1. What is a Quantum State?
In quantum mechanics, the state of a physical system is described by a vector $\ket{\psi}$ in a complex Hilbert space $\mathcal{H}$. For a single qubit, $\mathcal{H} = \C^2$, and the most general (pure) state is
\begin{equation} \ket{\psi} = \alpha \ket{0} + \beta \ket{1}, \qquad \alpha, \beta \in \C, \quad |\alpha|^2 + |\beta|^2 = 1. \label{eq:qubit} \end{equation}The computational basis $\{\ket{0}, \ket{1}\}$ corresponds to the eigenstates of the Pauli $Z$ operator. Physical observables are represented by Hermitian operators, and the Born rule tells us that measuring an observable $A$ yields eigenvalue $a_i$ with probability $|\braket{a_i}{\psi}|^2$.
2. Two Qubits and the Tensor Product
For a composite system of two qubits $A$ and $B$, the Hilbert space is the tensor product $\mathcal{H}_A \otimes \mathcal{H}_B \cong \C^4$. A general two-qubit state is
\begin{equation} \ket{\Psi} = \sum_{i,j \in \{0,1\}} c_{ij}\, \ket{i}_A \otimes \ket{j}_B, \qquad \sum_{i,j} |c_{ij}|^2 = 1. \label{eq:two-qubit} \end{equation}A state is called separable if it can be written as a product $\ket{\Psi} = \ket{\phi}_A \otimes \ket{\chi}_B$, and entangled otherwise.
3. Bell States
The four Bell states (or EPR pairs) form a maximally entangled orthonormal basis for $\C^4$:
\begin{align} \ket{\Phi^+} &= \frac{1}{\sqrt{2}}\bigl(\ket{00} + \ket{11}\bigr), \label{eq:phi+} \\ \ket{\Phi^-} &= \frac{1}{\sqrt{2}}\bigl(\ket{00} - \ket{11}\bigr), \label{eq:phi-} \\ \ket{\Psi^+} &= \frac{1}{\sqrt{2}}\bigl(\ket{01} + \ket{10}\bigr), \label{eq:psi+} \\ \ket{\Psi^-} &= \frac{1}{\sqrt{2}}\bigl(\ket{01} - \ket{10}\bigr). \label{eq:psi-} \end{align}Consider the singlet state $\ket{\Psi^-}$ in \eqref{eq:psi-}. If Alice measures her qubit in the $Z$ basis and obtains $+1$ (i.e., $\ket{0}$), Bob's qubit immediately collapses to $\ket{1}$, regardless of the spatial separation between them. This is the phenomenon that troubled Einstein, Podolsky, and Rosen.
4. The EPR Argument
Einstein, Podolsky, and Rosen (1935) argued that quantum mechanics must be incomplete. Their reasoning: if Alice's measurement can instantaneously determine Bob's outcome, and if no faster-than-light signaling is allowed (locality), then Bob's outcome must have been determined in advance by some hidden variable $\lambda$.
"If, without in any way disturbing a system, we can predict with certainty the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity." — Einstein, Podolsky, Rosen (1935)
This motivated the search for local hidden variable (LHV) theories: models where outcomes are determined by pre-existing variables $\lambda$, with Alice's result depending only on her setting and $\lambda$, and similarly for Bob.
5. Bell's Theorem and the CHSH Inequality
In 1964, John Bell showed that LHV theories make experimentally distinguishable predictions from quantum mechanics. The modern version, due to Clauser, Horne, Shimony, and Holt (CHSH), proceeds as follows.
Alice and Bob each choose one of two measurement settings: $a, a'$ for Alice and $b, b'$ for Bob. Each measurement outcome is $\pm 1$. Define the CHSH correlator
\begin{equation} S = \expect{AB} - \expect{AB'} + \expect{A'B} + \expect{A'B'}, \label{eq:chsh-def} \end{equation}where $\expect{AB}$ denotes the expectation value of the product of Alice's and Bob's outcomes when Alice uses setting $a$ and Bob uses $b$.
Bell's inequality (CHSH form): For any LHV theory,
\begin{equation} |S_{\rm LHV}| \leq 2. \label{eq:bell} \end{equation}Proof sketch. In any LHV model, the outcomes $A(\lambda), A'(\lambda), B(\lambda), B'(\lambda) \in \{-1, +1\}$ are determined by $\lambda$. Note that
\begin{equation} AB - AB' + A'B + A'B' = A(B - B') + A'(B + B'). \label{eq:chsh-algebra} \end{equation}Since $B, B' \in \{-1,+1\}$, we have either $B - B' = 0, B + B' = \pm 2$ or $B - B' = \pm 2, B + B' = 0$. In both cases the right-hand side of \eqref{eq:chsh-algebra} has absolute value $2$, so averaging over $\lambda$ gives $|S| \leq 2$. $\square$
6. The Quantum Violation
Quantum mechanics can violate the CHSH inequality \eqref{eq:bell}. For the Bell state $\ket{\Phi^+}$ and the optimal measurement settings (Alice: $0°$ and $45°$; Bob: $22.5°$ and $67.5°$), one obtains
\begin{equation} S_{\rm QM} = 2\sqrt{2} \approx 2.828. \label{eq:tsirelson} \end{equation}The value $2\sqrt{2}$ is known as the Tsirelson bound — the maximum quantum violation. To see this, consider the state $\ket{\Phi^+}$ and define Alice's observables $A = \sigma_z$, $A' = \sigma_x$, and Bob's $B = -(\sigma_z + \sigma_x)/\sqrt{2}$, $B' = (\sigma_z - \sigma_x)/\sqrt{2}$. A direct computation gives
\begin{align} \expect{AB} &= \braket{\Phi^+}{A \otimes B}{\Phi^+} = \frac{1}{\sqrt{2}}, \\ \expect{AB'} &= -\frac{1}{\sqrt{2}}, \\ \expect{A'B} &= \frac{1}{\sqrt{2}}, \\ \expect{A'B'} &= \frac{1}{\sqrt{2}}, \end{align}which yields $S = 4/\sqrt{2} = 2\sqrt{2}$, confirming \eqref{eq:tsirelson}.
7. Experimental Tests
Experiments by Aspect et al. (1982) and, more conclusively, the loophole-free Bell tests of Hensen et al. have firmly established that nature violates Bell inequalities. The measured values of $|S|$ are consistent with the quantum prediction $2\sqrt{2}$ and rule out all local hidden variable theories.
This has profound implications: quantum entanglement is a real, non-classical resource. It underlies applications such as quantum key distribution, quantum teleportation, and quantum computing.
8. Reduced Density Matrices
Even for a pure entangled state $\ket{\Psi^-}$, the local state of each subsystem is a mixed state, described by a density matrix. The reduced density matrix for Alice is obtained by tracing out Bob's subsystem:
\begin{equation} \rho_A = \Tr_B \bigl(\ket{\Psi^-}\!\bra{\Psi^-}\bigr) = \frac{I}{2}, \label{eq:reduced} \end{equation}where $I/2$ is the maximally mixed state. This confirms that local measurements on an entangled pair reveal no information about the global state — all correlations are nonlocal.
Summary
- Quantum states of composite systems live in tensor product Hilbert spaces (Eq. \eqref{eq:two-qubit}).
- Entangled states cannot be written as product states; the Bell states \eqref{eq:phi+}–\eqref{eq:psi-} are canonical examples.
- The EPR argument suggests hidden variables, but Bell's theorem (CHSH form, \eqref{eq:bell}) rules out local hidden variable theories.
- Quantum mechanics predicts $|S| = 2\sqrt{2}$ \eqref{eq:tsirelson}, exceeding the classical bound — confirmed by experiment.
- Locally, each half of an entangled pair looks maximally mixed (Eq. \eqref{eq:reduced}).