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The classical mixture has density matrix $\rho =\frac{1}{2}|HH⟩⟨HH|+\frac{1}{2}|VV⟩⟨VV|$. Average the probabilities of its two alternatives:Alice's single-detector transmission probability is $1/2$ at every angle. Dividing by it givesAt $45^{\circ }$, Alice's click does not distinguish the two classical alternatives. At zero, it selects the horizontally polarized alternative.

A bosonic coherent state is an eigenstate of its annihilation operator,Its mean occupation is $|\alpha {|}^2$ and its number variance is also $|\alpha {|}^2$, giving a Poisson number distribution. It has a well-defined complex mean field but fluctuating particle number. Its overlap with another coherent state is $\exp [-(|\alpha {|}^2+|\beta {|}^2)/2+{\alpha }^{\ast }\beta ]$, so distinct amplitudes need not be orthogonal. For large occupation, a phase-space wave packet has small relative number fluctuations; this does not require an exact self-adjoint phase operator conjugate to a number bounded below.

For a bosonic field, a useful coherent-field condensate variational state isIt obeys $⟨\hat{\Psi }(\mathbf{r})⟩=\psi (\mathbf{r})$. For a contact-interacting dilute Bose gas, its normal-ordered energy expectation isMinimizing $E-\mu \bar{N}$ gives the stationary Gross–Pitaevskii equation, $[-{\hslash }^2{\nabla }^2/(2m)+V+g|\psi {|}^2]\psi =\mu \psi$. Its phase gradient determines superfluid flow, while weak-link phase differences give Josephson currents. The complex microscopic expectation supplies the order field used by a mean-field description of a bosonic superfluid.

An exactly fixed-number condensate can instead use the Hartree product state $(a_{\phi }^{†}{)}^N|0⟩/\sqrt{N!}$. Its one-point field expectation vanishes by number conservation, but its one-body density matrix has a macroscopic eigenvalue. Off-diagonal long-range order expresses that number-conserving criterion. Selecting a phase is a useful thermodynamic symmetry-broken description; it is not a claim that every finite isolated condensate literally has indefinite total number. A plain coherent product also misses interaction-induced quantum depletion, which needs correlations such as those in the Bogoliubov quasiparticle vacuum.

In a spin-singlet superconductor, the corresponding variational state is the BCS ground stateEach block superposes an empty and an occupied pair state. Where $u_{\mathbf{k}}\ne 0$, it can be rewritten as $(\prod u_{\mathbf{k}})\exp [\sum (v_{\mathbf{k}}/u_{\mathbf{k}})a_{\mathbf{k}\uparrow }^{†}{a}_{-\mathbf{k}\downarrow }^{†}]|0⟩$, treating any fully occupied blocks separately. The exponential truncates within each block because a fermionic pair cannot be created twice in the same two modes. This is a Cooper-pair pseudospin coherent state, rather than a coherent state of unrestricted ideal bosons.

The one-electron expectation remains zero in a fermion-parity-even BCS state. Its nonzero microscopic order is the BCS anomalous average $F_{\mathbf{k}}=⟨a_{-\mathbf{k}\downarrow }{a}_{\mathbf{k}\uparrow }⟩=u_{\mathbf{k}}^{\ast }{v}_{\mathbf{k}}$. For an attractive interaction written with matrix elements $V_{\mathbf{k}{\mathbf{k}}^{\prime }}$, the gap is ${\Delta }_{\mathbf{k}}=-{\sum }_{{\mathbf{k}}^{\prime }}{V}_{\mathbf{k}{\mathbf{k}}^{\prime }}{F}_{{\mathbf{k}}^{\prime }}$. Minimization yields the coherence factors and gap equation. Fermionic coherent states using Grassmann variables are valuable integration tools, but they are not ordinary complex-eigenvalue electron states replacing this physical paired wavefunction.^[ref]

Near the transition, a slowly varying pair field connects the microscopic anomalous average to Ginzburg-Landau theory of superconductivity. The phenomenological complex field has pair charge $-2e$, a phase that controls supercurrent and a magnitude whose normalization is model dependent. It is not the expectation of a single-electron field. Microscopic derivation of Ginzburg-Landau theory supplies this connection under its stated near-critical assumptions. Pair phase coherence, Josephson interference and flux quantization then become different manifestations of the same macroscopic paired order. Paired-fermion superfluids likewise use a pair order field, while bosonic atomic superfluids can use a one-particle condensate field.

An ensemble with normalized quantum states $|{\psi }_a⟩$ and probabilities $p_a\ge 0$, ${\sum }_a{p}_a=1$, has density matrixEach rank-one projector is Hermitian, so ${\rho }^{†}=\rho$. Taking the trace in any orthonormal basis givesIt is also positive: $⟨v|\rho |v⟩={\sum }_a{p}_a|⟨{\psi }_a|v⟩{|}^2\ge 0$. Its eigenvalues ${\lambda }_a$ are therefore nonnegative and sum to one. Hencewith equality exactly when one eigenvalue is one and the others are zero. Thus the purity of a density operator distinguishesEquivalently, a pure state has ${\rho }^2=\rho$ and rank one. In a $d$-dimensional Hilbert space, purity is at least $1/d$, attained by the maximally mixed state $I/d$.

Fix the mode ordering with the $b^{†}$ operator before the $c^{†}$ operator, as in the given state. In the sector with one fermion in each subsystem, use the ordered basisThe state vector is $v=(1/2,1/\sqrt{2},0,1/2{)}^T$, whose squared norm is one. Thus the global density matrix is the rank-one projectorIt has trace one and satisfies ${\rho }_{BC}^2={\rho }_{BC}$, so the global state is pure.

Take the partial trace over $B$. The conditional vectors in $C$ are $v_1=(1/2,1/\sqrt{2}{)}^T$ for $B_1$ and $v_2=(0,1/2{)}^T$ for $B_2$. Therefore the reduced density matrix isIts determinant is $1/16$, giving eigenvaluesBoth are positive; in particular, $\mathrm{Tr}{\rho }_C^2=7/8<1$, so the reduced state is mixed despite global purity. The entanglement entropy is obtained by diagonalizing this reduced matrix and evaluatingFor a globally pure bipartite state this equals the entropy of either reduced subsystem. No numerical evaluation is needed.

Direct multiplication of the three Pauli matrices gives $X^2=Y^2=Z^2=I$, $XY=iZ$ and their cyclic permutations, with reversed products changing sign. Consequently $(\mathbf{a}\cdot \mathbf{\sigma })(\mathbf{b}\cdot \mathbf{\sigma })=(\mathbf{a}\cdot \mathbf{b})I+i(\mathbf{a}\times \mathbf{b})\cdot \mathbf{\sigma }$. In particular $(\mathbf{s}\cdot \mathbf{\sigma }{)}^2=|\mathbf{s}{|}^{2}I$, which gives the eigenvalues $(1\pm |\mathbf{s}|)/2$ of a qubit density matrix.

The identity and the three Pauli matrices form an orthogonal basis for the complex vector space of two-by-two matrices, with Hilbert-Schmidt inner product $\mathrm{Tr}(P^{†}Q)=2{\delta }_{PQ}$. Thus every operator has the expansion $A=\frac{1}{2}{\sum }_{P\in \{I,X,Y,Z\}}\mathrm{Tr}(PA)P$. For a density matrix, the identity coefficient is $1/2$ and the remaining real coefficients are half the components of its Bloch vector. This converts unitary conjugation into transformations of three real components.

In 1998, Jian-Wei Pan, Dik Bouwmeester, Harald Weinfurter and Anton Zeilinger created two photon pairs by two passages of a pulsed pump through a crystal using spontaneous parametric down-conversion. Interfering the inner photons at a nonpolarizing beam splitter implemented a partial Bell-basis measurement. Narrow optical filters and spatial filtering removed distinguishing information; changing the outer polarizer bases checked the conditional correlations. Fourfold detections selected the entanglement swapping events. The reported interference supported conditional quantum entanglement, but the paper reserved a statistically significant Bell inequality violation for improved visibility; it was not a demonstration closing the detection loophole.^[ref]

The ideal calculation makes the distinction between conditional and unconditional quantum states explicit. Start with spin singlet states $|{\psi }^{-}{⟩}_{12}|{\psi }^{-}{⟩}_{34}$, where $|{\psi }^{-}⟩=(|HV⟩-|VH⟩)/\sqrt{2}$ and $H,V$ denote orthogonal photon polarizations. Taking the inner product with $⟨{\psi }^{-}{|}_{23}$ gives $-|{\psi }^{-}{⟩}_{14}/2$: expanding the four terms leaves precisely the two antisymmetric outer terms. That Bell-basis measurement outcome has probability $1/4$ and leaves the outer pair in a spin singlet state. Ignoring the inner outcome instead leaves the outer density matrix $I_4/4$, a maximally mixed state. A conditional preparation flag is useful for later Bell tests only with the timing and setting conditions in setting-independent heralding preserves Bell locality.

An explicit original ideal meter model explains how photon number survives while field quantum coherence changes. Let $\hat{N}$ be the number operator and prepare an atomic qubit in $|+⟩=(|e⟩+|g⟩)/\sqrt{2}$. The dispersive unitary operatorcommutes with $\hat{N}\otimes I$. For a field quantum superposition ${\sum }_n{c}_n|n⟩$, it produces ${\sum }_n{c}_n|n⟩|a_n⟩$, where $|a_n⟩=(e^{-in\chi /2}|e⟩+e^{in\chi /2}|g⟩)/\sqrt{2}$. The atomic meter states encode $n$ through relative quantum phase and can become entangled with the field. The second Ramsey interferometry pulse and detector turn those correlations into the outcome probabilities above when $C=1$.

Tracing out the atom multiplies a field density matrix entry ${\rho }_{nm}$ by $⟨a_m|a_n⟩=\cos ((n-m)\chi /2)$. Hence the nonselective quantum measurement preserves every photon number population ${\rho }_{nn}$ while changing off-diagonal quantum coherence. For $\chi =\pi /2$, coherence between $|0⟩$ and $|1⟩$ is multiplied by $1/\sqrt{2}$ per fresh meter, even with no photons absorbed. Field quantum phase information can therefore be disturbed by an energy-preserving probe. No universal self-adjoint phase operator or naive canonical number–phase commutator is needed for this conclusion.

One possible finite-contrast quantum instrument makes the distinction sharper. Define diagonal Kraus operators $K_e=\cos ((\varphi +\chi \hat{N})/2)$ and $K_g=\sin ((\varphi +\chi \hat{N})/2)$. After their ideal readout, flip the reported label with probability $r=(1-C)/2$. The unnormalized selective maps areTheir traces reproduce $L_e,L_g$ on a Fock state, and their sum reproduces the displayed dephasing quantum channel, independently of $r$. At $C=0$ the recorded labels carry no information although the physical meter interaction can still cause measurement backaction. This is an illustrative instrument, not a reconstruction of all laboratory imperfections; equal likelihoods alone do not determine the disturbance.

For a photon in a fixed spatial mode, the two transverse photon polarizations span a qubit space. A pure photon polarization state is $\alpha |H⟩+\beta |V⟩$ with $|\alpha {|}^2+|\beta {|}^2=1$, and a general state is a two-dimensional density matrix. Its Bloch vector gives the normalized Stokes parameters. Polarization-dependent path separation can correlate this internal degree of freedom with a path qubit.

// Target: quantum-error-correction.bigb

Photon statistics is the probability distribution of the number operator in specified field modes, or the distribution of registered counts in a specified gate. Those distributions agree only for an appropriate ideal measurement. Mode selection, losses, and detector dead time matter. For a one-mode density matrix, $p_n=⟨n|\rho |n⟩$ andThese two sums already distinguish preparations with the same expected value.

Let $H,V$ be orthogonal linear polarizations, and prepare the Bell state $|{\Phi }^{+}⟩=(|HH⟩+|VV⟩)/\sqrt{2}$. The $+1$ analyzer vector is $|a⟩=\cos a|H⟩+\sin a|V⟩$; its orthogonal companion has outcome $-1$. Direct inner products with the Bell state give $P_{++}=P_{--}={\cos }^2(a-b)/2$ and $P_{+-}=P_{-+}={\sin }^2(a-b)/2$. Thus the expectation value of the outcome product is $E(a,b)=\cos 2(a-b)$. These are physical polarizer angles; their corresponding Bloch sphere angles are doubled.

For a stipulated isotropic admixture, use the density matrixThe maximally mixed state contributes zero correlation, so $P_{st}(a,b)=[1+stv\cos 2(a-b)]/4$ for $s,t\in \{-1,1\}$. Each local marginal is $1/2$, independently of the other setting, illustrating quantum no-signalling. In this model $v$ is the interferometric visibility of the coincidence fringe; other noise models need not have one common visibility for every basis.

Choose $a_0=0^{\circ }$, $a_1=45^{\circ }$, $b_0=22.5^{\circ }$ and $b_1=-22.5^{\circ }$. The four correlations in the displayed CHSH inequality are $v/\sqrt{2},v/\sqrt{2},v/\sqrt{2},-v/\sqrt{2}$. Consequently $S=2\sqrt{2}v$, exceeding the local bound exactly when $v>1/\sqrt{2}$. Perfect correlation at a single common setting would not yield this comparison between incompatible quantum measurements.

To connect probabilities with expected counts, stipulate $M$ emitted pairs per setting pair and independent detection efficiencies ${\eta }_A,{\eta }_B$, independent also of settings, outcomes and hidden variables. The expected joint counts are $n_{st}=M{\eta }_A{\eta }_B{P}_{st}$. Their normalized correlation isFor original teaching inputs $M=1000$, ${\eta }_A=0.8$, ${\eta }_B=0.75$ and $v=0.8$, the $(a_0,b_0)$ expected counts are $n_{++}=n_{--}\simeq 234.85$ and $n_{+-}=n_{-+}\simeq 65.15$. They total $600$ and give $E_{\det }\simeq 0.56569$, hence $S\simeq 2.26274$. Fractional expected counts are not measured data. Efficiency cancels here because the loss model was specified to be independent; this algebra supplies no justification for discarding failures in an arbitrary Bell test with a detection loophole.

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]

The quantum state associated with the two transverse polarization degrees of freedom of a photon in a specified propagation mode. In a horizontal-vertical basis, a pure state is $\alpha |H⟩+\beta |V⟩$ with $|\alpha {|}^2+|\beta {|}^2=1$; a density matrix also describes mixtures. Linear polarizers test real combinations of these basis states, while phase-sensitive elements allow elliptical and circular polarization measurements.

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.

Write each pure density matrix as $(I+r\cdot \sigma )/2$. The trace identities of the Pauli matrices, $\mathrm{Tr}{\sigma }_i=0$ and $\mathrm{Tr}({\sigma }_i{\sigma }_j)=2{\delta }_{ij}$, make the trace of their product equal to the displayed overlap. Antipodal unit vectors represent orthogonal states; vectors at right angles correspond to squared overlap one half. For mixed states the same trace formula gives Hilbert–Schmidt overlap, not in general quantum fidelity.

Quantum detection infers a classical hypothesis from measurements of a quantum system. If hypothesis $j$ prepares a density matrix ${\rho }_j$, a measurement with POVM elements $E_k$ gives conditional probabilities $\mathrm{Tr}({\rho }_j{E}_k)$. Detector design chooses the measurement and decision rule for a stated error criterion.

For two equally probable states, the Holevo–Helstrom theorem gives minimum error $P_e=[1-\Vert {\rho }_0-{\rho }_1{\Vert }_1/2]/2$. Orthogonal states permit zero error; identical states force error $1/2$. The optimal binary measurement separates the positive and negative spectral parts of ${\rho }_0-{\rho }_1$, explaining why state geometry limits detection even with ideal apparatus.

A quantum dynamical semigroup is a continuous family $({\Phi }_t{)}_{t\ge 0}$ of linear maps on density matrices satisfying ${\Phi }_0=I$ and ${\Phi }_{t+s}={\Phi }_t\circ {\Phi }_s$, with each map completely positive and trace preserving. The composition law expresses time-homogeneous evolution with no dependence on how the interval is divided.

In finite dimensions, continuity gives ${\Phi }_t=e^{t\mathcal{L}}$, whose generator has Lindblad equation form. For a qubit dephasing example,Off-diagonal decay factors multiply under composition, while populations stay fixed. The map is the mixture $(1+e^{-\gamma t})\rho /2+(1-e^{-\gamma t})Z\rho Z/2$, which verifies complete positivity. A semigroup need not have physical inverse maps at positive times: undoing this decay would generally amplify coherence beyond valid states.^[ref]

Quantum dynamics describes the time evolution of a quantum state and its observable predictions. For an isolated system with Hamiltonian operator $H$, the Schrödinger equation is $i\hslash {\partial }_t|\psi ⟩=H|\psi ⟩$. Equivalently, a density matrix satisfies $i\hslash \dot{\rho }=[H,\rho ]$.

For time-independent $H$, the evolution operator is $U(t)=e^{-iHt/\hslash }$. An energy eigenstate changes only by a global phase, whereas a superposition develops relative phases and can display oscillating observables. Interaction with an unobserved environment generally requires open quantum system dynamics rather than a unitary evolution for the subsystem alone.

The Quantum Fisher information measures the local sensitivity of a parametrized density matrix ${\rho }_{\theta }$. The symmetric logarithmic derivative $L_{\theta }$ satisfies ${\partial }_{\theta }\rho =(L_{\theta }\rho +\rho L_{\theta })/2$, and $F_Q=\mathrm{Tr}(\rho L_{\theta }^2)$.

For $n$ independent preparations, a locally unbiased estimator obeys $\mathrm{Var}(\hat{\theta })\ge 1/(n{F}_Q)$ under the usual regularity assumptions. For a pure family $|{\psi }_{\theta }⟩=e^{-i\theta G}|{\psi }_0⟩$, $F_Q=4\mathrm{Var}(G)$. A state with definite $G$ acquires only a global phase and has zero information about $\theta$; a superposition can have observable relative-phase sensitivity.^[ref]

A quantum state describes a preparation through its predicted measurement in quantum mechanics probabilities. In the Hilbert space formulation, a pure state is a quantum state ray: a normalized vector representative $|\psi ⟩$ satisfies $⟨\psi |\psi ⟩=1$, and representatives differing by a global phase describe the same quantum state. ^[ref]

Both pure states and mixed states are represented by a density operator on the chosen complex Hilbert space:Thus $\rho$ is Hermitian, positive semidefinite and of unit trace; in infinite dimension it must also be a trace-class operator. For a normalized pure state, $\rho =|\psi ⟩⟨\psi |$ is a rank-one orthogonal projection. A mixed state has no single such vector representative. Its density matrix is the matrix of $\rho$ in a chosen orthonormal basis. ^[ref] For a discrete POVM with effects $E_a\ge 0$ and ${\sum }_a{E}_a=I$, the Born rule readsThese constraints and Born rule probabilities describe arbitrary finite-dimensional preparations. ^[ref]

Randomly choosing normalized pure states with probabilities $w_j$ and forgetting the label givesThis quantum state ensemble averages probabilities; a quantum superposition adds probability amplitudes. Several ensembles can give the same density operator. Quantum entanglement can make a reduced density matrix mixed while the joint quantum state is pure. ^[ref]

The preparation, the choice of basis, and the measurement are separate ingredients. A wavefunction $\psi (x)=⟨x|\psi ⟩$ is the position representation of a pure state; a general mixed state requires a density operator. For a closed quantum system, a unitary operator evolves it as $\rho \mapsto U\rho U^{†}$, preserving its eigenvalues and purity of a density operator. ^[ref] The Bloch sphere and Bloch ball give the qubit geometry; quantum state tomography connects repeated preparations to an estimated density matrix; projective measurement treats the conditional state after an outcome.