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A one-photon polarization qubit can conditionally rotate another mode's coherent amplitude: the horizontal branch leaves $\alpha$ unchanged and the vertical branch gives $\alpha e^{i\Theta }$. The output is generally entangled. Measuring the polarization in a superposition basis can erase the branch label and prepare a coherent superposition of the two amplitudes.

A 2018 photonic experiment makes the distinction between populations and quantum coherence controllable. A 400-nm laser pumped a crystal to generate paired photons, filtered near 800 nm. One photon triggered collection; the other entered the apparatus. A half-wave plate and polarizing beam splitter selected horizontal photon polarization; another half-wave plate prepared balanced $H,V$ quantum superpositions. Separated polarization paths and diffraction gratings let a spatial light modulator control their frequency-dependent quantum phases and spectral weights. Recombined paths passed through quartz plates whose adjustable thickness controlled polarization–frequency coupling. Quarter- and half-wave plates plus a polarizing beam splitter selected tomography bases; jointly registered photodetector events supplied the readout. Quantum state tomography inferred polarization density matrices at each thickness. The dephasing model preserves diagonal populations while changing off-diagonal quantum coherence. Here the unobserved photon frequency acts as the environment for the polarization qubit. Finite modulator resolution and imperfect spectral shaping limited agreement with the intended dynamics; this experiment does not establish exact $I/2$ preparation or the ideal probabilities above. ^[ref]

A positive operator-valued measure is a family of positive operators $E_i$ satisfying ${\sum }_i{E}_i=I$. Each $E_i$ is called an effect and determines a probability $p_i=\mathrm{Tr}(E_i\rho )$ for input density operator $\rho$. The effects need not be orthogonal projections or commute. This permits measurements with more outcomes than the dimension of the Hilbert space, such as the trine qubit POVM. Coupling to a quantum ancilla and measuring the larger system supplies a physical route to generalized measurements. ^[ref].

For an elementary unsharp $Z$ example on a qubit, takeOn the eigenstate $|0⟩$, $p_{\pm }=(1\pm \eta )/2$; at $\eta =1/2$ they are $3/4$ and $1/4$. At $\eta =0$ the outcome is a fair coin independent of the input, and at $\eta =1$ these effects specify a sharp $Z$ projective measurement. Choosing $K_{\pm }=\sqrt{E_{\pm }}$ specifies one quantum instrument, but $U_{\pm }\sqrt{E_{\pm }}$ for outcome-dependent unitary operators has the same effects and different state updates. Thus POVM does not determine the post-measurement state.

Normalize the initial polarization qubit and its analysing wave-plate transformation. For a conditional phase rotation of $\pi$, a vertical detection has the displayed probability and prepares $N(|\alpha ⟩+e^{i\varphi }|-\alpha ⟩)$. At $\alpha =0,\varphi =\pi$, that outcome has zero probability, explaining the singular normalization of the absent branch.

A probability amplitude is a complex number that describes the contribution of a pure state to a specified measurement outcome. For a normalized quantum state $|\psi ⟩$ and a rank-one outcome associated with a normalized vector $|j⟩$, it is the inner product $a_j=⟨j|\psi ⟩$. The Born rule gives the corresponding probability $p_j=|a_j{|}^2=a_j^{\ast }{a}_j$, using the complex conjugate. Thus an amplitude can be negative or complex, while a probability is real and nonnegative. A wavefunction is the analogous position amplitude: its squared modulus gives a probability density, which must be integrated over a region to obtain a probability. ^[ref].

The extra information is the quantum phase. If $a=r{e}^{i\theta }$, its modulus $r$ fixes $|a{|}^2=r^2$, but its angle $\theta$ controls how it adds to another contribution. In the complex plane, picture amplitudes as arrows: add the arrows, then square the resulting length. For coherent contributions $A$ and $B$ to the same outcome,The last term is interference. Adding probabilities first loses it. Contributions with different relative quantum phases can reinforce or cancel even when their individual squared moduli are fixed.

For the qubit quantum superposition $|{\psi }_{\varphi }⟩=(\sqrt{3}/2)|0⟩+(e^{i\varphi }/2)|1⟩$, project onto the plus vector of the Hadamard basis. By linearity, the two contributions combine asHere $|A{|}^2=3/8$ and $|B{|}^2=1/8$, while the interference term is $(\sqrt{3}/4)\cos \varphi$. At $\varphi =0$, the arrows align and the plus probability is $1/2+\sqrt{3}/4\simeq 0.9330127$. At $\varphi =\pi$, they oppose and that probability is $1/2-\sqrt{3}/4\simeq 0.0669873$; cancellation is incomplete because the lengths differ. At $\varphi =\pi /2$, the arrows are perpendicular and the cross term vanishes. The minus amplitude uses $A-B$, reversing the cross term. These are alternate projective measurements on identically prepared copies, rather than a sequence of measurements on one copy.

A global phase multiplies every contribution by the same factor $e^{i\gamma }$, givingIt therefore changes no outcome probability. Changing a relative quantum phase can change the angle between contributions and hence their interference. Keeping amplitudes until the last step makes quantum circuits and matter-wave interference calculable; discarding their angles prematurely discards physically testable information. The Born rule example compares two measurement bases explicitly.

For $\Phi (\rho )=p\rho +(1-p)I/2$, every output qubit has Von Neumann entropy at least $h((1-p)/2)$. A product input therefore has output entropy at least $nh((1-p)/2)$. The average output has entropy at most $n$, so its Holevo quantity is bounded by the displayed expression. This allows correlated classical message labels and joint output measurements, but requires product quantum inputs. The Holevo bound and Fano's inequality turn it into the corresponding asymptotic rate bound.

The SWAP gate and local unitary operators preserve product states. The displayed identity moves every exchange past a pair of local gates, reducing any such two-qubit quantum circuit to a local operation, optionally followed by SWAP gate. On more wires the same argument leaves a permutation of the tensor factors. By induction, a fully factored input remains factored. Since universal quantum computation includes operations that create quantum entanglement from a product state, these gates cannot form a universal quantum gate set.

A projective measurement has self-adjoint orthogonal projections $P_i$ satisfying $P_i{P}_j={\delta }_{ij}{P}_i$ and ${\sum }_i{P}_i=I$. Measuring an observable with discrete spectral decomposition $A={\sum }_i{a}_i{P}_i$ reports eigenvalue $a_i$ with Born rule probability $p_i=\mathrm{Tr}(P_i\rho )$. A projector can have rank greater than one: the outcome then identifies an eigenspace, rather than a particular eigenvector inside it. Projectors specify the outcome statistics; the usual ideal conditional update is the additional Lüders rule.

In the qubit example from measurement in quantum mechanics, $Z=|0⟩⟨0|-|1⟩⟨1|$. At $\theta =\pi /3$, the zero and one probabilities are $3/4$ and $1/4$, regardless of the relative quantum phase $\varphi$. Under the Lüders rule, obtaining zero prepares $|0⟩$; obtaining one prepares $|1⟩$, up to global phase. Repeating the same measurement returns the same label with certainty, provided no intervening evolution or operation changes the measured sector. Repeatability concerns the conditional output, and does not mean the first outcome was predetermined.

Change to $X$ and its Pauli X eigenstates $|\pm ⟩=(|0⟩\pm |1⟩)/\sqrt{2}$. Adding the two probability amplitudes before squaring givesAt $\theta =\pi /3$ and $\varphi =0$, $p_{+}=(2+\sqrt{3})/4\simeq 0.9330$ and $p_{-}\simeq 0.0670$. At $\varphi =\pi /2$, both are $1/2$; at $\varphi =\pi$, the first pair is exchanged. The $Z$ statistics remain $3/4,1/4$ throughout. This is the same addition-and-cancellation principle as wave interference: relative quantum phase changes how amplitudes reinforce or cancel in another basis.

The beauty of this basis dependence is that information invisible to one measurement becomes accessible to another. A $Z$ measurement of an ensemble estimates its quantum state populations, while $X$ probes one component of its quantum coherence. These are experiments on separate identically prepared copies. Sequentially measuring $Z$ and then $X$ on one copy instead probes the disturbed post-measurement state. Quantum state tomography combines suitable ensemble statistics to reconstruct the original state.

A pseudo-pure state is a density matrix of the formwhere $|\psi ⟩$ is a normalized pure state, $I$ is the identity matrix on a $d$-dimensional space and here $0\le p\le 1$. The term $I/d$ is the maximally mixed state. Under a unitary transformation $U$, this term stays fixed and the selected pure state becomes $U|\psi ⟩$. This is why liquid-state NMR quantum computing can follow pure-state-like transformations in part of a highly mixed ensemble.^[ref]

For a traceless matrix representing an observable $O$, direct evaluation of the trace givesThe identity contribution vanishes because $\mathrm{tr}O=0$. Thus $p$ scales the detectable expectation. It is not itself the purity: expanding ${\rho }^2$ and taking its trace yieldsFor example, for two qubits with $p=1/10$, the purity is $103/400$, close to the maximally mixed value $1/4$, although the selected pure state still controls the traceless signal.

The preparation penalty can also be derived explicitly. Take $n$ independent equal spin-one-half systems with one-spin density matrix $(I+\eta Z)/2$, where $Z$ is a Pauli matrix and $0\le \eta \le 1$ is their individual polarization. Their largest population is $a=((1+\eta )/2{)}^n$. A preparation that preserves this population and averages all other populations to equality has $d=2^n$ andIndeed the selected population of $\rho$ is $[1+(d-1)p]/d=a$. In the weak-polarization regime $n\eta \ll 1$, the binomial theorem gives $p\simeq n\eta /(2^n-1)$. If the energy separation is $\hslash \omega$, set $ϵ=\hslash \omega /(k_{B}T)$; a thermal two-level system has $\eta =\tanh (ϵ/2)\simeq ϵ/2$. This gives $p\simeq nϵ/[2(2^n-1)]$, with every parameter defined. Here $\hslash$ is the reduced Planck constant, $k_B$ the Boltzmann constant and $T$ the temperature. The exponential signal loss concerns this specified averaging preparation; it is not a universal bound on every initialization method.

The purity of a density operator is $\gamma =\mathrm{Tr}({\rho }^2)$. It equals one exactly for a pure state; in finite dimension $d$ it lies between $1/d$ and $1$. The minimum occurs exactly at the maximally mixed state $I/d$.

A worked qubit family interpolates between the mixed state $I/2$ and the pure state ${\rho }_{+}=|+⟩⟨+|$, where $|+⟩=(|0⟩+|1⟩)/\sqrt{2}$:The eigenvectors are $|\pm ⟩=(|0⟩\pm |1⟩)/\sqrt{2}$, with eigenvalues ${\lambda }_{\pm }=(1\pm p)/2$. Squaring and summing these eigenvalues givesHere $P_X(+)$ is the Born rule probability of the positive outcome in an $X$ projective measurement. The Bloch vector is $(p,0,0)$, since ${\rho }_p=(I+p{\sigma }_x)/2$ with ${\sigma }_x$ one of the Pauli matrices. At $p=0$ the purity is $1/2$; at $p=1/2$ it is $5/8$, with $P_X(+)=3/4$; at $p=1$ it is one. Within this family, $p=1$ is a pure state and every $p<1$ is a mixed state. Extending the formula to $p>1$ produces a negative eigenvalue $(1-p)/2$, so it ceases to be a density operator despite remaining Hermitian with trace one.

The Bloch vector length measures polarization: for any qubit density operator, $\gamma =(1+|\mathbf{r}{|}^2)/2$. This connects the interior of the Bloch ball with mixed states and its boundary, the Bloch sphere, with pure states. ^[ref]

For a short proof of the general bound, use the spectral theorem: the nonnegative eigenvalues ${\lambda }_j$ sum to one, so $\gamma ={\sum }_j{\lambda }_j^2$. ThenThe lower bound is attained only when all eigenvalues equal $1/d$; the upper bound is attained only when exactly one eigenvalue is nonzero, and hence equals one. Purity is therefore a simple quantitative test of mixedness, invariant under a unitary operator. It does not specify the entire density operator: even distinct pure states all have purity one. ^[ref]

A quantum algorithm specifies quantum state preparation, a sequence of quantum gates and quantum measurements, and any classical computation controlling them. For a pure quantum state $|\psi ⟩={\sum }_x{\alpha }_x|x⟩$, computational-basis measurement returns $x$ with probability $|{\alpha }_x{|}^2$, as prescribed by the Born rule. David Deutsch formulated a universal quantum computer using unitary operators and measurement.^[ref]

For example, one Hadamard gate prepares $H|0⟩=(|0⟩+|1⟩)/\sqrt{2}$. Computational-basis measurement gives either bit with probability $1/2$. Applying another Hadamard gate before measuring instead gives $H^2|0⟩=|0⟩$: the amplitudes for outcome one cancel by quantum interference. This illustrates why manipulating probability amplitudes matters; merely preparing a superposition does not reveal every component.

An analysis of algorithms must state the counted resources: quantum gate count, number of qubits, quantum state preparation cost and repetitions needed for the required probability of a correct result. Quantum query complexity counts quantum oracle calls separately; one such call need not be one elementary quantum gate.

A quantum circuit specifies a sequence of operations on a quantum register, usually drawn with wires for qubits and boxes for quantum gates. Reading from input to output gives the order of composition; two gates acting on separate registers combine through a tensor product.

For example, a Hadamard gate on the first qubit followed by a controlled-NOT gate prepares $(|00⟩+|11⟩)/\sqrt{2}$ from $|00⟩$, making a Bell state. The circuit describes evolution of amplitudes, while the Born rule predicts the statistics of the final measurement.

General circuits can include preparations, measurements, discarded subsystems and classically controlled later operations, as well as unitary gates. Circuit size and depth measure different costs: many independent gates can occupy one layer, whereas a chain of dependent gates requires successive layers. Translating an ideal circuit into hardware also requires a noise and connectivity model.

Starting from $|000⟩$, apply a Hadamard gate to the first qubit and then controlled-NOT gates from the first qubit to each of the other two qubits.

Quantum communication transmits or distributes quantum states and the correlations they carry between separated systems. An optical photon's polarization or an excitation shared between stationary qubits can serve as a carrier. Loss and noise are represented by a quantum channel and can degrade the transmitted state or its entanglement.

Quantum teleportation transfers an unknown state using shared entanglement, a local measurement and a classical message. The classical message is necessary, so the protocol does not send usable information faster than light. The no-cloning theorem prevents repairing an arbitrary unknown state by simply making redundant copies, motivating quantum error correction and carefully designed communication protocols.

Physical Review A 57, 120 (1998); preprint cond-mat/9701055. Section II develops coupled single-electron spin qubits and electrically controlled exchange.

A quantum computer is a physical device that performs controlled quantum computation and extracts an output through quantum measurement. A programmable quantum computer combines physical qubits, control hardware and classical processing that specifies operations and interprets repeated measurements.

David P. DiVincenzo organized the central physical requirements into five capabilities: scalable and characterized qubits; initialization in a known quantum state; sufficiently long relevant decoherence times compared with operation times; a universal set of quantum gates; and quantum measurement of selected qubits. These requirements connect an abstract quantum circuit with an experiment: knowing the intended gate sequence is useful only if the apparatus can initialize, control and read the encoded system accurately.^[ref]

The encoding can use different physical systems. Two selected internal levels of a trapped ion, for example, can represent the basis states of a qubit. The remaining levels and environmental couplings then become part of the physical control problem.

Quantum computing processes quantum information through controlled physical transformations and quantum measurements. In the quantum circuit model, an initial quantum state evolves through quantum gates, and the Born rule predicts the final outcome probabilities. Quantum interference makes the relative quantum phases of amplitudes computationally useful. David Deutsch's 1985 paper developed a model of universal quantum computation.^[ref]

A one-qubit example shows why amplitudes matter. Starting from $|0⟩$, a Hadamard gate gives $(|0⟩+|1⟩)/\sqrt{2}$; a second Hadamard gate returns $|0⟩$. Insert a Pauli Z gate between the two Hadamard gates and the final state is instead $|1⟩$: the intermediate relative sign changes how amplitudes combine. Measuring the intermediate state alone would give equal probabilities in both cases. A useful quantum algorithm arranges such transformations to make information about the required answer accessible through the final quantum measurement.

A qubit depolarizing channel shrinks every Bloch vector by the same factor: ${\Lambda }_p(\rho )=p\rho +(1-p)I/2$. In this retention-parameter convention it is completely positive for $-1/3\le p\le 1$; the usual mixture of unchanged and fully randomized states uses $0\le p\le 1$.

Errors on a known subset of $m$ qubits are correctable by a distance-$d$ code whenever $m<d$. Every product $E_a^{†}{E}_b$ of errors supported on that subset still has support in the same subset, so its compression is scalar. The Knill--Laflamme condition therefore supplies one recovery for every channel on that subset. Equivalently, erasing the subset leaks no logical information to it: if a reference is maximally entangled with the encoded logical system, its joint density operator with the erased subsystem is a product state.

Quantum error correction protects a logical qubit by encoding its quantum state into a larger Hilbert space, where a specified family of physical errors can be diagnosed and reversed by a common recovery quantum channel. The requirement is to restore every encoded quantum superposition, including its relative phase, rather than only the computational basis states. A quantum error-correcting code specifies which errors have this protection; it does not promise recovery from every possible disturbance.

The beautiful idea is to measure relations between physical qubits that reveal an error syndrome while revealing nothing about the protected logical qubit. For example, the bit-flip repetition code compares neighboring qubits without reading their individual values. Its parity checks can locate one Pauli X gate error while leaving the unknown coefficients intact. Encoding is a linear isometry, consistent with the no-cloning theorem: redundancy in a shared quantum state supplies protection without making independent copies of the unknown input.

Peter W. Shor's 1995 paper ^[ref], published on 1 October, introduced the nine-qubit Shor code to reduce quantum decoherence in memory. Andrew M. Steane's ^[ref], published on 29 July 1996 and ^[ref], connected quantum error correction to classical linear codes; the seven-qubit Steane code protects against arbitrary errors on one physical qubit. These constructions protect relative phase as well as computational basis values, going beyond the elementary bit-flip repetition code.

For an authored account of the mechanism, see John Preskill's ^[ref], especially section 7.1 on collective parity measurements and section 7.9 on stabilizer codes. Andrew Steane's A tutorial on quantum error correction develops the distinction between correcting a stored quantum state and operating with imperfect quantum gates.