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If basis information and test positions are announced before the receiver confirms receipt of quantum carriers, an eavesdropper can store the carriers until learning that information. In a scheme sending one half of each Bell pair after a Hadamard gate encoding, the eavesdropper then forwards test qubits untouched and measures all data qubits in their now-known correct bases. Every test passes, while data are changed to classically correlated product states whose bit labels the eavesdropper knows. This invalidates the sampling argument needed to justify entanglement purification and quantum key distribution. A sound additional purification certification could detect this attack and abort; it cannot generate private Bell pairs from the separable attacked data. Announcing bases only after receipt prevents this particular delayed-carrier attack.

A unitary operation on a fixed number of qubits used as a component of a quantum circuit. Examples include the Hadamard gate and Controlled-Z gate. Measurements are described separately by quantum measurement; a gate is a reversible operation on its full input state.

A quantum superposition is a linear combination of quantum state vectors in a complex Hilbert space. In a specified orthonormal basis, a normalized pure state has the displayed expansion, where $a_j$ are probability amplitudes. A nonzero linear combination can always be normalized; an arbitrary sum of normalized vectors need not already have norm one. Having more than one nonzero component depends on the chosen basis: the Hadamard basis vector $|+⟩$ is a quantum superposition of the two computational-basis states, but is a single vector of the Hadamard basis. ^[ref].

The distinction from classical ignorance is quantitative. Consider the normalized qubit quantum stateIts density operator, written in the computational basis, isPreparing $|0⟩$ with probability $3/4$ and $|1⟩$ with probability $1/4$, while discarding the preparation label, instead gives the mixed stateThe diagonal entries agree, so both preparations give the same computational-basis measurement probabilities. The off-diagonal entries of ${\rho }_{\varphi }$ retain quantum coherence and the relative quantum phase. In the Hadamard basis, the Born rule givesThe density matrix representation retains this distinction between a pure state and a mixed state. ^[ref].

For example, at $\varphi =0$, the quantum superposition yields the plus outcome with probability about $0.9330$, whereas the mixed state yields it with probability $0.5$. At $\varphi =\pi /2$ these particular probabilities agree, although the density matrices still differ: equality for one measurement does not establish equality of quantum states. The probability amplitude figure shows the comparison throughout a full cycle of relative quantum phase.

This is why a quantum superposition is useful: quantum gates act linearly on its probability amplitudes, and unitary operators preserve the full pure state information, including relative quantum phases. A Hadamard gate can turn otherwise hidden quantum coherence into different outcome probabilities. The beautiful feature is that the same basis weights can support different predictions when amplitudes recombine. The expansion specifies a preparation and its measurement predictions; it does not assign every unmeasured outcome a simultaneous classical value. See projective measurement for the separate question of the conditional state after a measurement.

Quantum theory supplies a common language for quantum mechanics, measurement in quantum mechanics, and quantum information: prepare a quantum state, transform it, and ask which experimental outcomes occur. In a finite-dimensional idealization, a pure state is a normalized vector of complex numbers. Its entries are probability amplitudes; the Born rule turns their squared magnitudes into probabilities. Between measurements, an isolated system evolves by a unitary operator, preserving its norm and inner products. A quantum circuit makes this evolution explicit as a sequence of quantum gates.

The extra information carried by quantum phases is essential. Relative quantum phases affect how probability amplitudes add when different contributions reach the same output. Their interference can enhance one outcome and cancel another. A global phase, multiplying the entire quantum state by the same complex factor of magnitude one, changes neither these relative phases nor observable predictions. The Hadamard gate below provides a small experiment in which this distinction becomes a deterministic change of output.

Quantum coherence describes this capacity to interfere relative to a specified orthonormal basis. In a density matrix, it appears in off-diagonal entries; a diagonal mixed state retains the populations but lacks that basis coherence. Thus matching the probabilities of one measurement need not match the predictions of a later experiment. Baumgratz, Cramer, and Plenio made this basis-dependent viewpoint precise in their resource framework for quantum coherence. ^[ref].

This makes quantum information an operational question: which distinctions can a preparation, an available unitary operator, and measurement in quantum mechanics reveal? Amplitude amplification illustrates how carefully arranged interference converts a weakly represented answer into a likely observable result. Preserving the relevant quantum coherence throughout the computation is part of the physical task. ^[ref].

For four uniformly distributed Bell states, measure one qubit along Bloch vector $\mathbf{n}$ and the other along $z$. The sign within each ${\Phi }^{\pm }$ or ${\Psi }^{\pm }$ pair is invisible and the displayed mutual information is at most one bit. A local Hadamard gate before a computational detector changes $n_z$ from one to zero and destroys the remaining information. An entangling Bell decoder falls outside this product-measurement restriction and recovers two bits.

Prepare a qubit in $|+⟩$, apply a phase gate $P(\varphi )=\mathrm{diag}(1,e^{i\varphi })$, and correlate its computational basis states with normalized environment states $|e_0⟩,|e_1⟩$. Write $⟨e_0|e_1⟩=v{e}^{i\alpha }$. After a final Hadamard gate, the environment amplitude accompanying output zero is $(|e_0⟩+e^{i\varphi }|e_1⟩)/2$. Its squared norm is $P_0=[1+v\cos (\varphi +\alpha )]/2$. Thus $v$ is the interference visibility, while $\alpha$ shifts the fringe. This quantum decoherence map commutes with the phase gate, so placing it immediately before that gate gives the same probabilities.

Since $HZH=X$, unitary conjugation gives $H{e}^{-i\beta Z/2}H=e^{-i\beta X/2}$. Multiplying by the two outer $R_z(\alpha )$ rotations givesThe case $\alpha =\beta =\pi /2$ equals $-iH$. Equal output probabilities for one input do not by themselves identify an optical component with the Hadamard gate; relative wave phases also matter.

The inversion about the mean for four computational-basis states can be implemented using Hadamard gates, Pauli X gates and one controlled-NOT gate, up to global phase. Let $Q=H\otimes H$ and $P=X\otimes X$. Since $CZ=(I\otimes H)\mathrm{CNOT}(I\otimes H)=I-2|11⟩⟨11|$, one has $PCZP=I-2|00⟩⟨00|$ and therefore $QPCZPQ=I-2|++⟩⟨++|=-D$. This gives an exact quantum circuit representative of the four-entry Grover search algorithm diffusion step modulo its irrelevant overall minus sign.

In an $N$-dimensional computational basis, the uniform superposition state isMeasurement in that basis gives the uniform distribution on a finite set. For $N=2^n$, Hadamard gates prepare it from $|0^n⟩$. Its reflection $2|s_N⟩⟨s_N|-I$ is a Grover diffusion operator.