Advanced Qubit Notes

These notes start from one sharp question: a qubit’s state is parameterized by two complex numbers, so if you know $\alpha$, is $\beta$ determined? Following the logic of that question leads through an entire stretch of the single-qubit map: why phase is real physical information, how it takes physical form in polarized light, how qubits relate to hardware, the three properties of the Pauli matrices, rotation gates and circuit notation, and finally the formal mathematics of measurement. The first section reviews the prerequisites in one line apiece; everything after that is self-contained.

0. Prerequisites at a glance

Only the following facts are needed, each in one sentence:

• A complex number is an arrow: length times direction, $z = r\,e^{i\phi}$. Here $e^{i\phi}$ is the point on the unit circle at angle $\phi$; its modulus is always 1, so multiplying by it changes direction but never length.
• A qubit’s state: $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$, where $\alpha,\beta$ are complex numbers (called amplitudes), also writable as the column vector $\binom{\alpha}{\beta}$.
• The Born rule: measuring in the 0/1 basis gives 0 with probability $|\alpha|^2$ and 1 with probability $|\beta|^2$; the normalization condition is $|\alpha|^2+|\beta|^2=1$.
• Bras and inner products: $\langle\psi| = (\alpha^*\;\;\beta^*)$, obtained from the ket by transposing and conjugating (the combined operation is written $\dagger$, dagger); $\langle\phi|\psi\rangle$ is a row times a column, and the result is one complex number.
• The standard superposition states: $|\pm\rangle = \tfrac{1}{\sqrt2}(|0\rangle\pm|1\rangle)$, orthogonal to each other.
• The Bloch sphere: the map of single-qubit states. $|\psi\rangle = \cos\frac{\theta}{2}|0\rangle + e^{i\phi}\sin\frac{\theta}{2}|1\rangle$, where the latitude $\theta$ governs probabilities and the longitude $\phi$ governs phase; two orthogonal states sit at antipodal points (directly opposite), not at right angles.

1. The seed question: knowing $\alpha$, can you determine $\beta$?

The answer splits in two. Normalization pins down the length:

But it cannot pin down the direction. There are infinitely many complex numbers of length $\sqrt{1-|\alpha|^2}$, a full circle of them:

This free $\phi$ is the phase, and it is real, measurable physical information. A counterexample is the most convincing argument: fix $\alpha = \tfrac{1}{\sqrt2}$ (so $|\beta| = \tfrac{1}{\sqrt2}$ is also fixed) and change only $\phi$:

Same $\alpha$, same $|\beta|$, yet different $\phi$ means entirely different points on the equator. $|+\rangle$ and $|-\rangle$ are even orthogonal to each other, two states that can be distinguished perfectly. Measure in another basis, or pass the state through a gate, and the outcomes differ completely.

In the two-arrows picture: $\alpha$ and $\beta$ are each an arrow (length plus direction). Physically there is no absolute phase, just as there is no absolute position; saying ”$\beta$ points at $30°$” means nothing without a reference, and the only reference in sight is the other arrow, $\alpha$. So exactly one quantity carries physical meaning: the angle between the two arrows.

• Rotate both together (global phase): the angle is unchanged, no experiment can detect it, it means nothing;
• Rotate only one (relative phase): the angle changes, and both interference and measurements in other bases can see it.

An analogy: tilt the whole clock by $30°$ and the angle between the hour and minute hands is unchanged, so the time is unchanged; move only the minute hand by $30°$ and the time has changed. The convention “take $\alpha$ real” is hanging the clock straight: point $\alpha$ at $0°$, and all phase information compresses into a single angle $\phi$.

The bookkeeping of degrees of freedom now matches the Bloch sphere: two complex numbers carry 4 real parameters, normalization eats one, global phase eats another, and 2 remain, exactly $(\theta,\phi)$. Knowing $\alpha$ gives you only the latitude; the longitude is an independent second piece of information.

2. Where the phase hides: why $\alpha|0\rangle+\beta|1\rangle$

A common confusion: “I don’t see any phase in this formula.” Answer: the phase has been there all along, packed inside the complex numbers. Every complex number carries a direction by birth, so writing $\alpha|0\rangle+\beta|1\rangle$ actually writes down four pieces of information:

The lengths $r_\alpha, r_\beta$ govern probabilities; the directions $\phi_\alpha,\phi_\beta$ are the phases. The $\alpha\beta$ notation is the packed form (easy to write and compute with); the Bloch form $\cos\frac\theta2|0\rangle + e^{i\phi}\sin\frac\theta2|1\rangle$ is the unpacked form (the redundant global phase already discarded, the one physical phase $\phi$ laid out in the open). They are two notations for the same thing.

Why is “a complex two-dimensional unit vector” the right container for a state? Because it is the least common multiple of three experimental facts:

1. Quantum states superpose (diagonal polarization really exists), so states need an additive structure: linear combinations of basis states;
2. Quantum states interfere (two paths can cancel), so coefficients must carry direction. Real numbers cannot cancel naturally; complex numbers do it exactly: $1 + e^{i\pi} = 0$;
3. Measurement yields probabilities, so the squared modulus of a coefficient serves as probability, $|\alpha|^2+|\beta|^2=1$ is precisely “the vector has length 1”, and since quantum operations are all length-preserving rotations, conservation of probability comes for free.

Addable, direction-carrying, squared-modulus-as-probability: exactly one object satisfies all three at once, the unit-length complex two-dimensional vector.

3. The physical incarnation of phase: polarized light

3.1 What light is, what polarization is

Light is an electromagnetic wave: an electric field oscillating as it travels, with the oscillation perpendicular to the direction of travel. The rope analogy: the rope stretches away from you (the direction of travel), and your hand can shake it up and down, side to side, or at a slant; the wave runs forward either way, but the direction of shaking differs. That direction is the polarization: horizontal oscillation is $|H\rangle$, vertical is $|V\rangle$. Looking into the beam, the tip of the electric field traces a line segment back and forth, and the tilt of that line is the polarization angle.

3.2 45° polarization = vector decomposition = superposition demystified

What is the electric field of light oscillating at $45°$? Middle-school vector decomposition:

that is, an equal horizontal component plus an equal vertical component. Translated into quantum notation (with $\cos45° = \tfrac{1}{\sqrt2}$):

“Superposition state” sounds mystical, but for polarization it is just vector decomposition. The $45°$ field is really and simultaneously composed of a horizontal oscillation and a vertical one, and the coefficient $\tfrac{1}{\sqrt2}$ is not an abstract symbol but the length of a geometric projection. This also answers whether $|D\rangle$ is “secretly already H or V”: neither. It is one definite vector pointing at $45°$, and asking “but is it really horizontal or vertical” is like asking “is northeast really east or north”. The question itself is malformed.

3.3 Arbitrary angles and Malus’s law

A photon polarized at physical angle $\theta$:

Example: at $30°$ polarization, $P(H) = \cos^2 30° = 0.75$ and $P(V) = 0.25$. Note that $\theta$ and $\theta+180°$ are the same polarization (the field oscillates back and forth anyway), so the distinct linear polarizations live in $[0°,180°)$.

A polarizer passes only the component along its transmission axis. Classical optics gives Malus’s law (1809): transmitted intensity $I = I_0\cos^2(\Delta\theta)$. That $\cos^2$ and the Born rule’s $\cos^2$ are the same one: at the single-photon level a photon is indivisible, so it either passes whole (with probability $\cos^2\Delta\theta$) or is absorbed whole, and classical light is the statistical average over myriad photons. Malus’s law is the classical shadow of the Born rule; it is just that in 1809 nobody knew photons were hiding in the denominator.

3.4 The “angle doubling” mystery, resolved physically

Compare with the Bloch formula $\cos(\theta_B/2)|0\rangle+\cdots$: the physical polarization angle plays exactly the role of $\theta_B/2$:

Half a physical turn sweeps a full circle on the sphere: one-to-one, nothing repeated, nothing missed. The design decision “why $\theta/2$” acquires a visible incarnation in polarization: H and V differ physically by only $90°$, yet they are perfectly distinguishable (orthogonal), so the map must draw them as the two poles.

3.5 Circular polarization: the physical identity of $i$

So far the field has oscillated along a line. If the H component and the V component fall out of step, offset by a quarter of a period, the field tip no longer traces a line but draws a circle: circular polarization. “Delayed by a quarter period” translates into complex language as multiplication by $i$ (since $i = e^{i\pi/2}$, a $90°$ turn):

These are the two ends of the Bloch sphere’s $y$ axis (and the eigenvectors of Pauli $Y$; see §5). The sphere is now completely filled by polarization: the $x$–$z$ great circle holds all linear polarizations (real coefficients), the two ends of the $y$ axis are left and right circular polarization, and every remaining point is elliptical. Here ”$i$ means rotate by $90°$” stops being a metaphor: it is literally a quarter-period delay, implemented in the lab by a quarter-wave plate.

3.6 An experiment you can do by hand: three polarizers

1. Light passes an H polarizer, so everything that exits is $|H\rangle$;
2. Add a V polarizer after it: $\langle V|H\rangle = 0$, total darkness, as expected;
3. Insert a $45°$ polarizer between them, and light comes through (about $1/4$).

More obstacles, more light? Because the middle sheet is a measurement: $|H\rangle$ hits the $45°$ sheet and passes with probability $\cos^2 45° = \tfrac12$, and having passed, its state is projected onto $|D\rangle$; then $|D\rangle$ hits the V sheet, another $\tfrac12$. Total $\tfrac14$. This is the tabletop demonstration of “measuring in the wrong basis changes the state”. Quantum key distribution (the BB84 protocol) catches eavesdroppers with exactly this law of physics, and three lenses from polarized sunglasses let you watch it happen.

4. A qubit is not a photon: three layers of abstraction

A natural misconception: “qubit = photon.” The best correction starts with the classical question: what is a bit made of? In a CPU it is a voltage, on a hard drive the orientation of a magnetic domain, on an optical disc a pit, on a punched card a hole. The carriers could hardly differ more, and algorithms care about none of it. “Bit” is an abstract unit of information; the voltages are merely its physical implementations.

A qubit is the same story, quantum version. Its definition: any two-level quantum system. Whenever there are two mutually orthogonal (perfectly distinguishable) quantum states that can be prepared, manipulated, and measured, you have a qubit. $|0\rangle,|1\rangle$ are logical labels that deliberately say nothing about hardware. The correct picture has three layers:

1. The logical layer: the qubit itself. $|0\rangle/|1\rangle$, algorithms, protocols, and all the mathematics in these notes live here;
2. The degree-of-freedom layer: the particular pair of orthogonal states chosen to carry the information — a photon’s polarization, an electron’s spin ($|{\uparrow}\rangle=|0\rangle,\ |{\downarrow}\rangle=|1\rangle$), an atom’s energy levels (ground/excited), a superconducting circuit’s current states;
3. The carrier layer: the physical entities, photons, electrons, atoms, circuits. One entity can offer several degrees of freedom, hence several potential qubits.

Strictly, even “a photon’s polarization is a qubit” is half a step off; the right sentence is “a photon’s polarization implements a qubit”, just as “the voltage on this wire implements a bit”. This layering is why quantum computing works as an engineering discipline: theorists design algorithms at the logical layer, experimentalists swap hardware at the carrier layer, and the interface contract is “two orthogonal states, unitary operations, measurement”.

Two practical footnotes. First, labels collide: a single photon has polarization, path, arrival time, photon number, several degrees of freedom, each encodable. In polarization encoding $|0\rangle=|H\rangle$; in photon-number encoding $|0\rangle$ is the vacuum (no photon present) and $|1\rangle$ means one photon. Identical symbols on paper, entirely different physics. The first thing to do with an experimental paper is find the authors’ definition of the encoding. Labels are cheap; always ask what physical states they name. Second, for a spin-$\tfrac12$ particle the Bloch sphere almost stops being an abstract map: the arrow’s direction on the sphere really is the direction the spin magnetic moment points in laboratory space. That is also the context in which the sphere was invented (Felix Bloch, studying spins in nuclear magnetic resonance).

5. Pauli matrices: one set of matrices, three properties

The three protagonists:

They are gates: they act by matrix-times-vector. $X$ swaps the two components ($X|0\rangle=|1\rangle$, the quantum NOT); $Z$ flips the sign of the second component; $Y$ does both and throws in an $i$. Their geometric identity: each is a $180°$ rotation of the Bloch sphere about the $x$, $y$, or $z$ axis respectively.

5.1 Property one: self-inverse

“Doing it twice equals doing nothing”: rotating $180°$ twice is a full turn, just like classical NOT. It looks trivial and is the hidden workhorse in several places: it forces the eigenvalues to satisfy $\lambda^2=1$ (see 5.3); it collapses the rotation operator’s power series into $\cos$ and $\sin$ (see §6); and it will appear again in the proof of the uncertainty principle (see 5.4).

5.2 Property two: both Hermitian and unitary — two credentials

Two definitions, both built from $\dagger$:

• Unitary: $U^\dagger U = I$. These matrices preserve length ($\|Uv\| = \|v\|$, provable in two lines). Physical meaning: states must stay normalized and probabilities must keep summing to 1, so the legal quantum gates are exactly the unitary matrices — the license to be a gate.
• Hermitian: $A^\dagger = A$. These matrices have all-real eigenvalues. Physical meaning: measurement readouts must be real numbers, so observables are represented by Hermitian matrices — the license to be an instrument (details in §8).

The two credentials usually cannot be held at once (a rotation gate $R_z(\theta)$ is unitary but generally not Hermitian, so it can only be a gate). The Paulis hold both, and the reason is property one: Hermitian ($P^\dagger = P$) plus self-inverse ($P^2=I$) immediately gives $P^\dagger P = P^2 = I$. Consequence: the same matrix $Z$ is a gate inside a circuit box and an instrument in the sentence “measure $Z$”. Beginners trip hardest over “measuring a matrix”; it refers to the second identity.

5.3 Eigenvectors, and eigenvalues $\pm1$

Definition: if $Mv = \lambda v$ — the matrix leaves the direction unchanged and only multiplies by a number — then $v$ is an eigenvector of $M$ and the factor $\lambda$ is its eigenvalue. Plainly: eigenvectors are the directions the matrix cannot move, and the eigenvalue is the only thing it can do in that direction (stretch or flip).

For the Paulis the factor takes only two values. Verify $Z$:

$X$ swaps components: the two components of $|+\rangle$ are equal, so swapping changes nothing ($+1$); $|-\rangle$ picks up an overall minus sign ($-1$). The complete dictionary:

Why the factor can only be $\pm1$: self-inverse forces $\lambda^2=1$; Hermitian forces $\lambda$ real. The list is locked.

Why the eigenvectors sit exactly on the corresponding axis: a Pauli gate is a $180°$ rotation about its axis. When a globe spins about its north–south axis, the only points that do not move are the two on the axis, the north and south poles. “Direction unchanged under the action” equals “position fixed under the rotation” equals being on the rotation axis. So each row’s eigenvectors are precisely the two endpoints of that axis, and an old rule gets confirmed in passing: the two ends of any diameter are a pair of orthogonal (perfectly distinguishable) states.

The two physical identities of $\pm1$ (matching the matrix’s two roles):

• As a gate, $-1$ is a phase. $Z|1\rangle = -|1\rangle$: on $|1\rangle$ alone the $-1$ is a global phase, no motion on the sphere; but inside a superposition, the two components pick up different factors ($+1$ versus $-1$), which is a relative phase: $Z|+\rangle = \tfrac{1}{\sqrt2}(|0\rangle - |1\rangle) = |-\rangle$, a $180°$ turn along the equator, in agreement with ”$Z$ = half-turn about the $z$ axis”.
• As an instrument, $\pm1$ are the dial readings. The measurement rules (§8) say readouts must be eigenvalues, so “measuring $Z$” outputs $+1$ or $-1$: reading $+1$ means the state collapsed to $|0\rangle$, reading $-1$ means it collapsed to $|1\rangle$. Choosing $\pm1$ rather than $0/1$ has a convenience: the expectation value $\langle Z\rangle = P(+1)-P(-1)$ expresses “which way the state leans” in a single number.

5.4 Property three: cyclic non-commutation — the algebraic fingerprint of quantum weirdness

The commutator is defined as

It measures how much the order of operations matters: $[A,B]=0$ means order is irrelevant (they commute); otherwise order counts. Ordinary numbers always commute; matrices generally do not. Compute one by hand:

The three relations rotate cyclically (advance the letters $X\to Y\to Z\to X$; remember one, turn the crank twice):

The structure is identical to the vector cross product $\hat x\times\hat y = \hat z$, and that is no coincidence: it is the algebra of three-dimensional rotations. And rotations are inherently non-commuting: take your phone, flip it $90°$ about the horizontal axis then $90°$ about the vertical axis, and compare with the reverse order; the final orientations differ. Pauli non-commutation is that everyday geometric fact in matrix form.

The main event: non-commutation = the uncertainty principle. The notebook version of the statement is that the observables which fail to commute are exactly the ones that cannot be known simultaneously, and it can be proven in five lines with tools you now fully possess.

Step one, translate “known simultaneously” into mathematics: “the state has a definite value of observable $A$” means the outcome of measuring $A$ is 100% certain, which means the state is an eigenvector of $A$. So “definite $Z$ and definite $X$ at once” means a common eigenvector exists.

Step two, prove none exists, by contradiction. Suppose a nonzero $v$ satisfies both $Zv=\lambda v$ and $Xv=\mu v$ (with $\lambda,\mu=\pm1$). Then

(numbers commute, so the two terms cancel). But $[Z,X]=2iY$, hence $2iYv=0$, that is, $Yv=0$. And property one says $Y^2=I$: $Y$ is invertible, and an invertible matrix sends only the zero vector to zero, so $v=0$, contradiction. $\blacksquare$

Conclusion: $Z$ and $X$ share no eigenvector at all. Not “hard to know both at once”: there exists no state that is definite for both. All three properties appear within the five lines: the commutator supplies $2iY$, self-inverseness makes $Y$ invertible, and “definite value = eigenvector” rests on Hermiticity.

Step three, the physical face. $|0\rangle$ is an eigenvector of $Z$ (its $Z$ value is definite), but measure $X$: $P(\pm) = |\langle\pm|0\rangle|^2 = \tfrac12$ — maximal uncertainty. On the Bloch sphere it is obvious: knowing $Z$ means the arrow lies on the $z$ axis, knowing $X$ means it lies on the $x$ axis, and one arrow cannot lie on two perpendicular axes at once — a seesaw whose two ends cannot both be up. One more layer deserves puncturing: this is not “the $X$ value exists but we are ignorant of it”. For a qubit in $|0\rangle$, the $X$ value is simply not defined (like asking whether northeast is really east or north); and this is experimentally distinguishable from classical ignorance — distinguishing exactly that is what Bell-inequality experiments do.

Step four, recognize an old acquaintance. Heisenberg’s $\Delta x\,\Delta p \ge \hbar/2$ starts from the same algebra: $[\hat x,\hat p] = i\hbar$. The quantitative general form (the Robertson inequality):

The product of two uncertainties is bounded below by the commutator. Classical physics commutes everywhere, so everything can be known at once; all of quantum “weirdness” grows out of the single algebraic fact that matrix multiplication does not commute. A cryptographic application: BB84’s two encoding bases are precisely the eigenvectors of $Z$ and $X$, and $[Z,X]\ne0$ means no common eigenvector, which means no measurement an eavesdropper can make reads both bases reliably — the five-line mathematical backbone of the protocol’s security.

6. Rotation gates: Euler’s formula in matrix form

Pauli gates are “violent” ($X$ teleports the north pole to the south pole). To move continuously on the Bloch sphere, feed a Pauli into the matrix exponential:

where $P_a$ is the Pauli of the rotation axis (rotating about $x$ uses $X$, and so on) and $\theta$ is the angle you want to turn on the sphere. The left side is the definition, the right side the computed closed form; the step between them is this section’s main course.

6.1 The matrix exponential: an old formula with a widened domain

The Taylor series $e^x = 1 + x + \frac{x^2}{2!} + \cdots$ uses only multiplication and addition, both of which matrices can do. So define directly:

6.2 The series splits itself into two piles

Substitute $A = -\tfrac{i\theta}{2}P$; the $n$-th term is $\frac{1}{n!}\big({-}\tfrac{i\theta}{2}\big)^n P^n$. Two cycles run simultaneously: the powers of $(-i)$ cycle with period four ($-i,-1,+i,+1$), and the powers of $P$ cycle with period two ($P^2=I$ enters: $P, I, P, I,\ldots$). So every even term carries $I$ and every odd term carries $P$, and the series groups into two piles:

The two brackets are exactly the Taylor series of $\cos(\theta/2)$ and $\sin(\theta/2)$. Closed form obtained. The derivation parallels the derivation of Euler’s formula word for word: there $i^2=-1$ splits the series, here $P^2=I$ does the same job:

One observation runs deeper: $(-iP)^2 = (-i)^2P^2 = -I$ — a square equal to $-I$, which is the matrix version of the defining property of $i$ (namely $i^2=-1$). $-iX, -iY, -iZ$ are three different “matrix versions of $i$”, one per rotation axis; the rotation formula is Euler’s formula replayed with a new engine.

6.3 Verification and use

Sanity checks at special angles: $\theta=0$ gives $I$ (no rotation) ✓; $\theta=\pi$ gives $-iP$, which up to a global phase is the Pauli gate itself, so “Pauli = $180°$ rotation about its axis” turns from slogan into corollary ✓; $\theta=2\pi$ gives $-I$ (see 6.4).

The explicit matrix of $R_z$ (running Euler’s formula backwards to fold the diagonal entries into exponentials):

(the last step factors out a global phase and discards it). $R_z$ adds $\theta$ to the relative phase and leaves the probabilities untouched — it is the knob that turns the angle between the two arrows; changing probabilities takes $R_x$ or $R_y$. Verify one more:

North pole to equator ✓. Unitarity in one line: dagger turns $-i$ into $+i$, so $R_a(\theta)^\dagger = R_a(-\theta)$, hence $R_a(\theta)^\dagger R_a(\theta) = R_a(0) = I$ — “rotating back” is the inverse.

Why $\theta/2$: the sphere runs at double speed (third appearance in these notes). The matrix lives in state space, the sphere is a map, map angle = state-space angle × 2, so to turn the map by $\theta$ the matrix must contain $\theta/2$. Why the minus sign: a convention fixing the positive rotation direction (right-hand rule), inherited from the solution of the Schrödinger equation, $e^{-iHt/\hbar}$.

6.4 A full 360° turn is not “doing nothing”

Substitute $\theta=2\pi$: $R_a(2\pi) = \cos(\pi)I = -I$. The arrow on the sphere returns home, but the state has been multiplied by $-1$; to truly return home mathematically takes $\theta = 4\pi$, that is, $720°$.

An apparent contradiction arrives: isn’t $-1$ a global phase, the thing that cannot be measured? The resolution is exactly §1’s principle: multiplying the whole state means nothing; multiplying one branch means everything. The trick: do not rotate the entire system, rotate only one branch of a superposition. Implement it with an interferometer — split one particle into a superposition of two paths and rotate the spin by $2\pi$ on the lower path only:

The $-1$ hangs on one branch only and becomes a relative phase of $\pi$: at recombination the interference flips from constructive to destructive, the fringes invert, and the detectors see it. This is no thought experiment: in 1975 the groups of Rauch and of Werner each did it with neutron interferometers — a silicon crystal splits the neutron beam, a magnetic field precesses the spin on one arm, the fringes cycle with period $4\pi$ in the rotation angle, and a $360°$ turn lands exactly in antiphase. All spin-$\tfrac12$ particles (electrons, protons, neutrons) behave this way; the technical term is spinor.

A shadow of it exists in daily life: the plate trick — hold a plate palm-up and rotate your arm $360°$, and your arm ends up twisted; continue another $360°$ in the same direction and the arm untwists by itself, $720°$ total to return to the start. The mathematical root is a group-theoretic fact: $SU(2)$ is the double cover of the rotation group $SO(3)$ — every rotation of the sphere corresponds to two matrices $\pm U$, and that $\pm$ is the shared ID card behind the $\theta/2$ mystery, the $2\pi\to-1$ phenomenon, and the plate trick.

6.5 The hardware identity

In the lab this formula is not abstract notation: hit a superconducting qubit with a microwave pulse, or an ion with a laser, and the hardware is solving the Schrödinger evolution $e^{-iHt/\hbar}$; when the Hamiltonian is proportional to some Pauli, the evolution is exactly $R_a(\theta)$, and $\theta$ is proportional to the pulse duration. “Calibrating a gate” is, to a large extent, tuning the pulse length until $\theta$ hits the target angle.

7. Circuit diagrams and the unitary world

7.1 Circuit notation: time reads left to right, matrices right to left

Three rules for reading a circuit: each horizontal wire is one qubit (its time axis, not a spatial wire); each box is a gate; left to right = earlier to later. The trap is converting to matrices: applying matrices is function composition, the first gate hugs $|\psi\rangle$ and later gates wrap around the outside, so a matrix string reads right to left:

Walk $|0\rangle$ through it and see how fatal reading the wrong way is:

• Correct ($X$ first): $X|0\rangle = |1\rangle$, then $H|1\rangle = |-\rangle$. Output $|-\rangle$.
• Reversed ($H$ first): $H|0\rangle = |+\rangle$, then $X|+\rangle = |+\rangle$ (as $|+\rangle$ is the $+1$ eigenvector of $X$, it does not move). Output $|+\rangle$.

$|-\rangle$ and $|+\rangle$ are orthogonal — as wrong as wrong gets. The algebraic reason order matters: $H$ and $X$ do not commute. The mnemonic: the state is fed into the matrix string from the right, and whatever it hits first acts first. ($H$ is the Hadamard gate, $H|0\rangle=|+\rangle,\ H|1\rangle=|-\rangle$: the standard tool for creating superposition and realizing interference.)

7.2 The complete roster of gates = the unitary matrices

Fact: the legal single-qubit operations are exactly the $2\times2$ unitary matrices. “Exactly” is a two-way promise. (⟹) They must be unitary: probability has to survive, and the unitaries are all of the length-preservers. (⟸) Unitary suffices: any $2\times2$ unitary decomposes into three rotations ($U = e^{i\delta}R_z(\gamma)R_y(\beta)R_z(\alpha)$), and rotations are just pulses. The roster is neither more nor less.

7.3 Closed versus open: where unitarity applies

• Closed system: zero contact with the outside — the Schrödinger equation takes over and the evolution is always unitary;
• Open system: the environment gets involved — non-unitary things start happening, and measurement is the most extreme example (random, irreversible, superposition-destroying).

An ideal quantum computer stays closed between measurements, unitary throughout, with measurement as the single sanctioned exit. This also explains hardware’s enemy number one, decoherence: any “peek” by the environment is an uninvited little measurement, quietly collapsing your superposition.

7.4 Bonus one: every gate can be undone

Unitary implies invertible, and the antidote is dagger: $U^\dagger U|\psi\rangle = |\psi\rangle$; undoing a stretch of circuit means running it backwards with every box daggered. Contrast the classical world: an AND gate takes two bits in and puts one bit out — seeing output 0, you cannot tell whether the input was 00, 01, or 10; information is destroyed, irreversibly. Quantum circuits have no destruction option (before measurement everything is unitary), so compiling classical logic into a quantum circuit requires reversibilization first (introducing reversible gates such as the Toffoli). This is not philosophical trivia; it is a design constraint you actually hit when writing circuits.

7.5 Bonus two: any point to any point on the sphere

Unitaries preserve inner products, hence the “angle” between any two states, hence they act on the Bloch sphere as rigid rotations, able to carry any point to any other point. This is the license on the first line of every algorithm: hardware natively prepares only $|0\rangle$, but any desired initial state is just ”$|0\rangle$ plus a suitable $U$“.

8. Measurement: the mathematical identity of the instrument

8.1 Observables: one Hermitian matrix per instrument

Postulate: every measurable physical quantity (an “observable”: energy, spin along some direction, polarization at some angle…) corresponds to a Hermitian matrix $A$. Here $A$ is a placeholder for “whatever instrument”; you already know three concrete members: $Z, X, Y$. The dictionary between matrix and instrument has exactly two entries:

• Eigenvalues = the list of possible readings on the dial (Hermiticity guarantees they are all real);
• Eigenvectors = the states for which each reading is certain, and also where the system lands after the measurement.

The three clauses of measuring $A$: the reading is always one of $A$‘s eigenvalues; the state projects (collapses) onto the eigenvector of that reading; which reading occurs is random, with probabilities determined by the state (Born). Corollary: only states already sitting on an eigenvector give predictable outcomes; for every other state, no one can predict a single shot.

Readings are not always $\pm1$: the energy observable is the Hamiltonian matrix $\hat H$, whose eigenvalues are the energies of the levels (arbitrary real numbers) and whose eigenvectors are the levels themselves — the same clauses, word for word.

8.2 The key clarification: measurement is not computing $A|\psi\rangle$

A misconception that almost everyone produces: “multiply the matrix into the state and get a classical bit.” No. Demonstrate why the “hard multiply” must be wrong: take $\hat H = \mathrm{diag}(2,5)$ (eigenvalues 2 and 5) and the state $|\psi\rangle = \tfrac{1}{\sqrt2}(|0\rangle+|1\rangle)$, and actually multiply:

Three sins: the squared length is $(4+25)/2 = 14.5 \ne 1$, not a legal quantum state; no number was output; and matrix multiplication is fully deterministic, while real measurement is random. The real outcome: with probability 50% read 2 and the state becomes $|0\rangle$; with probability 50% read 5 and the state becomes $|1\rangle$ — bearing no resemblance to the product.

8.3 The correct picture: a codebook plus dice

The true identity of $A$ is a package of two things: an orthogonal basis (the eigenvectors) plus a reading label glued to each basis state (the eigenvalues). The actual procedure of measurement has four steps, and multiplication never appears:

1. Consult the codebook: solve for $A$‘s eigenvectors and eigenvalues (done once, when the instrument was designed);
2. Decompose: expand $|\psi\rangle$ in the eigenbasis, $|\psi\rangle = c_1|v_1\rangle + c_2|v_2\rangle$;
3. Roll the dice: select the outcome by the Born rule, reading $\lambda_i$ with probability $|c_i|^2$;
4. Collapse: the state jumps to the winning $|v_i\rangle$.

The physical process is carried out by apparatus (beam splitters, magnets…); the matrix describes not the machinery but the interface specification — $A$ is the instrument’s API signature: the list of return values (eigenvalues) plus the set of deterministic inputs (eigenvectors). Implementation belongs to hardware; the signature belongs to the matrix. The spectral theorem guarantees that a Hermitian matrix is exactly equivalent to this data: $A = \sum_i \lambda_i |v_i\rangle\langle v_i|$ — the matrix is the codebook compressed into a single object.

Matrix multiplication has three legitimate uses, all paper bookkeeping: making the codebook (solving $Av=\lambda v$, multiplication as a diagnostic tool); computing the average reading $\langle A\rangle = \langle\psi|A|\psi\rangle$ (in the example above, $\tfrac12\cdot2+\tfrac12\cdot5 = 3.5$ ✓ — a statistical tool, not the measurement itself); and testing compatibility (the commutator machine needs matrices to subtract).

8.4 The dual identity, side by side

Matrix multiplication belongs to the left column’s world (unitary evolution); measurement is the right column’s world, two independent rules in the axiom list, neither reducible to the other.

Finally, state the “Hermitian” credential in full: a qualified instrument needs real readings (Hermitian ⟹ all eigenvalues real), distinguishable outcomes (spectral theorem ⟹ eigenvectors of distinct eigenvalues are automatically orthogonal — $|0\rangle\perp|1\rangle$ was never a coincidence), and every state measurable (the eigenvectors form a complete basis, so every state has a page in the codebook). All three in one package: the Hermitian matrix is the mathematical name of the concept “measuring instrument”.

9. Projective measurement: the formal uniform

9.1 A new part: the outer product — brackets facing outward

You know the inner product $\langle 0|\psi\rangle$: row times column, a number. Reverse the order and you get the outer product: column times row, a matrix:

The mnemonic: brackets facing inward, $\langle\cdot|\cdot\rangle$, close up into a number; brackets facing outward, $|\cdot\rangle\langle\cdot|$, open out into a matrix.

9.2 The projector: casting a shadow

Write $\Pi_0 = |0\rangle\langle 0|$. Act on $|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$ and use the sandwich reading — the $\langle 0|$ on the right bites $|\psi\rangle$ first and spits out a number:

The $|1\rangle$ part is killed, the $|0\rangle$ part kept: $\Pi_0$ asks “how much $|0\rangle$ does $|\psi\rangle$ contain” and keeps only that. The geometric picture: push the vector down onto the $|0\rangle$ axis and take its shadow.

Two properties, one line each. Hermitian: $\mathrm{diag}(1,0)$ is real symmetric ✓. Idempotent ($\Pi^2=\Pi$) — “the shadow of a shadow is the shadow”:

9.3 Walking the formula chain

Read it segment by segment: the starting point is the actual law, $P = \|\Pi|\psi\rangle\|^2$ — the squared length of the shadow; expanding by the definition of the norm ($\|v\|^2 = \langle v|v\rangle$ with $|v\rangle=\Pi|\psi\rangle$) gives $\langle\psi|\Pi^\dagger\Pi|\psi\rangle$; Hermitian plus idempotent collapses $\Pi^\dagger\Pi$ into $\Pi$; and the sandwich splits into two numbers multiplied, $\langle\psi|0\rangle = \alpha^*$ and $\langle 0|\psi\rangle = \alpha$. The whole chain is the Born rule wearing the projector uniform; no new physics anywhere.

The post-measurement state falls out along the way: the shadow $\Pi_0|\psi\rangle = \alpha|0\rangle$ has length $|\alpha|$, not a legal state — renormalize (divide by the length) to get $(\alpha/|\alpha|)|0\rangle$, and the unit-modulus factor in front is a global phase, discarded, leaving $|0\rangle$. Two old friends, renormalization and discarding global phase, take the stage together.

9.4 Spectral decomposition: the codebook is a stack of projectors

One projector per page, one reading label glued on — the simplest instance of the spectral decomposition $A = \sum_i\lambda_i|v_i\rangle\langle v_i|$ mentioned in §8.3. The expectation value also turns transparent at once: $\langle\psi|Z|\psi\rangle = P(0) - P(1)$, the probability-weighted average reading.

9.5 Why it is worth the trouble: it transplants unchanged to many qubits

On a single qubit this machinery is overkill ($|\alpha|^2$ can be read off by eye). The real payoff comes with multiple qubits: when measuring only the first qubit of an entangled pair, “the amplitude” is no longer a single number, yet the projector recipe carries over without changing a word:

(Here $\otimes$ is the tensor product, the operation that joins subsystems into a composite system; $|0\rangle\langle 0|\otimes I$ reads “project the first qubit, do nothing to the second”.) “Probability $=\langle\psi|\Pi|\psi\rangle$, post-state $=$ normalized $\Pi|\psi\rangle$” is the one version that upgrades losslessly, and the form worth memorizing.

9.6 True randomness

Three worked cases: $|0\rangle$ gives 0 with probability 1 (already on the eigenvector); $|1\rangle$ likewise; $|+\rangle$ gives 50/50 — and nobody in the universe can tell you in advance which outcome this particular run will produce; only ensemble statistics are lawful. This is not the ignorance-style randomness of a coin hidden under a cup (the kind in a complexity class like BPP, where peeking at the random tape would in principle let you predict), but true randomness: there is no coin, the value simply was not defined before the measurement (Bell experiments ruled out the hidden coin). Two direct consequences: quantum random number generators are the best randomness sources because their randomness is backed by physical law rather than algorithmic disguise; and the quantum complexity class BQP is defined with bounded error (error at most $1/3$, then amplified) precisely because quantum algorithms are born with random output — the “two-thirds, then amplify” machinery built for BPP works off the shelf.

10. A checklist of common misconceptions

1. “Measurement means computing $A|\psi\rangle$” — wrong. The matrix is never multiplied into the state; measurement = decompose → roll dice → collapse, and $A$ is the codebook, not the action (the three-sins counterexample of §8.2).
2. “Eigenvalue $-1$ means the state changed into another state” — on its own it is only a global phase, motionless on the sphere; only inside a superposition does it become a consequential relative phase (§5.3).
3. “Rotating $360°$ returns you to the start” — off by a factor of $-1$; only $720°$ truly returns home. Rotate just one branch of a superposition and that $-1$ shows up in interference fringes (§6.4, the neutron experiments).
4. “Circuits and matrices read in the same direction” — circuits read left to right (time), matrices right to left (function composition); reading backwards can be wrong by as much as orthogonal ($HX|0\rangle = |-\rangle$ versus $XH|0\rangle = |+\rangle$).
5. “A qubit is a photon” — the qubit is a role; the photon (one of its degrees of freedom) is one of the actors; keep the three layers straight: logical, degree-of-freedom, carrier (§4).
6. “Superposition is mysterious” — for polarization it is vector decomposition: a $45°$ field really is composed of simultaneous horizontal and vertical oscillation (§3.2).
7. “Quantum randomness is like classical randomness” — BPP’s randomness is an unflipped coin; quantum randomness is the coin not existing; Bell experiments are the evidence (§9.6).
8. “Global phase can never be measured” — needs sharpening: acting on the entire system, it cannot; acting on only part of a superposition, it demotes itself to a relative phase and becomes measurable (§6.4).

11. Self-test (answers included)

Problems

1. Verify by hand that $[Y,Z] = 2iX$.
2. What does $R_y(\pi)$ do to $|0\rangle$? Which Pauli gate is it equivalent to?
3. A photon linearly polarized at $30°$ enters an H/V polarizing beam splitter. What are the probabilities at the two exits?
4. Do $Z$ and $X$ share a common eigenvector? Explain using the five-line proof.
5. Define $\Pi_+ = |+\rangle\langle+|$. For $|\psi\rangle = |0\rangle$, compute $P(+) = \langle 0|\Pi_+|0\rangle$ and the post-measurement state.
6. Why can a classical AND gate not be used directly as a quantum gate?
7. What is the output of the circuit $|0\rangle —[H]—[Z]—[H]—$? (Hint: multiply right to left, or walk it step by step.)

Answers

1. $YZ = \begin{pmatrix}0&-i\\i&0\end{pmatrix}\begin{pmatrix}1&0\\0&-1\end{pmatrix} = \begin{pmatrix}0&i\\i&0\end{pmatrix} = iX$; $ZY = -iX$; subtracting gives $2iX$ ✓.
2. $R_y(\pi) = \cos\frac\pi2 I - i\sin\frac\pi2 Y = -iY$; and $-iY|0\rangle = -i\cdot i|1\rangle = |1\rangle$. North pole to south pole; up to a global phase it is the $Y$ gate (which also sends $|0\rangle$ to $|1\rangle$; the difference from $X$ lies in phase details).
3. $P(H) = \cos^2 30° = 3/4$, $P(V) = \sin^2 30° = 1/4$.
4. No. If $v$ were an eigenvector of both, then $[Z,X]v = 0$; but $[Z,X]=2iY$ and $Y^2=I$ makes $Y$ invertible, so $v=0$, contradiction — a state definite in $Z$ is necessarily maximally uncertain in $X$.
5. $P(+) = |\langle+|0\rangle|^2 = 1/2$; the shadow $\Pi_+|0\rangle = \tfrac{1}{\sqrt2}|+\rangle$ renormalizes to $|+\rangle$.
6. AND takes two bits to one bit, destroying information, hence irreversible; quantum gates must be unitary (reversible), so classical logic must first be rewritten with reversible gates such as the Toffoli.
7. $H|0\rangle = |+\rangle$; $Z|+\rangle = |-\rangle$; $H|-\rangle = |1\rangle$. Output $|1\rangle$. (In matrix language: $HZH = X$ — “a $Z$ sandwiched between two $H$‘s becomes an $X$”, a classic example of basis change.)

12. Symbol quick reference (new in these notes)

The whole article in one sentence: knowing $\alpha$ pins down the length of $\beta$ but not the angle between the two arrows — that angle is real physical information, taking bodily form as an angle you can verify with polarizers, turned by the $R_z$ knob, and exposed by the $-1$ of a full turn inside an interferometer; gates work by multiplying unitary matrices, instruments roll dice from a Hermitian codebook, and the projector’s “squared shadow length” formula is the one piece of luggage that travels to the many-qubit world unaltered.