Multi-Qubit Systems & Entanglement

When we combine two qubits, their joint state lives in the tensor product of their individual Hilbert spaces. A general two-qubit state can be written as $\|psi\rangle = \alpha|00\rangle + \beta|01\rangle + \gamma|10\rangle + \delta|11\rangle$, where the basis states $|00\rangle$, $|01\rangle$, $|10\rangle$, and $|11\rangle$ are tensor products of the single-qubit computational basis.

The normalization condition requires $|\alpha|^2 + |\beta|^2 + |\gamma|^2 + |\delta|^2 = 1$, ensuring the total probability of finding the system in one of the four basis states is unity. For separable states, we can factor the state as $|a\rangle \otimes |b\rangle$, but most states in the Hilbert space cannot be factored in this way.

The dimensionality of an $n$-qubit system scales as $2^n$, which is the root of both quantum computing's power and the difficulty of simulating quantum systems classically.

• • The tensor product $|a\rangle \otimes |b\rangle$ is often abbreviated as $|a\rangle|b\rangle$ or $|ab\rangle$.
• • A state is separable if and only if it can be written as a tensor product of individual qubit states.
• • The number of amplitudes grows exponentially: $n$ qubits require $2^n$ complex coefficients.

The Controlled-NOT (CNOT) gate is the fundamental two-qubit gate in quantum computing. It acts on two qubits: a control qubit and a target qubit. In the computational basis, the CNOT gate flips the target qubit if and only if the control qubit is $|1\rangle$.

Mathematically, the action of CNOT is $|a,b\rangle \rightarrow |a, b \oplus a\rangle$, where $\oplus$ denotes addition modulo 2. The matrix representation in the computational basis $\{|00\rangle, |01\rangle, |10\rangle, |11\rangle\}$ is a $4 \times 4$ unitary matrix.

One of the most important applications of CNOT is generating entanglement. When applied to $|+\rangle|0\rangle$, where $|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$, the CNOT gate produces the maximally entangled Bell state $\|Phi^+\rangle$.

• • CNOT is its own inverse: applying it twice returns the original state.
• • The gate is universal for entanglement generation when combined with single-qubit gates.
• • In circuit diagrams, the control qubit is marked with a filled dot and the target with a $\oplus$ symbol.

The four Bell states form a maximally entangled orthonormal basis for the two-qubit Hilbert space. They are defined as superpositions of two basis states with equal amplitudes and specific relative phases. Any two-qubit state can be expressed as a linear combination of these four states.

Maximal entanglement means that measuring one qubit completely determines the state of the other, with no information gained about the individual qubits beforehand. This is reflected in the reduced density matrix of either subsystem being the maximally mixed state $I/2$.

• • The four Bell states are $\|Phi^+\rangle$, $\|Phi^-\rangle$, $\|Psi^+\rangle$, and $\|Psi^-\rangle$.
• • They are generated from the computational basis by applying a Hadamard gate followed by a CNOT.
• • Bell states are invariant under local unitary operations up to a change of basis.

In 1935, Einstein, Podolsky, and Rosen (EPR) presented a thought experiment challenging the completeness of quantum mechanics. They considered two particles in an entangled state shared between Alice and Bob, spatially separated. If Alice measures her qubit and obtains $|0\rangle$, Bob's state instantaneously collapses to $|0\rangle$, and similarly for $|1\rangle$.

EPR argued that since no signal can travel faster than light, the outcome of Bob's measurement must be predetermined by hidden variables. They introduced the principle of local realism: physical properties are real and independent of measurement (realism), and influences cannot propagate faster than light (locality).

Quantum mechanics predicts correlations that cannot be explained by any local hidden variable theory, setting the stage for Bell's theorem.

• • The collapse appears instantaneous, but no usable information is transmitted faster than light.
• • EPR concluded that quantum mechanics must be incomplete and hidden variables are needed.
• • The paradox highlights the tension between quantum entanglement and classical intuitions about reality.

Bell's theorem provides a way to experimentally distinguish quantum mechanics from local hidden variable theories. The CHSH inequality is the most commonly tested version. Consider two parties, Alice and Bob, each choosing between two measurement settings. The CHSH quantity $S$ is constructed from the expectation values of their outcomes.

For any local hidden variable theory, the absolute value of $S$ is bounded by 2: $|S| \leq 2$. However, quantum mechanics allows correlations strong enough to violate this bound. The maximum quantum value, known as the Tsirelson bound, is $|S| \leq 2\sqrt{2}$.

Experimentally observing $|S| > 2$ rules out local realism. To calculate $S$, one evaluates the expectation values $\langle AB \rangle$ for different measurement settings and combines them according to the CHSH operator.

• • The CHSH operator is $\hat{S} = \hat{A}_1 \otimes \hat{B}_1 + \hat{A}_1 \otimes \hat{B}_2 + \hat{A}_2 \otimes \hat{B}_1 - \hat{A}_2 \otimes \hat{B}_2$.
• • Optimal violation is achieved with maximally entangled states and appropriate measurement angles.
• • Numerous experiments have confirmed the quantum prediction, closing loopholes in recent years.