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Measuring the transmitted qubit in the computational basis and forwarding its measured state produces a dephasing channel for users who do not know the outcome. For a Bell state $|{\Psi }^{+}⟩$, the reduced state becomes $(|01⟩⟨01|+|10⟩⟨10|)/2$. It retains perfect $Z$ anticorrelation while destroying $X$ correlation. Thus a single perfectly correlated key basis need not reveal an intercept-resend attack; incompatible-basis correlation tests can.

A dephasing channel suppresses off-diagonal entries of a density matrix in a preferred basis while preserving quantum state populations. Complete dephasing is identical to a projective measurement in that basis with its outcome discarded.

For the Bell correlations versus a classical mixture example, completely dephasing Alice's qubit in the computational basis maps ${\rho }_{\mathrm{Bell}}$ to ${\rho }_{\mathrm{mix}}$. It preserves their perfect $Z$ agreement and each reduced density matrix, yet destroys the $X$ correlation. This is why testing only populations can miss damage to a quantum memory: the stored quantum coherence may have disappeared while every preferred-basis measurement still looks correct. The joint Von Neumann entropy rises from zero to one bit, although each local entropy stays fixed.

Different ensemble members accumulating different deterministic phases can shorten their averaged Bloch vector even when each member remains in a pure state. Averaging static frequency offsets gives an effective dephasing channel. It need not represent irreversible phase loss in each individual system; an appropriate refocusing pulse can reverse static offsets. Changing offsets during the sequence requires a temporal noise model.

A random Pauli Z gate with equal application and nonapplication probabilities givesThis is exactly the same dephasing channel as the unrecorded measurement, for every input density operator, including one entangled with an auxiliary system. Individual trials differ: conditioned on the classical random choice, the state is the pure state $|\psi ⟩$ or $Z|\psi ⟩$, generally neither a computational basis vector. Knowledge of that choice permits recovery by the same Pauli Z gate; a measurement outcome does not restore an unknown destroyed superposition.

The normalized input density operator isA quantum measurement in the computational basis has projectors $P_j=|j⟩⟨j|$. Discarding its recorded outcome gives the dephasing channelThe original off-diagonal coherence disappears. If the outcome is retained instead, the conditional state is $|0⟩$ or $|1⟩$ and the record carries information about which occurred. Thus discarding the record is an essential part of this channel specification.

Let $c=⟨R_0|R_1⟩$. The joint state after the environment interaction is $\alpha |0⟩|R_0⟩+\beta |1⟩|R_1⟩$. Taking its partial trace over the environment gives the exact reduced density operatorThe conjugate in the upper off-diagonal entry follows from tracing $|R_0⟩⟨R_1|$. Normalization of the environment states and the Cauchy-Schwarz inequality imply $|c|\le 1$.

For comparison, a phase-flip channel sends $\rho$ to $(1-p)\rho +pZ\rho Z$, multiplying its off-diagonal entries by the real number $1-2p$. Consequently the claimed representation is valid, for a general input, provided the environment overlap is real, withThis is a genuine probability between zero and one. The environment maps printed in the question do not themselves imply that $c$ is real. For example, choose $|R_1⟩=i|R_0⟩$, so $c=i$, and take $\alpha =\beta =1/\sqrt{2}$. The qubit becomes $(|0⟩+i|1⟩)/\sqrt{2}$, whose imaginary off-diagonal entries cannot come from any probabilistic mixture of $\rho$ and $Z\rho Z$. These environment maps are a valid isometry and can be extended to a unitary interaction, so this is an actual counterexample to the unrestricted premise.

The general complex environment overlap in qubit dephasing has an exact corrected description. Write $c=r{e}^{i\phi }$, $0\le r\le 1$, and let $D_{\phi }=\mathrm{diag}(1,e^{i\phi })$. ThenAt $r=0$, choose any phase. The dephasing channel includes both random phase errors and, when the overlap is complex, a coherent diagonal rotation. The subsequent probability formulas use the intended real-overlap phase-flip model.

A quantum channel is a linear, completely positive, trace-preserving map taking input density operators to output density operators. Trace preservation keeps total probability one. Complete positivity requires positivity even when the map acts on one part of an entangled larger system, with the other part unchanged.

In finite dimensions, a Kraus representation writes $\mathcal{E}(\rho )={\sum }_a{K}_a\rho K_a^{†}$, with ${\sum }_a{K}_a^{†}{K}_a=I$. A unitary channel has one Kraus operator; a dephasing channel suppresses coherences in a specified basis. Different physical implementations can yield the same channel after unobserved degrees of freedom are traced out. A conditional measurement outcome alone is generally trace decreasing; only the sum over all outcomes gives a trace-preserving channel.

For the pure qubit $\alpha |0⟩+\beta |1⟩$, subtracting the output of dephasing channel leaves a Hermitian matrix with zero diagonal and off-diagonal modulus $|\alpha \beta ||1-c|$. Its two eigenvalues are that number and its negative, giving the displayed trace distance. For real $c=1-2p$, this is $2p|\alpha \beta |$.