2},$$

and

$$S(\rho_A)=S(\rho_B)=\log 2.$$

This is a characteristic pure-state signature: the composite can be known as sharply as possible while neither subsystem has its own pure state.

Correlation Is Not Enough

The separable state

$$\rho_{\mathrm{cc}} =\frac12 \lvert00\rangle\langle00\rvert +\frac12 \lvert11\rangle\langle11\rvert$$

has perfect correlations in the computational basis but is not entangled. It is explicitly a convex mixture of product states.

Entanglement is therefore not synonymous with correlation. Classical correlations, quantum discord, entanglement, Einstein–Podolsky–Rosen steering, and Bell nonlocality are distinct notions. Bell-inequality violation implies entanglement, but many entangled mixed states do not violate a given Bell inequality.

Classical Correlation versus Entanglement owns the detailed comparison.

Entanglement Witnesses and Tests

For pure bipartite states, Schmidt rank or reduced-state purity gives a complete test. For mixed states, no single elementary test works in all dimensions.

Useful criteria include:

• a negative partial transpose, which certifies entanglement;
• the positive-partial-transpose criterion, which is necessary and sufficient only in $2\otimes2$ and $2\otimes3$ dimensions;
• entanglement witnesses tailored to a state family;
• concurrence, negativity, or entanglement of formation in appropriate settings.

A positive partial transpose does not prove separability in general. The criterion’s dimensional scope is part of the result.

Local Operations

Local operations and classical communication, abbreviated LOCC, cannot create entanglement from a separable state. They can redistribute, consume, concentrate, or probabilistically reveal entanglement already present, subject to the relevant resource monotones.

Local unitary operations preserve all entanglement measures for a fixed subsystem split. Local measurements can reduce average entanglement while producing conditionally selected branches with different amounts.

Entanglement is consequently treated as a resource in quantum information, but the useful resource depends on the task, allowed operations, noise model, and whether the state is pure or mixed.

No Faster-Than-Light Signaling

Entangled states can produce correlations that have no local hidden-variable explanation, but they do not permit controllable faster-than-light communication.

If subsystem $B$ undergoes a local trace-preserving quantum operation $\mathcal E_B$, then the unconditioned state of $A$ remains

$$\begin{aligned} \rho_A' &=\operatorname{Tr}_B \left[ (I_A\otimes\mathcal E_B)(\rho_{AB}) \right]\\ &=\rho_A. \end{aligned}$$

Conditional states of $A$ can depend on an outcome recorded at $B$, but identifying that condition requires the classical outcome record. Correlation and signaling are different operational questions.

Superposition and Entanglement

Superposition follows from linearity in one Hilbert space. Entanglement requires a composite split. For example,

$$\frac{ \lvert0\rangle_A+\lvert1\rangle_A }{\sqrt2} \otimes\lvert0\rangle_B$$

is a superposition in a product basis but remains a product state. Entanglement is not simply “more superposition.”

Common Aliases

• nonseparability;
• quantum entanglement;
• bipartite or multipartite entanglement, when the partition is stated;
• EPR correlation, in some historical contexts, though that phrase is not a precise synonym.

“Quantum correlation” is broader than entanglement and should not be used as an exact replacement.

Common Confusions

• Entanglement is defined relative to a subsystem decomposition.
• Correlation alone is not sufficient to identify quantum entanglement.
• Mixed-state entanglement requires more care than pure-state non-factorization.
• A pure joint state can have mixed reduced states.
• A mixed state can be separable without being a product state.
• A decomposition into entangled pure states does not prove a mixed state is entangled, because ensemble decompositions are not unique.
• A local basis change cannot create or remove entanglement.
• Entanglement is not a force, interaction, or signal.
• Bell nonlocality, steering, discord, and entanglement are not equivalent.
• Positive partial transpose proves separability only in limited dimensions.
• Identical-particle and field-theory entanglement requires a carefully declared subsystem notion.

Related Entries

• Density Matrix
• Superposition
• Hilbert Space
• Tensor Products
• Product States
• Partial Trace: First Encounter
• Bell States
• Local Unitary Equivalence

References

• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th anniversary ed., Cambridge University Press, 2010, secs. 2.4 and 12.5.
• M. M. Wilde, Quantum Information Theory, 2nd ed., Cambridge University Press, 2017, chs. 4 and 19.
• J. Watrous, The Theory of Quantum Information, Cambridge University Press, 2018, chs. 2 and 6.
• R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Reviews of Modern Physics 81, 865–942 (2009).