Entanglement, EPR, and Bell's Theorem

Entanglement is the feature of quantum mechanics with no classical counterpart: composite systems can be in states that are not products of states of their parts, so that measurements on separated subsystems are correlated in ways no local theory can reproduce. Einstein, Podolsky, and Rosen argued in 1935 that this made QM "incomplete"; Bell showed in 1964 that the disagreement is experimentally decidable — and experiment has decisively favored quantum mechanics. This page develops entanglement, the EPR argument, Bell's inequality and its violation, and the no-signalling and no-cloning theorems that police what entanglement can and cannot do.

1. Entangled States

By Postulate 6, a bipartite system lives in ${\mathcal{H}}_A\otimes {\mathcal{H}}_B$. A state is separable if it factorizes, $|\psi ⟩=|{\varphi }_A⟩\otimes |{\chi }_B⟩$, and entangled otherwise. The paradigm is the singlet (one of the four Bell states) of two spin-$\frac{1}{2}$ particles:

$$|{\Psi }^{-}⟩=\frac{1}{\sqrt{2}}(|\uparrow {⟩}_A|\downarrow {⟩}_B-|\downarrow {⟩}_A|\uparrow {⟩}_B).$$

No choice of single-particle states reproduces it. Its hallmark: measuring $A$'s spin along any axis yields a random result, but instantly determines $B$'s outcome along that axis to be opposite. The subsystems individually are in maximally mixed states (reduced density operator ${\hat{\rho }}_A=\frac{1}{2}\hat{1}$; see open systems) — all the information is in the correlations, none in the parts.

2. The EPR Argument

EPR assumed locality (no instantaneous action at a distance) and a criterion of reality (if a value can be predicted with certainty without disturbing a system, it is an element of reality). In the singlet, measuring $A$ lets one predict $B$'s spin along the chosen axis with certainty; by locality the distant $B$ was not disturbed, so that spin component is "real." But this holds for any axis, and QM forbids simultaneous sharp values of non-commuting spin components. EPR concluded the wavefunction must be an incomplete description — that "hidden variables" fix the outcomes in advance. The argument is logically valid; Bell showed its premises are testable.

3. Bell's Theorem

Consider measurements on the two particles along adjustable directions, $a,a^{\prime }$ for $A$ and $b,b^{\prime }$ for $B$, each yielding $\pm 1$. Any local hidden-variable theory — outcomes determined by a shared variable $\lambda$ with $A$'s result independent of $B$'s setting — obeys the CHSH inequality:

$$\boxed{S\equiv |E(a,b)-E(a,b^{\prime })+E(a^{\prime },b)+E(a^{\prime },b^{\prime })|\le 2,}$$

where $E(a,b)$ is the correlation (expectation of the product of outcomes). This is a constraint on correlations alone, derived without any quantum mechanics.

Quantum mechanics violates it. For the singlet, $E(a,b)=-\cos {\theta }_{ab}$, and choosing the four directions at successive $45^{\circ }$ gives

$$S_{\text{QM}}=2\sqrt{2}\approx 2.83>2$$

(Tsirelson's bound, the maximum QM allows). The predictions are numerically different, so the question "is nature locally realistic?" is empirical.

4. The Verdict of Experiment

From Aspect (1980s) through the loophole-free experiments of 2015 (Delft, NIST, Vienna) — closing the locality and detection loopholes simultaneously — the CHSH inequality is violated in agreement with QM. Nature is not describable by any local hidden-variable theory. One must give up locality or realism (definite pre-existing values); the interpretations differ in which. The 2022 Nobel Prize recognized this line of work.

5. What Entanglement Cannot Do: No-Signalling and No-Cloning

Entanglement produces correlations but cannot be used to communicate. Two theorems make this precise and police what entanglement can and cannot do. No-signalling says that nothing Bob does to his half of an entangled pair — which observable he measures, or whether he measures at all — has any effect on the statistics Alice sees locally; the correlations surface only after the two parties compare their records over an ordinary, luminally-limited classical channel. No-cloning says an arbitrary unknown state cannot be duplicated. We state and prove each in turn.

No-signalling theorem. Setup and definitions. Model the two wings as a bipartite quantum system on the (separable) Hilbert space $\mathcal{H}={\mathcal{H}}_A\otimes {\mathcal{H}}_B$, Alice owning the tensor factor ${\mathcal{H}}_A$ and Bob ${\mathcal{H}}_B$. Their joint state is a density operator ${\hat{\rho }}_{AB}$ on $\mathcal{H}$ — Hermitian, positive semidefinite, $\mathrm{Tr}{\hat{\rho }}_{AB}=1$ (a pure state $|\psi ⟩⟨\psi |$ is the special case). Fix:

• Local measurements and their outcomes. A measurement on Alice's wing is a POVM $\{{\hat{E}}_a{\}}_a$ on ${\mathcal{H}}_A$ — operators ${\hat{E}}_a\ge 0$ with ${\sum }_a{\hat{E}}_a={\hat{1}}_A$ — acting globally as ${\hat{E}}_a\otimes {\hat{1}}_B$. Its outcome is a random variable $A$ with Born distribution $$\mathrm{Pr}(A=a)=\mathrm{Tr}[({\hat{E}}_a\otimes {\hat{1}}_B){\hat{\rho }}_{AB}].$$ Bob's measurement $\{{\hat{F}}_b{\}}_b$ on ${\mathcal{H}}_B$ and its outcome $B$ are defined symmetrically, $\mathrm{Pr}(B=b)=\mathrm{Tr}[({\hat{1}}_A\otimes {\hat{F}}_b){\hat{\rho }}_{AB}]$.
• Alice's reduced state. The partial trace over $B$, ${\hat{\rho }}_A:={\mathrm{Tr}}_B{\hat{\rho }}_{AB}$, is the unique operator on ${\mathcal{H}}_A$ satisfying $${\mathrm{Tr}}_A\left[\hat{X}{\hat{\rho }}_A\right]=\mathrm{Tr}[(\hat{X}\otimes {\hat{1}}_B){\hat{\rho }}_{AB}]\text{for every bounded }\hat{X}\text{ on }{\mathcal{H}}_A;$$ equivalently, in any orthonormal basis $\{|j{⟩}_B\}$ of ${\mathcal{H}}_B$, $${\hat{\rho }}_A=\sum _j({\hat{1}}_A\otimes ⟨j{|}_B){\hat{\rho }}_{AB}({\hat{1}}_A\otimes |j{⟩}_B).$$ Setting $\hat{X}={\hat{E}}_a$ in the defining property gives $\mathrm{Pr}(A=a)={\mathrm{Tr}}_A[{\hat{E}}_a{\hat{\rho }}_A]$: ${\hat{\rho }}_A$ is a sufficient statistic for everything Alice can observe.

Claim. No local operation Bob performs on ${\mathcal{H}}_B$ changes ${\hat{\rho }}_A$ — hence none changes any Alice probability $\mathrm{Pr}(A=a)$.

• Bob applies a local unitary ${\hat{U}}_B$ (e.g. rotates his measurement basis). The new global state is $(\hat{1}\otimes {\hat{U}}_B){\hat{\rho }}_{AB}(\hat{1}\otimes {\hat{U}}_B^{†})$, and $${\hat{\rho }}_A^{\prime }={\mathrm{Tr}}_B[(\hat{1}\otimes {\hat{U}}_B){\hat{\rho }}_{AB}(\hat{1}\otimes {\hat{U}}_B^{†})]={\mathrm{Tr}}_B[{\hat{\rho }}_{AB}(\hat{1}\otimes {\hat{U}}_B^{†}{\hat{U}}_B)]={\mathrm{Tr}}_B{\hat{\rho }}_{AB}={\hat{\rho }}_A,$$ using cyclicity of the trace inside the $B$ factor and ${\hat{U}}_B^{†}{\hat{U}}_B={\hat{1}}_B$.
• Bob performs a measurement but does not reveal the outcome (a non-selective measurement with Kraus operators $\{{\hat{M}}_k\}$, ${\sum }_k{\hat{M}}_k^{†}{\hat{M}}_k={\hat{1}}_B$). Averaging over his unknown result, $${\hat{\rho }}_A^{\prime }={\mathrm{Tr}}_B[{\sum }_k(\hat{1}\otimes {\hat{M}}_k){\hat{\rho }}_{AB}(\hat{1}\otimes {\hat{M}}_k^{†})]={\mathrm{Tr}}_B\left[{\hat{\rho }}_{AB}{\sum }_k{\hat{M}}_k^{†}{\hat{M}}_k\right]={\mathrm{Tr}}_B{\hat{\rho }}_{AB}={\hat{\rho }}_A.$$

In both cases ${\hat{\rho }}_A^{\prime }={\hat{\rho }}_A$, so every Alice probability $\mathrm{Pr}(A=a)={\mathrm{Tr}}_A[{\hat{E}}_a{\hat{\rho }}_A]$ is unchanged: Bob's choice of setting is invisible on Alice's side, and no message crosses. The load-bearing step is Bob's completeness relation ${\sum }_k{\hat{M}}_k^{†}{\hat{M}}_k=\hat{1}$ — i.e. that some outcome certainly occurs and Alice is not told which. Signalling would require Bob to select a sub-ensemble (post-select on $k$) and communicate that fact, which needs a classical channel and so respects relativity. $■$

Equivalently, in probabilistic form: $P(a\mid x)={\sum }_{b}P(a,b\mid x,y)$ is independent of Bob's setting $y$, the operational no-signalling condition that defines the no-signalling correlation set (see Bell Nonlocality). Note this is only parameter independence (no-signalling); the outcome correlations themselves remain, and can still violate a Bell inequality.

No-cloning theorem (Wootters–Zurek, Dieks, 1982). There is no fixed device that makes a perfect copy of an arbitrary unknown quantum state.

Setup and assumptions. Fix a Hilbert space $\mathcal{H}$ with $\dim \mathcal{H}\ge 2$ (a single qubit already suffices). A universal cloning machine would consist of:

• a blank register in $\mathcal{H}$ prepared in a fixed, input-independent state $|b⟩$ (the "paper" onto which the copy is written);
• an optional ancilla / machine in a fixed, input-independent state $|M⟩\in {\mathcal{H}}_M$;
• a single unitary $\hat{U}$ on $\mathcal{H}\otimes \mathcal{H}\otimes {\mathcal{H}}_M$, the same for every input (this is the universality assumption), such that for every $|\psi ⟩\in \mathcal{H}$, $$\hat{U}(|\psi ⟩\otimes |b⟩\otimes |M⟩)=|\psi ⟩\otimes |\psi ⟩\otimes |M_{\psi }⟩,$$ where $|M_{\psi }⟩$ (possibly $\psi$-dependent) is the final machine state.

The four operative hypotheses are thus: exactness (the copy is perfect, not approximate), determinism (success with probability $1$), universality (one $\hat{U}$ independent of the input), and unitarity/linearity (closed-system quantum evolution). We show they are jointly inconsistent unless the inputs are restricted to a preselected orthonormal set.

Proof 1 (inner-product / unitarity). Unitary maps preserve inner products. Apply $\hat{U}$ to inputs built from two states $|\psi ⟩,|\varphi ⟩\in \mathcal{H}$ and equate the input and output overlaps. On the input side, since $|b⟩,|M⟩$ are normalized, $$⟨\psi |\varphi ⟩⟨b|b⟩⟨M|M⟩=⟨\psi |\varphi ⟩.$$ On the output side, $$⟨\psi |\varphi ⟩⟨\psi |\varphi ⟩⟨M_{\psi }|M_{\varphi }⟩=⟨\psi |\varphi {⟩}^2⟨M_{\psi }|M_{\varphi }⟩.$$ Writing $x=⟨\psi |\varphi ⟩$ and using $|⟨M_{\psi }|M_{\varphi }⟩|\le 1$, $$x=x^2⟨M_{\psi }|M_{\varphi }⟩⟹|x|=|x{|}^2|⟨M_{\psi }|M_{\varphi }⟩|\le |x{|}^2,$$ so either $x=0$ or $|x|\ge 1$, i.e. $|x|=1$ (Cauchy–Schwarz bounds $|x|\le 1$). Hence any two clonable states are orthogonal ($x=0$) or equal up to a phase ($|x|=1$). A universal cloner would therefore have to succeed on non-orthogonal states — impossible. $■$

Proof 2 (linearity). Suppose $\hat{U}$ clones two states, $\hat{U}|\psi ⟩|b⟩=|\psi ⟩|\psi ⟩$ and $\hat{U}|\varphi ⟩|b⟩=|\varphi ⟩|\varphi ⟩$ (suppressing $|M⟩$). Linearity of $\hat{U}$ forces, on the superposition $|\chi ⟩=\frac{1}{\sqrt{2}}(|\psi ⟩+|\varphi ⟩)$, $$\hat{U}|\chi ⟩|b⟩=\frac{1}{\sqrt{2}}(|\psi ⟩|\psi ⟩+|\varphi ⟩|\varphi ⟩).$$ But cloning $|\chi ⟩$ demands $$|\chi ⟩|\chi ⟩=\frac{1}{2}(|\psi ⟩|\psi ⟩+|\psi ⟩|\varphi ⟩+|\varphi ⟩|\psi ⟩+|\varphi ⟩|\varphi ⟩),$$ which differ by the cross terms unless $⟨\psi |\varphi ⟩=0$. Contradiction. $■$

Scope — what is not forbidden. The theorem constrains only the four hypotheses above; relaxing any one restores a copying operation:

• Known or mutually orthogonal states can be copied: on a fixed orthonormal basis $\{|k⟩\}$ the map $|k⟩|b⟩\mapsto |k⟩|k⟩$ extends to a unitary, which is exactly the classical CNOT/fan-out. No-cloning bites only for unknown, generically non-orthogonal inputs.
• Approximate cloning is allowed: the optimal universal qubit cloner reaches fidelity $F=5/6$ (Bužek–Hillery), below the perfect $F=1$.
• Probabilistic cloning of a set of linearly independent states is possible with success probability $<1$ (Duan–Guo).
• For mixed states the sharper no-broadcasting theorem holds: a set of states can be broadcast iff they mutually commute.

Consistency with relativity. No-cloning and no-signalling are linked: if perfect cloning existed, Alice could read out Bob's reduced state ${\hat{\rho }}_A$ by making many copies, inferring his distant measurement basis and enabling superluminal signalling. Forbidding cloning is exactly what keeps entanglement compatible with relativistic causality; it is also the structural fact underwriting the security of quantum key distribution (an eavesdropper cannot copy the transmitted qubits undetected). Together these ensure quantum correlations coexist with relativistic causality.

6. Entanglement as a Resource

Far from a paradox, entanglement is the fuel of quantum information:

• Teleportation transfers an unknown state using a shared Bell pair plus two classical bits (consuming the entanglement; no cloning, no signalling).
• Superdense coding sends two classical bits in one qubit via a shared pair.
• Bell violation certifies device-independent randomness and key distribution.
• Entanglement entropy $S({\hat{\rho }}_A)=-\mathrm{Tr}({\hat{\rho }}_A\ln {\hat{\rho }}_A)$ quantifies it and links to decoherence, thermodynamics, and black-hole physics.

See also

• Postulates — composite systems (Postulate 6).
• Open quantum systems — reduced states and the partial trace.
• Decoherence — how entanglement with an environment yields classicality.
• Interpretations — locality vs. realism.
• general/quantum-mechanics.md — conceptual overview.