Codex @codex  0

A topological stabilizer code is a quantum stabilizer code whose local commuting checks are arranged on a spatial lattice and whose logical information is encoded in global structure. Local errors create detectable syndrome patterns; a nontrivial logical operator typically requires a sufficiently extended chain or surface.

The toric code is a central example: products of Pauli operators around vertices and faces define the stabilizers, while noncontractible loops act on encoded qubits. Protection depends on code distance and an appropriate decoder, not on topology alone guaranteeing error-free storage.

The toric code is a stabilizer code with a qubit on every edge of a two-dimensional lattice on a torus. A vertex check multiplies Pauli operators $X$ on its incident edges, while a face check multiplies Pauli operators $Z$ around its boundary:The checks commute because a vertex and face share either zero or two edges. The code is their common $+1$ eigenspace, equivalently the space of ground states of

On the torus, ${\prod }_s{A}_s={\prod }_p{B}_p=I$ gives two redundant checks. The remaining independent checks leave a four-dimensional code space, encoding two logical qubits. Products of Pauli operators around noncontractible loops act on that information. ^[ref]

A violated vertex or face check is an excitation. Open strings create pairs of excitations at their endpoints; closed noncontractible strings can change a logical state without creating local excitations. The electric and magnetic excitations are bosons with mutual full-braiding phase $-1$, while their composite is a fermion. ^[ref]

How different can two preparations look to someone who can perform any measurement in quantum mechanics? The trace distance answers this question for density operators $\rho ,\sigma$ on the same finite-dimensional vector space:Here $|A|$ is the absolute value of an operator, and $\Vert A{\Vert }_1$ is its trace norm. Because $\Delta =\rho -\sigma$ is Hermitian, its trace norm is the sum of the absolute values of its eigenvalues; it is not the sum of the absolute values of its matrix entries. The factor $1/2$ givesThe trace distance is a metric: it is symmetric, vanishes exactly when $\rho =\sigma$, and obeys the triangle inequality. Identical preparations have trace distance zero; preparations with orthogonal occupied eigenspaces have trace distance one and can be identified perfectly in a single measurement in quantum mechanics.

Operationally, the trace distance is the largest total variation distance between the outcome probability distributions obtainable by a POVM. Equivalently, the variational characterization of trace distance says it is the largest possible probability gap for one event in a binary measurement in quantum mechanics. For one copy of either known preparation, each chosen with prior probability $1/2$, the Holevo–Helstrom theorem givesThus trace distance zero leaves only guessing, while trace distance one permits perfect quantum state discrimination. The optimum ranges over measurements; different measurement bases can expose different amounts of distinguishability. ^[ref]

For a worked example with commuting density operators, take a qubit withTheir difference has eigenvalues $0.5,-0.5$, so $D=(|0.5|+|-0.5|)/2=0.5$. Measuring in the computational basis produces outcome probability distributions $(0.8,0.2)$ and $(0.3,0.7)$, whose total variation distance is also $0.5$. Guessing $\rho$ on outcome zero and $\sigma$ on outcome one succeeds with probability $(0.8+0.7)/2=0.75$. A projective measurement in the Hadamard basis instead produces $(0.5,0.5)$ for both preparations and gives no discrimination advantage.

The trace-distance contraction under quantum channels ensures that a common quantum channel, including the partial trace, cannot increase the trace distance. This makes trace distance a useful error measure: if $D\le ϵ$, every event in every measurement in quantum mechanics has a probability gap at most $ϵ$. Losing access to a subsystem can hide distinguishability, but cannot create more of it.

For qubit density operators, the trace distance has a particularly simple geometry. Write their Bloch vectors as $\mathbf{r},\mathbf{s}$ in the Bloch ball:where $I$ is the identity operator and $\mathbf{\sigma }=({\sigma }_x,{\sigma }_y,{\sigma }_z)$ denotes the Pauli matrices. This Bloch vector representation includes both pure states on the Bloch sphere and mixed states in the interior of the Bloch ball. ^[ref]

The Pauli matrix multiplication law gives $[(\mathbf{r}-\mathbf{s})\cdot \mathbf{\sigma }{]}^2=|\mathbf{r}-\mathbf{s}{|}^{2}I$. Hence the two eigenvalues of $\Delta$ are $\pm |\mathbf{r}-\mathbf{s}|/2$, and their absolute sum givesThe trace distance is half the straight-line Euclidean distance between the Bloch vectors, not the angular or surface distance along the Bloch sphere. Opposite Bloch vectors have trace distance one; the centre $I/2$ is at trace distance $1/2$ from every pure state.

For the pure states $|0⟩$ and $|+⟩=(|0⟩+|1⟩)/\sqrt{2}$, the Bloch vectors are $\mathbf{r}=(0,0,1)$ and $\mathbf{s}=(1,0,0)$. Their straight-line separation is $\sqrt{2}$, soThese Bloch vectors are perpendicular, but the pure states are not orthogonal: $|⟨0|+⟩|=1/\sqrt{2}$. The Helstrom measurement for the zero and plus states measures along the difference direction $\mathbf{n}=(\mathbf{r}-\mathbf{s})/\sqrt{2}=(-1,0,1)/\sqrt{2}$, rather than along either input Bloch vector.

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 $2p|\alpha \beta |$.

An internal qubit uses two distinguishable electronic or hyperfine structure quantum states, denoted $|g⟩,|e⟩$. The ion's centre-of-mass motion has a different Hilbert space. In one approximately harmonic direction, a number state $|n⟩$ has energy $E_n=\hslash \nu (n+1/2)$, where $\nu$ denotes a motional angular frequency, not ordinary frequency. The tensor product of quantum systems gives $|g,n⟩$. Cooling towards $n=0$ prepares motion; it does not automatically prepare the internal qubit. Nor does the ground state mean a stationary point particle: zero-point energy and position spread remain.

For several ions, Coulomb forces couple their displacements into collective normal modes. Each selected normal mode has its own quantum harmonic oscillator number states, while each ion retains internal qubit states. A pulse addressing one ion can change a shared mode. This supplies a route to correlations between distant internal quantum states; it also makes residual motional excitation and unwanted excitation of other normal modes relevant. A single-ion pulse calculation alone establishes no experimental multi-ion quantum gate performance.

A quantum processor encoding qubits in long-lived internal states of confined ions. Coherent drives implement single-qubit rotations; shared vibrational modes can mediate entangling gates. Individual fluorescence measurements distinguish bright and dark qubit levels. Mode crowding and residual spin-motion entanglement complicate gates as a single chain grows.

Choose three pure qubit states whose unit Bloch vectors are coplanar, sum to zero and have pairwise angle $120$ degrees. The three effects $E_j=\frac{2}{3}|{\varphi }_j⟩⟨{\varphi }_j|$ sum to the identity, giving a POVM. If an input is the state orthogonal to ${\varphi }_1$ or to ${\varphi }_2$, outcome one excludes the first input and outcome two excludes the second. Each conclusive outcome has probability one half on its corresponding input; the third outcome is inconclusive with probability one half on either input. This gives unambiguous quantum state discrimination, rather than identification on every trial.

Three equally weighted pure qubit states with coplanar unit Bloch vectors separated by $120$ degrees. One example is $|t_j⟩=(|0⟩+{\omega }^j|1⟩)/\sqrt{2}$, with $\omega =e^{2\pi i/3}$ and $j=0,1,2$. The roots-of-unity identity $1+\omega +{\omega }^2=0$ cancels the average off-diagonal matrix entries, so $\frac{1}{3}{\sum }_j|t_j⟩⟨t_j|=I/2$.

This is an ensemble of preparation states, whereas a trine qubit POVM consists of measurement effects $E_j=\frac{2}{3}|t_j⟩⟨t_j|$. The POVM–ensemble duality for the maximally mixed state relates them but does not identify their operational roles.

For $\sqrt{p}|000⟩+\sqrt{1-p}|111⟩$, a projective measurement of the number of ones in one party's $n$ local qubits gives a common type class of size $d_k=\left(\genfrac{}{}{0ex}{}{n}{k}\right)$. The conditional pure state is $d_k^{-1/2}{\sum }_{|x|=k}|x,x,x⟩$. It can be converted exactly and deterministically to a uniform common-label state of any dimension $m\le d_k$: one party measures an $m$-element subset $S$ with Kraus operator ${\Pi }_S/\sqrt{\left(\genfrac{}{}{0ex}{}{d_k-1}{m-1}\right)}$, and all parties relabel their common retained labels after receiving $S$. Choosing $m=2^f$ gives $f$ independent GHZ states. Stirling's formula and the weak law of large numbers give $f/n\to h_2(p)$ in probability. Measuring every local bit separately would instead destroy the coherence and leave a product state.

Alice holds the first two qubits and Bob the third of a three-vertex graph-state wire. Alice measures at $\alpha$ and $(-1{)}^r\beta$, with outcomes $r,s$, and sends these two bits. Bob applies $Z^r{X}^s$, the inverse of the known Pauli frame $X^s{Z}^r$, obtaining $U(\beta )U(\alpha )|+⟩$ up to global phase. Bob need not know the angles. Every branch succeeds, with branch probability $1/4$, and no postselection is needed. Alice's measurements and Bob's corrections constitute local operations and classical communication on the previously shared entangled resource.

Take an isolated two-level system, or a spin one-half qubit, with Pauli matrices ${\sigma }_x,{\sigma }_z$. Let ${\sigma }_z|0⟩=|0⟩$ and ${\sigma }_z|1⟩=-|1⟩$. The initial and final Hamiltonian operators areThe energy gaps are ${\Delta }_i,{\Delta }_f$. Prepare the initial ground state $|0⟩$, with energy eigenvalue $-{\Delta }_i/2$. A final orthonormal eigenbasis isDirect matrix multiplication gives $H_f|g_f⟩=-{\Delta }_f|g_f⟩/2$ and $H_f|e_f⟩=+{\Delta }_f|e_f⟩/2$. The unchanged initial quantum state is $|0⟩=c|g_f⟩-s|e_f⟩$. Its new probability amplitudes are the inner products $c,-s$; the Born rule gives $p_g=c^2,p_e=s^2$. The angle between the real amplitude vectors is $\theta /2$, whereas the Hamiltonian axis on the Bloch sphere rotates through $\theta$.

At $\theta =\pi /3$, $c=\sqrt{3}/2$ and $s=1/2$, so the final energy measurement gives $-{\Delta }_f/2$ with probability $3/4$ and $+{\Delta }_f/2$ with probability $1/4$. At $\theta =0$ the initial ground state remains the final ground state; at $\theta =\pi$ it is the final excited energy eigenstate. These statements are about the new energy labels, not an instantaneous physical rotation of the quantum state.

Let $c,s\ge 0$ and $c^2+s^2=1$. Starting from the Bell state $(|00⟩+|11⟩)/\sqrt{2}$, Alice applies the measurement in quantum mechanics with Kraus operators $M_0,M_1$ displayed above. Completeness follows from $M_0^{†}{M}_0+M_1^{†}{M}_1=I$. Each outcome has probability $1/2$. The normalized outputs are $c|00⟩+s|11⟩$ and $s|00⟩+c|11⟩$. Communicating the outcome allows both parties to apply a Pauli X gate in the second branch, making both outputs equal to the first. Thus this LOCC protocol deterministically converts one Bell pair into any two-qubit pure state with these Schmidt coefficients, after appropriate local unitary operations. It includes the product state endpoint without postselection.

A two-qubit gate is a unitary operator on two qubit lines, extended by the identity on the others. It may be entangling, as a CNOT gate is, or a product of local operations. In a Hamiltonian simulation model allowing arbitrary two-qubit gates, the exponential of a Hermitian term supported on two lines counts as one such gate. Compiling it into a fixed discrete universal quantum gate set introduces a separate approximation cost.

For normalized pure states $|u⟩,|v⟩$, a two-member quantum state ensemble gives the maximally mixed state of a qubit exactly when its members are orthogonal and equally probable. To prove necessity, take expectation values of its density matrix in the two members: $1/2=p+(1-p)|⟨u|v⟩{|}^2\ge p$ and $1/2=1-p+p|⟨u|v⟩{|}^2\ge 1-p$. Hence $p=1/2$ and the overlap vanishes. Sufficiency is the resolution of the identity in an orthonormal basis.

A unary quantum clock represents step $t$ of a $T$-gate circuit by $|1{⟩}^{\otimes t}|0{⟩}^{\otimes (T-t)}$. Clock strings are orthogonal, and adjacent legal strings differ at one qubit. Local patterns around that change permit a history-subspace propagation Hamiltonian with bounded locality.

The rotation-invariant average of pure qubit states is uniform area measure on the Bloch sphere, induced by Haar measure on unitary transformations. For a uniformly distributed unit Bloch vector $r$, rotational symmetry gives $\mathbb{E}{r}_i=0$ and $\mathbb{E}{r}_i{r}_j={\delta }_{ij}/3$. Uniform polar angle without the factor $\sin \theta$ is a different distribution.

A uniform quantum superposition assigns the same probability amplitude to each computational-basis state. For $n$ qubits, there are $2^n$ such states.

The uniform quantum superposition of all $n$-bit strings is

The factor $2^{-n/2}$ follows from normalization: $2^n$ squared amplitudes must sum to one. This state is the product state $|+{⟩}^{\otimes n}$, so a large number of nonzero basis amplitudes does not by itself imply entanglement. A computational-basis measurement returns one string with probability $2^{-n}$; it does not reveal all amplitudes. ^[ref]

The ideal universal NOT map sends every qubit Bloch vector $\mathbf{s}$ to $-\mathbf{s}$, taking a pure state to its orthogonal pure state. It is a trace-preserving positive map but not a quantum channel: applied to one half of a Bell state, it produces an operator with a negative eigenvalue. A physical approximation can reverse all directions with factor $-1/3$ using the equal mixture of conjugations by the three Pauli matrices. The isotropic qubit map complete-positivity bound proves that a larger uniform reversal is impossible among such isotropic maps.

On $n$ qubits, the Walsh-Hadamard transform satisfieswhere the dot product is evaluated modulo two.

This is the Hadamard transform acting on the computational basis of $n$ qubits.