Ensembles and Preparation Procedures

An ensemble specifies a random preparation procedure. On each run, a classical label $i$ occurs with probability $p_i$, and the apparatus prepares a quantum state $\rho_i$. If that label is ignored or unavailable, the emitted system is described by the average state

$$\bar\rho = \sum_i p_i\rho_i, \qquad p_i\ge0, \qquad \sum_i p_i=1.$$

For a pure-state ensemble, $\rho_i=\lvert\psi_i\rangle\langle\psi_i\rvert$, so

$$\bar\rho = \sum_i p_i \lvert\psi_i\rangle\langle\psi_i\rvert.$$

The average density operator determines every unconditioned measurement probability on the emitted system. The particular ensemble decomposition is additional information about the preparation, and it is generally not unique.

Helpful background. Density operators supplies the trace rule, and pure versus mixed states separates a state from any chosen preparation ensemble. We use finite-dimensional systems and finite ensembles below.

Statistical preparations

Let a measurement have effects $\{E_a\}$. Conditional on preparation label $i$, the Born rule gives

$$p(a\mid i) = \operatorname{Tr}(\rho_i E_a).$$

The law of total probability then gives

$$\begin{aligned} p(a) &= \sum_i p_i p(a\mid i) \\ &= \sum_i p_i\operatorname{Tr}(\rho_i E_a) \\ &= \operatorname{Tr}(\bar\rho E_a). \end{aligned}$$

Thus averaging states before applying the trace rule produces the same unconditioned statistics as averaging the conditional probabilities afterward. This compatibility between classical probability and quantum measurement is why convex mixtures represent randomized preparations.

The components $\rho_i$ may themselves be mixed. A pure-state ensemble is important because every finite-dimensional density operator admits one, but it is not the only type of ensemble used in experiments.

Retained classical labels

Ignoring a preparation label means making predictions without using it. The record may still exist physically; neither averaging the state nor taking a partial trace requires destroying it. Introduce a classical register $X$ with orthonormal record states $\{\lvert i\rangle_X\}$. The joint classical–quantum state is

$$\rho_{XQ} = \sum_i p_i \lvert i\rangle\langle i\rvert_X \otimes \rho_i^{(Q)}.$$

Discarding the record gives

$$\operatorname{Tr}_X\rho_{XQ} = \sum_i p_i\rho_i = \bar\rho_Q.$$

If $X$ remains available, an experimenter can condition on $i$ and use $\rho_i$ for that subensemble. If $X$ is discarded, all predictions on $Q$ alone are made from $\bar\rho_Q$.

This distinction answers a common question: the average density operator is complete for measurements on the emitted system alone, but it is not a description of an accessible preparation record or of every larger system correlated with the emission.

Pure-state ensemble decompositions

An expression

$$\rho = \sum_i p_i \lvert\psi_i\rangle\langle\psi_i\rvert$$

is a pure-state ensemble decomposition of $\rho$. The vectors:

• need not be orthogonal;
• need not be linearly independent;
• may occur with repeated rays under different labels;
• may outnumber the dimension or rank of $\rho$.

The phases of individual $\lvert\psi_i\rangle$ do not matter because each term is a projector.

Every positive-probability ensemble vector lies in the support of $\rho$. If $\lvert\chi\rangle$ belongs to the kernel of $\rho$, then

$$\begin{aligned} 0 &= \langle\chi\rvert\rho\lvert\chi\rangle \\ &= \sum_i p_i \lvert \langle\chi\vert\psi_i\rangle \rvert^2. \end{aligned}$$

Each term is nonnegative, so every overlap must vanish. Therefore the ensemble vectors span the support, and any pure-state decomposition of a rank-$R$ state needs at least $R$ nonzero components.

The spectral ensemble

Diagonalizing $\rho$ gives

$$\rho = \sum_{r=1}^{R} \lambda_r \lvert r\rangle\langle r\rvert, \qquad \lambda_r>0.$$

This is a pure-state ensemble with orthonormal component states. It uses exactly $R=\operatorname{rank}\rho$ components, the smallest possible number.

The spectral decomposition is mathematically distinguished:

• the weights are the nonzero eigenvalues;
• the component vectors are orthogonal;
• nondegenerate eigenspaces fix their rays;
• a degenerate eigenspace permits basis rotations within that subspace.

It still need not be the preparation used in a laboratory. Diagonalizing a density matrix reveals its spectrum and eigenspaces, not a unique history of how the state was made.

Nonuniqueness for the maximally mixed qubit

The maximally mixed qubit has the computational-basis decomposition

$$\frac{I_2}{2} = \frac12 \lvert0\rangle\langle0\rvert + \frac12 \lvert1\rangle\langle1\rvert.$$

It also has the $x$-basis decomposition

$$\frac{I_2}{2} = \frac12 \lvert+\rangle\langle+\rvert + \frac12 \lvert-\rangle\langle-\rvert.$$

More generally, for any orthonormal qubit pair $\{\lvert\mathbf n,+\rangle,\lvert\mathbf n,-\rangle\}$,

$$\frac{I_2}{2} = \frac12 \lvert\mathbf n,+\rangle \langle\mathbf n,+\rvert + \frac12 \lvert\mathbf n,-\rangle \langle\mathbf n,-\rvert.$$

There are therefore infinitely many two-state decompositions. No measurement on the unconditioned qubit reveals which basis the source used.

Nonorthogonal decompositions of a biased qubit

Nonuniqueness is not restricted to the maximally mixed state. Let

$$\rho_r = \frac12 \left( I+rZ \right), \qquad 0<r<1.$$

Its spectral decomposition is

$$\rho_r = \frac{1+r}{2} \lvert0\rangle\langle0\rvert + \frac{1-r}{2} \lvert1\rangle\langle1\rvert.$$

Now define

$$\begin{aligned} \lvert\psi_+\rangle &= \sqrt{\frac{1+r}{2}}\, \lvert0\rangle + \sqrt{\frac{1-r}{2}}\, \lvert1\rangle, \\ \lvert\psi_-\rangle &= \sqrt{\frac{1+r}{2}}\, \lvert0\rangle - \sqrt{\frac{1-r}{2}}\, \lvert1\rangle. \end{aligned}$$

These states are normalized but not orthogonal:

$$\langle\psi_+\vert\psi_-\rangle=r.$$

Their equal mixture is

$$\rho_r = \frac12 \lvert\psi_+\rangle\langle\psi_+\rvert + \frac12 \lvert\psi_-\rangle\langle\psi_-\rvert.$$

The opposite off-diagonal terms cancel, while the diagonal populations remain. The same nonmaximally mixed state therefore has both an unequal orthogonal ensemble and an equal nonorthogonal ensemble.

Bloch-ball geometry

A pure qubit state has a unit Bloch vector $\mathbf n_i$, while a general qubit density operator has

$$\rho = \frac12 \left( I+\mathbf r\cdot\boldsymbol\sigma \right).$$

For an ensemble of pure qubit states,

$$\mathbf r = \sum_i p_i\mathbf n_i.$$

An ensemble decomposition is therefore a convex construction of an interior point from points on the sphere. Every interior point admits infinitely many such constructions. The density operator retains the average vector $\mathbf r$, not a preferred polygon of component vectors.

The full qubit geometry is developed in Bloch Sphere.

Operational equivalence

Two ensembles

$$\mathcal E = \left\lbrace (p_i,\rho_i) \right\rbrace_i, \qquad \mathcal F = \left\lbrace (q_j,\sigma_j) \right\rbrace_j$$

are operationally equivalent on the emitted system when

$$\sum_i p_i\rho_i = \sum_j q_j\sigma_j = \rho.$$

For every measurement effect $E_a$,

$$\begin{aligned} \sum_i p_i\operatorname{Tr}(\rho_iE_a) &= \operatorname{Tr}(\rho E_a) \\ &= \sum_j q_j\operatorname{Tr}(\sigma_jE_a). \end{aligned}$$

The equivalence includes every POVM and every data-processing strategy applied only to that system. It does not say the laboratory procedures are identical, consume the same resources, or leave the same external records.

What the average state does not specify

Side information

If different labels are stored in accessible registers, the joint classical–quantum states can differ even when their reductions on $Q$ are the same. Measurements on $XQ$ may then distinguish the preparations.

Correlations across emissions

A single-copy density operator does not determine temporal or multi-copy correlations. Compare two two-qubit sources. Independent fair emissions give

$$\rho_{\mathrm{iid}} = \frac{I_2}{2} \otimes \frac{I_2}{2} = \frac{I_4}{4}.$$

A source that chooses one fair bit and emits it twice gives

$$\rho_{\mathrm{corr}} = \frac12 \lvert00\rangle\langle00\rvert + \frac12 \lvert11\rangle\langle11\rvert.$$

Every individual qubit has marginal $I_2/2$ in both cases, but a joint $Z\otimes Z$ measurement distinguishes the sources. Claiming single-system operational equivalence does not license an independent and identically distributed assumption.

Ensemble-dependent averages

Linear averages of observables depend only on $\rho$. Nonlinear quantities assigned to the component states can depend on the decomposition. For example, the average of component-state variances need not equal the variance computed from $\rho$. With $m_i=\operatorname{Tr}(\rho_i A)$ and $\bar m=\sum_i p_i m_i$, the law of total variance gives

$$\operatorname{Var}_{\bar\rho}(A) =\sum_i p_i\operatorname{Var}_{\rho_i}(A) +\sum_i p_i(m_i-\bar m)^2.$$

For the $z$-basis ensemble of $I/2$, each component has zero $Z$ variance and the conditional means $+1,-1$ supply total variance one. For its $x$-basis ensemble, both conditional means vanish and each component has variance one. The observed total is the same; its split into quantum uncertainty within a preparation and classical variation between preparations depends on the ensemble.

This is why mixed-state constructions such as entanglement of formation optimize over all pure-state decompositions rather than selecting one arbitrary ensemble.

Updating a preparation label

Suppose outcome $a$ has positive probability and the ensemble model is known. Bayes’ rule updates the classical label:

$$p(i\mid a) = \frac{ p_i\operatorname{Tr}(\rho_iE_a) }{ \operatorname{Tr}(\bar\rho E_a) }.$$

This update concerns uncertainty about which preparation branch occurred. The quantum post-measurement state also depends on the measurement instrument, not only on the effect $E_a$. Classical label updating and quantum state update answer different questions and should not be conflated.

Purification and remote ensembles

Every finite-dimensional density operator can be realized as the reduced state of a larger pure state. Suitable POVMs on the purifying system can realize every finite pure-state ensemble decomposition of the same reduced state. Projective measurements on a fixed minimal reference alone need not suffice for ensembles with more labels than its dimension.

For the Bell state,

$$\lvert\Phi^+\rangle = \frac{ \lvert00\rangle+\lvert11\rangle }{\sqrt2} = \frac{ \lvert++\rangle+\lvert--\rangle }{\sqrt2}.$$

A computational-basis measurement on $B$ produces conditional $z$-basis states on $A$; communicating the outcome lets Alice identify the corresponding subensemble. An $x$-basis measurement instead produces the conditional $x$-basis ensemble. If the outcome is ignored, both average to

$$\rho_A=\frac{I_2}{2}.$$

The Hughston–Jozsa–Wootters theorem classifies this relationship between ensemble decompositions and measurements on a purification. Its proof and generality belong to Purification. Conditional States develops the steering and no-signaling distinction.

Classical uncertainty and entanglement

A mixture generated by a classical randomizer and a reduced state generated by entanglement can be identical as operators on the subsystem. Their global contexts differ: a classical record or a purifying system may carry different correlations.

The density operator is operationally complete for the subsystem alone. It need not encode every fact about its preparation history or environment. Conversely, the existence of one ensemble containing entangled vectors does not prove that a mixed bipartite state is entangled; another decomposition may use only product states. Mixed-state separability is a property of the density operator, addressed in Separable Mixed States.

Common Mistakes

Reading the preparation history from eigenvectors. The spectral ensemble is one possible realization. Its components need not be the states produced by the actual source.

Extending one-copy equality to a correlated source. Equal single-emission density operators do not specify the joint state of several emissions. State explicitly whether repetitions are independent.

Interpreting an ensemble decomposition as a coherent sum. Individual ket phases cancel in a mixture of projectors. A superposition instead retains relative amplitudes and phases before the outer product is taken.

Treating a classical posterior as a quantum output state. Bayes’ rule updates the preparation label using the outcome likelihood. The detector’s instrument is still needed to predict the state it leaves behind.

Where deeper treatment lives

Trace Rule for Expectation Values develops the measurement calculation. Purification Overview introduces the larger-system representation. Classical Mixtures vs Quantum Superpositions focuses on coherence and basis changes.

The canonical purification theorem is in Purification, and remote outcome conditioning is in Conditional States.

References

• L. P. Hughston, R. Jozsa, and W. K. Wootters, “A Complete Classification of Quantum Ensembles Having a Given Density Matrix”, Physics Letters A 183, 14–18 (1993).
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th anniversary ed., Cambridge University Press (2010), Secs. 2.4 and 2.5.
• A. Peres, Quantum Theory: Concepts and Methods, Kluwer Academic (1995), Ch. 6.
• J. Watrous, The Theory of Quantum Information, Cambridge University Press (2018), Sec. 2.1.

Exercises

1. A source prepares $\lvert0\rangle$ with probability $p$ and $\lvert1\rangle$ with probability $1-p$. Find the average density matrix and the probabilities of a computational-basis measurement.

Solution

The average state is

$$\rho = p\lvert0\rangle\langle0\rvert + (1-p)\lvert1\rangle\langle1\rvert.$$

In the computational basis,

$$\rho = \begin{pmatrix} p&0\\ 0&1-p \end{pmatrix}.$$

The computational-basis projectors are $P_0=\lvert0\rangle\langle0\rvert$ and $P_1=\lvert1\rangle\langle1\rvert$. Therefore

$$p(0)=\operatorname{Tr}(\rho P_0)=p,$$

and

$$p(1)=\operatorname{Tr}(\rho P_1)=1-p.$$

2. Verify explicitly that equal mixtures of the $z$-basis states and the $x$-basis states both give $I_2/2$.

Solution

Completeness gives

$$\lvert0\rangle\langle0\rvert + \lvert1\rangle\langle1\rvert = I_2,$$

and

$$\lvert+\rangle\langle+\rvert + \lvert-\rangle\langle-\rvert = I_2.$$

Multiplying either identity by $1/2$ gives the same average density operator. Every unconditioned qubit measurement therefore has the same statistics for the two ensembles.

3. For $0<r<1$, verify the nonorthogonal decomposition of $\rho_r$ given above.

Solution

Let

$$a=\sqrt{\frac{1+r}{2}}, \qquad b=\sqrt{\frac{1-r}{2}}.$$

Then

$$\lvert\psi_\pm\rangle = a\lvert0\rangle \pm b\lvert1\rangle.$$

Their projectors are

$$\lvert\psi_\pm\rangle\langle\psi_\pm\rvert = \begin{pmatrix} a^2&\pm ab\\ \pm ab&b^2 \end{pmatrix}.$$

The equal average cancels the off-diagonal entries:

$$\frac12 \lvert\psi_+\rangle\langle\psi_+\rvert + \frac12 \lvert\psi_-\rangle\langle\psi_-\rvert = \begin{pmatrix} a^2&0\\ 0&b^2 \end{pmatrix} = \rho_r.$$

Their overlap is $a^2-b^2=r$, so they are nonorthogonal.

4. Prove that every positive-probability vector in a pure-state ensemble for $\rho$ lies in $\operatorname{supp}\rho$.

Solution

Let $\lvert\chi\rangle\in\ker\rho$. Then

$$\begin{aligned} 0 &= \langle\chi\rvert\rho\lvert\chi\rangle \\ &= \sum_i p_i \lvert \langle\chi\vert\psi_i\rangle \rvert^2. \end{aligned}$$

Every summand is nonnegative. For each $p_i>0$, it follows that $\langle\chi\vert\psi_i\rangle=0$. Thus each ensemble vector is orthogonal to the kernel and lies in the support. Because the ensemble vectors must span the support, at least $\operatorname{rank}\rho$ of them are needed.

5. Show that tracing out the classical register in

$$\rho_{XQ} = \sum_i p_i \lvert i\rangle\langle i\rvert_X \otimes \rho_i^{(Q)}$$

gives the average state on $Q$.

Solution

Use

$$\operatorname{Tr}_X \left( \lvert i\rangle\langle i\rvert_X \otimes \rho_i^{(Q)} \right) = \rho_i^{(Q)}$$

for every normalized record state. Linearity gives

$$\operatorname{Tr}_X\rho_{XQ} = \sum_i p_i\rho_i^{(Q)} = \bar\rho_Q.$$

The discarded register contained the label needed to select a conditional subensemble.

6. A fair source prepares $\lvert0\rangle$ or $\lvert1\rangle$. A noisy detector reports the correct label with probability $1-\epsilon$ and the wrong label with probability $\epsilon$. If the detector reports $0$, find the posterior probability that $\lvert0\rangle$ was prepared.

Solution

Bayes’ rule gives

$$\begin{aligned} p(i=0\mid a=0) &= \frac{ \tfrac12(1-\epsilon) }{ \tfrac12(1-\epsilon)+\tfrac12\epsilon } \\ &= 1-\epsilon. \end{aligned}$$

The equal prior makes the posterior match the detector reliability. This updates the classical preparation label; a complete quantum state update would additionally require the detector’s measurement instrument.

7. Compare the two-emission states $\rho_{\mathrm{iid}}$ and $\rho_{\mathrm{corr}}$. Compute $\langle Z\otimes Z\rangle$ for each.

Solution

For the independent source,

$$\begin{aligned} \langle Z\otimes Z\rangle_{\mathrm{iid}} &= \operatorname{Tr} \left[ \left( \frac{I_2}{2}\otimes\frac{I_2}{2} \right) (Z\otimes Z) \right] \\ &= \frac14 \operatorname{Tr}Z\, \operatorname{Tr}Z =0. \end{aligned}$$

For the correlated source, both emitted bits are always equal, so each component has $Z\otimes Z$ eigenvalue $+1$. Hence

$$\langle Z\otimes Z\rangle_{\mathrm{corr}}=1.$$

The one-copy marginals agree, while the two-copy states and their joint statistics differ.

8. For $\lvert\Phi^+\rangle$, identify the conditional ensembles on $A$ when $B$ is measured in the $z$ basis and in the $x$ basis. Show that both unconditioned states equal $I_2/2$.

Solution

Using

$$\lvert\Phi^+\rangle = \frac{ \lvert00\rangle+\lvert11\rangle }{\sqrt2},$$

a $z$-basis measurement on $B$ gives $\lvert0\rangle_A$ or $\lvert1\rangle_A$, each with probability $1/2$. Its average is

$$\frac12 \lvert0\rangle\langle0\rvert + \frac12 \lvert1\rangle\langle1\rvert = \frac{I_2}{2}.$$

Using instead

$$\lvert\Phi^+\rangle = \frac{ \lvert++\rangle+\lvert--\rangle }{\sqrt2},$$

an $x$-basis measurement gives $\lvert+\rangle_A$ or $\lvert-\rangle_A$, again with equal probabilities. The average is

$$\frac12 \lvert+\rangle\langle+\rvert + \frac12 \lvert-\rangle\langle-\rvert = \frac{I_2}{2}.$$

The communicated remote outcome selects a conditional subensemble; without that outcome, the local state is unchanged.