A defining feature of the classical bit is that, in the ideal case, reading it does not change it: a light switch stays in whatever position it was in, however many times it is checked. This measurement-without-disturbance property is one of the first things that stops holding once qubits are introduced.

Because a classical bit's value is always definite, it can also be copied perfectly: read it, and write the same value elsewhere, as many times as needed. Storage, retrieval, and duplication of classical information all rest on this definiteness.

The two special vectors \(|0\rangle\) and \(|1\rangle\) — the computational basis states — are only two points in a space that also contains every possible combination of them. A qubit's state is not "0 or 1, and we just don't know which"; it is one specific vector, which can be a combination — a superposition — of \(|0\rangle\) and \(|1\rangle\). The key contrast: a qubit's state lives in a continuous space, while any single measurement of it still returns one of only two classical outcomes.

Two further properties separate qubits from bits immediately: the full state cannot be extracted from a single measurement (only one bit of outcome is obtained, not the whole state), and an unknown qubit state cannot be copied the way a classical bit can. Both are resolved properly later in the module.

"Qubit" is an abstraction, not a piece of hardware: spins, photons, superconducting circuits, and trapped ions are all different physical containers for the same mathematical object.

Classical information is encoded in definite states — a bit is simply in one state or the other. Quantum information is encoded in quantum states, a richer mathematical object that assigns a complex number, called a probability amplitude, to each possible measurement outcome.

The state itself is not what is observed. It is a predictive description: it gives the probabilities of each outcome under measurement, but is not identical to any single measurement result. Two different-looking states can predict the same statistics for one measurement while disagreeing about others — the full state carries more information than any one measurement can reveal.

A classical probability is a direct statement about likelihood: "70% chance of this outcome." A quantum probability amplitude is a different kind of object — a complex number, with both a magnitude and a phase — and is not itself a probability.

Many different amplitude combinations can produce identical probabilities — phase can change without changing any outcome's likelihood, yet phase still matters, because it controls how states interfere. This is a genuinely new kind of uncertainty, not a relabeling of classical probability, and is revisited in Classical vs Quantum Uncertainty.

A qubit requires exactly two distinguishable quantum states to serve as its logical basis, \(|0\rangle\) and \(|1\rangle\). Any physical system that has (or can be restricted to) two such distinguishable states is a candidate qubit — and real physical systems often have many more than two energy levels available. Building a qubit typically means selecting two of those levels and agreeing to treat everything else as outside the computation.

An electron carries an intrinsic angular momentum called spin, and a spin measured along a given axis takes one of two quantized values — informally "spin-up" and "spin-down." These two states can be used directly as \(|0\rangle\) and \(|1\rangle\). The Stern–Gerlach experiment is the classic evidence that spin really is quantized rather than continuous, and modern spin-qubit devices manipulate and measure these states electromagnetically.

A photon's polarization — horizontal versus vertical — is another natural two-level system, and it maps directly onto \(|0\rangle\) and \(|1\rangle\). Photons can also encode a qubit using which path they travel or which time-bin they arrive in. Because photons travel easily through optical fiber and free space without needing to be trapped, photonic qubits are particularly well suited to quantum communication, where the qubit has to move between two distant parties.

Superconducting qubits are engineered circuits — not naturally occurring two-level systems — cooled to cryogenic temperatures so that quantized energy levels emerge in the circuit. The lowest two energy levels are chosen as \(|0\rangle\) and \(|1\rangle\), and the qubit is controlled with precisely timed microwave pulses. This is currently one of the leading platforms in large-scale quantum computing hardware.

A trapped-ion qubit encodes information in two discrete internal energy states of an individual atom, held in place by electromagnetic fields. Lasers are used both to control the ion's state and to read it out, and trapped-ion systems are known for very high-fidelity operations and measurements — the physical implementation is precise, even though the logical qubit underneath is the same two-dimensional object as every other physical platform used to realize a qubit.

A vector space allows vectors to be added together and scaled by numbers, always landing on another valid vector. Quantum mechanics requires the scalars to be complex numbers, not just real ones — this is necessary to represent phase, which is physically meaningful. Vector addition and complex scalar multiplication are the only two operations needed to construct every qubit state.

A quantum state is represented as a vector in this complex space, written using Dirac's ket notation \(|\psi\rangle\). This notation is deliberately abstract and does not reference any particular hardware.

The inner product \(\langle\varphi|\psi\rangle\) is the operation used to compare two states by measuring their overlap. Normalization, orthogonality, and measurement probabilities are all inner products in disguise.

This condition is not cosmetic: it is tied directly to total probability summing to one, a connection made explicit in Pure States.

Orthogonality has a strong physical meaning: orthogonal states are perfectly distinguishable by measurement, a much stronger statement than simply "different." Two non-orthogonal states can never be told apart with certainty by any single measurement, however cleverly it is chosen. Direct calculation confirms the computational basis states satisfy \(\langle 0|1\rangle = 0\) and \(\langle 0|0\rangle = \langle 1|1\rangle = 1\).

A basis is complete when every valid state in the space can be written as a linear combination of the basis vectors, with nothing left unreachable. This spanning property is what makes a basis usable as a coordinate system for the whole space, not just for some states within it.

Summing the basis states' own projections reconstructs the full identity: any state can be rebuilt exactly from its components along a complete basis. Physically, this relation underlies what it means for a set of measurement outcomes to be exhaustive — together, they account for the entire state space.

Basis states form a complete, orthonormal reference framework: a fixed set of directions against which every other state can be described as a linear combination. Choosing a basis turns an abstract vector into a set of concrete numbers, and connects a state description to actual measurement outcomes — the basis measured in determines which outcomes are possible.

\(|0\rangle\) is a valid quantum state in its own right — interpretable as the outcome "measured 0" — and this basis is a hardware-independent abstraction: the same \(|0\rangle\) regardless of whether the underlying qubit is a spin, a photon, or a superconducting circuit.

Orthonormality makes a basis convenient to work with: extracting a state's coordinates and computing measurement probabilities both reduce to simple inner products, with no cross-terms to track.

Within a fixed, chosen basis, this expansion is unique: exactly one set of coefficients produces a given state.

Because the computational basis vectors are the column vectors \((1,0)^T\) and \((0,1)^T\), any qubit state can be written directly as a column vector of its amplitudes:

\[|\psi\rangle = \alpha|0\rangle + \beta|1\rangle = \begin{pmatrix}\alpha\\\beta\end{pmatrix}.\]

This is the same object as the Dirac-notation ket, written in coordinate form — often more convenient for direct calculation.

The computational basis is one convenient choice among many possible orthonormal bases. Re-expressing a state in a different basis changes its coordinates through a unitary transformation (developed fully in Quantum Evolution; for now, basis changes preserve length and orthogonality).

Pure single-qubit states form an infinite continuum — \(\alpha\) and \(\beta\) can be dialed to a continuum of values, subject only to the normalization condition. Yet a measurement of that qubit produces a discrete result: for a computational-basis measurement, either "0" or "1" and nothing in between. The state space and the outcome space are genuinely different objects, and keeping that distinction clear is essential for everything that follows in Superposition.

A quantum state contains complete predictive information about a system — everything there is to know, in the sense of predicting future measurement statistics. But a measurement reveals only limited information about that state: one measurement gives one outcome, not the full vector. It is, in fact, impossible to determine an arbitrary unknown state exactly from a single copy of it — there's a real gap between the state as a mathematical description and what's actually accessible through measurement. This gap reappears as the central idea of the No-Cloning Theorem.

A quantum state is the complete mathematical description required to predict the probabilities of measurement outcomes for a physical system. Its amplitudes encode that predictive information directly — this is the operational role the state plays.

A state predicts a distribution over possible outcomes, not a single definite result. This distinction matters: knowing the state exactly still doesn't tell you what any individual measurement will give — only what the statistics will look like if you repeat the same preparation and measurement many times.

Two broad ways of thinking about what a state "is" show up in discussions of quantum theory. The operational view treats the state purely as a mathematical tool for predicting measurement outcomes — a compact record of how a system was prepared and how it will behave under measurement. A more ontic view treats the state as a description of physical reality itself. Quantum theory does not by itself settle this interpretational debate; what matters practically is that the state is, at minimum, a complete carrier of information about possible future observations.

(The contrast with mixed states, which carry classical uncertainty on top of quantum uncertainty, is developed fully in Open Quantum Systems.)

This is the validity condition for a quantum state: total probability must sum to one, and normalization is exactly the algebraic statement of that fact.

This postulate ties together the vector-space structure, the inner product and norm, the computational basis, basis expansion, and normalization into a single formal statement of what a valid quantum state is. Every physical prediction about an isolated qubit follows from applying it correctly.

Given any two valid states, you can construct a new one through vector addition and scalar multiplication, so long as the result is renormalized afterward. This is exactly the same vector-space machinery from Hilbert Space, now used constructively: it's the recipe for building new quantum states out of existing ones.

Quantum evolution is linear: the physical operators that evolve quantum states preserve superpositions rather than collapsing or destroying them. This isn't a side detail: linearity is the single mathematical fact the proof of the No-Cloning Theorem rests on.

This is not a special or exotic case — it is the general form of a single-qubit state, of which \(|0\rangle\) and \(|1\rangle\) are just two particular examples (\(\alpha=1,\beta=0\) and \(\alpha=0,\beta=1\)).

Each coefficient in a state's basis expansion is a complex amplitude, with both a magnitude and a phase. Geometrically, an amplitude can be drawn as a point in the complex plane, at some distance from the origin (its magnitude) and some angle (its phase). These amplitudes are exactly the coefficients \(\alpha\) and \(\beta\) already introduced, and reading them correctly means treating magnitude and phase as distinct pieces of physical information.

Amplitudes are not directly observable — no experiment measures \(\alpha\) or \(\beta\) themselves. What's observable are probabilities, obtained from amplitudes via the Born rule, covered next.

The Born rule is a statement about statistics, not about any one trial. If you prepare many identical copies of the same state and measure each one, the frequency of each outcome converges to the Born-rule probability as the number of trials grows — this is the same frequentist logic used to check any probabilistic prediction experimentally, and it's how quantum probabilities are actually verified in a lab.

The same physical state can be a basis state in one basis and a full superposition in another. \(|+\rangle = \dfrac{|0\rangle+|1\rangle}{\sqrt2}\) is an equal superposition in the computational (Z) basis — but in the X basis, where \(|+\rangle\) and \(|-\rangle\) are themselves the basis vectors, it's simply one basis state, no superposition language required. "Superposition" describes a relationship between a state and a chosen basis, not a property some states have and others lack.

This is the same Change of Basis machinery from State Vector Formalism, now applied to a physical question rather than a purely mathematical one: the coordinate description changes with the basis, but the underlying physical state — and every prediction it makes — does not. Recognizing which parts of a description are coordinate artifacts and which are physically invariant is essential for interpreting any quantum calculation correctly.

Classical uncertainty comes from ignorance: a coin has already landed heads or tails, you simply haven't looked yet. The probability describes your incomplete knowledge of an already-definite fact.

Quantum uncertainty is different in kind: outcomes are intrinsically probabilistic, even when you have complete knowledge of the state. Knowing \(|\psi\rangle\) exactly — the maximal information a pure state provides — still doesn't tell you what a single measurement will yield if the state isn't already an eigenstate of that measurement. Uncertainty here isn't a gap in your knowledge; it's a feature of the physics itself.

A natural response to quantum uncertainty is to suspect it's secretly classical after all — that some hidden variable, not captured by \(|\psi\rangle\), already determines the outcome, and we just don't have access to it. The full Bell/CHSH argument against local hidden variables belongs to a later pillar. Hidden-variable theories run into real conceptual trouble under close examination, which is why the intrinsic nature of quantum uncertainty should be treated as an examined claim rather than an unexamined assertion.

This turns an abstract two-complex-number object into something that can be pointed at and rotated directly.

This mapping from amplitudes \((\alpha,\beta)\) to angles \((\theta,\varphi)\) is what lets every abstract object from State Vector Formalism — normalization, orthogonality, basis states — be read directly off the sphere's geometry.

\(|0\rangle\) sits at the north pole (\(\theta=0\)) and \(|1\rangle\) sits at the south pole (\(\theta=\pi\)). This is the geometric anchor for the whole sphere: everything else is located relative to these two fixed points.

States on the equator (\(\theta = \pi/2\)) are equal-magnitude superpositions of \(|0\rangle\) and \(|1\rangle\) — \(|+\rangle\), \(|-\rangle\), and every other equal-weight combination, distinguished from one another purely by their azimuthal angle \(\varphi\), i.e. their relative phase.

Orthogonal states — perfectly distinguishable, as established in Hilbert Space — sit at exactly antipodal points on the sphere, diametrically opposite each other. \(|0\rangle\) and \(|1\rangle\) are the clearest example, but any pair of antipodal points on the sphere represents an orthogonal pair.

This has no observable effect whatsoever, and correspondingly maps to the exact same point on the Bloch sphere. The Bloch sphere discards global phase by construction — a deliberate feature of the representation, not a gap in it.

The Bloch sphere represents pure single-qubit states only. It has no representation for mixed states — states that carry classical uncertainty on top of quantum uncertainty — which require the density-matrix formalism introduced in Open Quantum Systems.

Multi-qubit entangled states live in a composite Hilbert space that a single sphere simply cannot capture — there is no single-sphere picture for an entangled pair, because the correlations between qubits are exactly the information a per-qubit sphere would have to discard. The tensor-product formalism needed for multi-qubit systems is developed in Multi-Qubit Systems.

Classical information copies perfectly: read a bit, write the same value elsewhere, done. Quantum information does not share this property in general — an arbitrary unknown quantum state cannot be perfectly duplicated. The next concept proves exactly why, using nothing but linearity.

The precise claim is about unknown states specifically. If you already know what state you have, you can prepare as many fresh copies as you like — that's just state preparation, and it's unrestricted. What's impossible is taking a single copy of a state you don't already know and producing a second, independent copy of it.

Suppose a universal "cloning machine" exists: a unitary operator \(U\) (previewed here from Quantum Evolution — for now, all you need is that \(U\) is linear) acting on a qubit plus a blank "scratch" qubit, such that for every state \(|\psi\rangle\),

\[U\big(|\psi\rangle \otimes |0\rangle\big) = |\psi\rangle \otimes |\psi\rangle.\]

(The \(\otimes\) symbol is a tensor product, previewed here from Multi-Qubit Systems — for now, just read \(|\psi\rangle\otimes|0\rangle\) as "the state \(|\psi\rangle\) alongside a separate blank qubit.") By assumption, this cloner works correctly for the two basis states:

\[U(|0\rangle\otimes|0\rangle) = |0\rangle\otimes|0\rangle, \qquad U(|1\rangle\otimes|0\rangle) = |1\rangle\otimes|1\rangle.\]

Now test the same machine on \(|+\rangle = \dfrac{|0\rangle+|1\rangle}{\sqrt2}\), using nothing but the linearity of \(U\) — recall from Hilbert Space that this is exactly the property that lets an operator be pulled through a sum term by term:

\[U(|+\rangle\otimes|0\rangle) = \frac{1}{\sqrt2}\Big(U(|0\rangle\otimes|0\rangle) + U(|1\rangle\otimes|0\rangle)\Big) = \frac{1}{\sqrt2}\Big(|0\rangle\otimes|0\rangle + |1\rangle\otimes|1\rangle\Big).\]

But a cloning machine that actually worked on \(|+\rangle\) would have to produce \(|+\rangle\otimes|+\rangle\) instead:

\[|+\rangle\otimes|+\rangle = \frac{1}{2}\Big(|0\rangle|0\rangle + |0\rangle|1\rangle + |1\rangle|0\rangle + |1\rangle|1\rangle\Big).\]

These two results are not the same state — the linearity-derived output is missing the cross terms \(|0\rangle|1\rangle\) and \(|1\rangle|0\rangle\) entirely. A single unitary \(U\) cannot simultaneously clone \(|0\rangle\), \(|1\rangle\), and \(|+\rangle\) — contradiction. No universal cloning unitary exists, for any pair of non-identical, non-orthogonal states, not just this particular example.

Measuring an unknown state generally changes it, and a single measurement only reveals limited information — never enough to reconstruct the full original state. This was flagged, unresolved, back in Classical vs Quantum Information — this is where it's finally resolved: measurement cannot be used as a workaround for cloning, because the very act of measuring an unknown state destroys the information a clone would need to preserve.

Between the linearity argument and the measurement-disturbance argument, there's no loophole: measurement cannot replace cloning, and unitary evolution cannot achieve it either. This is a genuine information–disturbance trade-off, not a gap waiting to be closed by a cleverer protocol.

Because an eavesdropper cannot make a perfect, undetectable copy of an unknown quantum state in transit, any attempt to intercept quantum-encoded information necessarily disturbs it — and that disturbance is, in principle, detectable by the legitimate parties. This is the conceptual foundation of quantum key distribution (QKD): the no-cloning theorem is precisely what makes eavesdropping detection possible. The full protocol details belong to the Quantum Communication pillar; what matters here is the conceptual chain — a result about linearity and superposition, proved with nothing but algebra, turns out to guarantee the security of a real communication technology.