Quantum States and Operations

A quantum state predicts the statistics of observations on a physical system. In a finite-dimensional Hilbert space, a pure state is represented by a normalized vector $\lvert\psi\rangle$ , with vectors differing only by global phase representing the same state. A density matrix $\rho\succeq0$ , $\mathrm{tr}(\rho)=1$ also describes mixtures and subsystems of entangled states. The expectation of an observable $A$ is $\mathrm{tr}(A\rho)$ . These objects supply the inputs and outputs of quantum circuits.

Operations and measurement

A quantum channel is a completely positive, trace-preserving linear map. Its Kraus representation is

\mathcal E(\rho)=\sum_i K_i\rho K_i^\dagger, \qquad \sum_i K_i^\dagger K_i=I.

Complete positivity ensures that the map remains positive when applied to part of a larger, potentially entangled system. Replacing equality by $\sum_i K_i^\dagger K_i\preceq I$ gives a trace-nonincreasing operation.

For a measurement with one Kraus operator $M_m$ per outcome, outcome $m$ has probability $p_m=\mathrm{tr}(M_m\rho M_m^\dagger)$ . When $p_m>0$ , the conditional state is $M_m\rho M_m^\dagger/p_m$ . If the outcome is discarded, the output is the sum of the unnormalized branches. Keeping the outcome instead requires a classical register or an equivalent explicit branch structure.

Example: coherence and its loss

The Hadamard gate prepares $\lvert+\rangle=(\lvert0\rangle+\lvert1\rangle)/\sqrt2$ from $\lvert0\rangle$ . Measuring in the computational basis and forgetting the result gives

\sum_{m=0}^1 P_m\lvert+\rangle\langle+\rvert P_m=I/2, \qquad P_m=\lvert m\rangle\langle m\rvert.

The initial and final states give identical computational-basis probabilities. They differ in other measurements: measuring in the $\{\lvert+\rangle,\lvert-\rangle\}$ basis returns $+$ with probability one initially and one half afterward. Probabilities in one basis therefore do not determine a quantum state.

Limitations

Joint systems use tensor products; an entangled state need not factor into individual states. The no-cloning theorem rules out a universal physical operation that copies arbitrary unknown states. It permits copying distinguishable orthogonal states and classical measurement records, a distinction reflected in linear types.

This introductory demonstration assumes finite-dimensional quantum mechanics. Channels can describe noise, but a hardware claim requires a justified noise model and calibration data. State notation alone supplies neither an implementation nor a correctness proof.

References

Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information, 10th Anniversary Edition. Cambridge University Press. DOI: 10.1017/CBO9780511976667.
W. K. Wootters and W. H. Zurek. A single quantum cannot be cloned. Nature 299, 802-803 (1982). DOI: 10.1038/299802a0.