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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.

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 POVM $\{E_i\}$ in dimension $d$ corresponds to a quantum state ensemble with $p_i=\mathrm{Tr}(E_i)/d$ and ${\rho }_i=E_i/\mathrm{Tr}(E_i)$, omitting zero effects. Its average is the maximally mixed state $I/d$. Conversely any ensemble with that average defines a POVM by $E_i=d{p}_i{\rho }_i$. These transformations are inverse. A pure POVM corresponds exactly to a pure-state ensemble decomposition of $I/d$.

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

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

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

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

For a $d$-dimensional unitary operator $V={\sum }_{j=1}^d{e}^{i{\varphi }_j}|v_j⟩⟨v_j|$, a maximally mixed state $I/d$ on its target is the equal mixture of these orthonormal eigenvectors. Therefore quantum phase estimation with register size $L$ produces label $m$ with probabilityThis follows by linearity of unitary time evolution and the Born rule on the spectral mixture. If every eigenphase has an exact $L$-grid label, that label has probability equal to its multiplicity divided by $d$. Otherwise the exact distribution has tails, and the nearest labels are guaranteed only the nearest-integer success bound for quantum phase estimation weighted by the corresponding spectral probabilities.

For any qubit operator $A$,Expand $A$ in the Pauli operator basis. The identity term is fixed, while every nonidentity coefficient has two positive and two negative contributions and cancels. Thus forgetting a uniformly random Pauli key sends every density operator to the maximally mixed state. This is the dimension-two Heisenberg-Weyl twirling channel, since replacing $Y$ by $XZ$ changes only an irrelevant phase in the conjugation.

Superdense coding uses a shared Bell state to communicate two classical bits by transmitting one qubit. The sender selects one of four orthogonal Bell states with a local Pauli operator, and the receiver performs a Bell-basis measurement. For an ideal shared Bell pair, the transmitted qubit alone has a maximally mixed state for every message and reveals no information about the chosen encoding.

Prepare the auxiliary register in the maximally mixed state $I/N$ and estimate both mixed-state Hadamard-test quadratures. Multiplying the normalized expectation by $N$ estimates the trace of $U$. A uniformly randomized preparation of any orthonormal basis produces $I/N$; the eigenvectors of $U$ need not be known.

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.