Purification

Quantum theory speaks two dialects of "state." Sometimes a system is a vector $|\psi\rangle$ in Hilbert space — pure, maximally informative, a single ray. Sometimes it is a density operator $\rho$ — mixed, carrying classical uncertainty or the shadow of entanglement with something outside. Purification dissolves the apparent dichotomy by showing it was never fundamental: every mixed state is simply what a pure state of a larger system looks like once you cover part of it. Adjoin an auxiliary "reference" system $R$ , and any $\rho_A$ can be written as the partial trace of a pure entangled state $|\psi\rangle_{AR}$ . Mixedness becomes ignorance of correlations — entanglement with an environment you have chosen, or been forced, to ignore. The construction is explicit and tight. Diagonalize $\rho_A=\sum_i p_i|i\rangle\langle i|$ , take a reference at least as large as the number of nonzero eigenvalues, and form $|\psi\rangle_{AR}=\sum_i\sqrt{p_i}\,|i\rangle_A|i\rangle_R$ . That is precisely a Schmidt decomposition, so the eigenvalues of $\rho_A$ are the squared Schmidt coefficients, the reference inherits the very same spectrum, and the two halves always carry equal entropy. The minimal reference dimension is the rank of $\rho_A$ . Purifications are far from unique, but their non-uniqueness is fully characterized: any two purifications of the same $\rho_A$ are related by an isometry acting on the reference alone. This single fact — the operational core of the GHJW theorem — underlies the interchangeability of ensembles, the steering of remote systems, the worst-case eavesdropper of quantum cryptography, and the isometric dilation of every noisy channel. Purification is not a trick but a lens: it converts statements about mixed states and noisy dynamics into statements about pure states and unitary evolution, bridging the operator picture and the state-vector picture of quantum information.

Key takeaways

Every density operator $\rho_A$ can be written as $\rho_A=\mathrm{Tr}_R\!\big(|\psi\rangle\langle\psi|_{AR}\big)$ for a pure state $|\psi\rangle_{AR}$ on an enlarged Hilbert space; $|\psi\rangle_{AR}$ is a purification of $\rho_A$ .
A purification always exists with reference dimension $\dim\mathcal{H}_R\ge\mathrm{rank}(\rho_A)$ ; the eigendecomposition of $\rho_A$ gives the canonical Schmidt purification $\sum_i\sqrt{p_i}\,|i\rangle_A|i\rangle_R$ .
All purifications of one $\rho_A$ are related by an isometry (a unitary when the references match dimension) on $R$ alone — unique up to reference-side isometries, the operational GHJW theorem.
The reference shares the spectrum of $\rho_A$ , so the two halves have equal von Neumann entropy $S(\rho_A)=S(\rho_R)$ : how mixed the part is equals how entangled the whole is.
Measuring the reference steers $A$ into any ensemble consistent with $\rho_A$ (Hughston–Jozsa–Wootters), yet leaves $\rho_A$ itself untouched — the no-signalling guarantee.
Key idea: adjoin a reference system $R$ and write the mixed state $\rho_A$ as the partial trace of a pure entangled state $|\psi\rangle_{AR}$ ; the Schmidt decomposition makes the eigenvalues of $\rho_A$ the squared Schmidt coefficients, and any two purifications differ only by an isometry on $R$ .

The core insight

Adjoin a reference system $R$ and write the mixed state $\rho_A$ as the partial trace of a pure entangled state $|\psi\rangle_{AR}$ ; the Schmidt decomposition of $|\psi\rangle_{AR}$ makes the eigenvalues of $\rho_A$ the squared Schmidt coefficients, and any two purifications differ only by an isometry acting on the reference alone.

Where this lives today

Purification is the quiet engine under a great deal of working quantum information. In quantum key distribution one proves security by handing the eavesdropper the purification: because Alice and Bob's joint mixed state can always be purified, assuming Eve holds the reference is the provably worst case, and bounds on her knowledge follow from entropic quantities such as $S(\rho_R)=S(\rho_{AB})$ . In open-system physics, treating decoherence as entanglement with an environment is purification in reverse — a noisy reduced state is the partial trace of a pure system–environment evolution, and the Stinespring dilation of the resulting channel is exactly how one simulates arbitrary noise on a quantum computer by adjoining ancillas, applying a unitary, and discarding them. Numerical many-body methods store locally purified tensor-network states, guaranteeing positivity by construction and giving $S(\rho_A)=S(\rho_R)$ its meaning as the entanglement across a cut that controls simulability. And in high-energy theory the thermofield double — the canonical purification of a Gibbs state — ties two-sided eternal black holes to entanglement between two boundary copies, placing Schrödinger's 1936 steering at the center of the holographic study of spacetime.

The physical picture

A mixed state is what a pure entangled state looks like through a keyhole. Correlate two coins so that they always match, then hide one: your description of the coin you can see is a probabilistic mixture, even though the joint object sits in one definite, pure configuration. Purification reverses the keyhole. It hands you the rest of the wavefunction, the reference system $R$ , so that the whole becomes pure again and the mixedness of the part is re-read as entanglement across the cut. Nothing about $A$ has changed; you have only enlarged the description until it closes into a pure state. The eigenvalues $\{p_i\}$ of $\rho_A$ are not arbitrary weights. In the canonical purification they are exactly the squared Schmidt coefficients of the global pure state, so "how mixed $\rho_A$ is" becomes "how entangled the purification is." A pure $\rho_A$ has rank one and needs no reference at all; a maximally mixed $\rho_A=I/d$ demands a maximally entangled purification. The first figure fixes this with concrete numbers.

Figure

A mixed state is a pure entangled state seen through a keyhole: the global state ψ on A together with reference R is pure, yet discarding R by the partial trace Tr_R leaves ρ_A mixed with eigenvalues 0.7 and 0.3 — the squared Schmidt coefficients √0.7 = 0.837 and √0.3 = 0.548 — and entropy S(ρ_A) = S(ρ_R) = 0.881 bits

The freedom in purification is precisely a rotation on the unseen part. Because nothing done to the reference can be felt at $A$ , the many purifications of one $\rho_A$ all agree on every observable prediction for the system you can access; they differ only in how the inaccessible correlations are labeled. The deepest reading is quantitative — the entropy of $A$ equals the entropy of $R$ — so a single curve governs the whole story. The second figure is that curve for a qubit: the binary entropy tying the mixedness of the part to the entanglement of the whole.

Figure

How mixed equals how entangled: for a qubit ρ_A with eigenvalues p and 1−p the entanglement entropy of the purification is the binary entropy S(p), which is zero for a pure ρ_A giving a product-state purification and rises to a maximum at p = ½ where the purification is the Bell state with S = 1 bit; the marked point p = 0.3 gives S = 0.881 bits

The core equations

The whole subject rests on one definition. A purification is any pure state on an enlarged space whose restriction to the system — its partial trace over the reference — is the given mixed state, so the defining equation reads a mixed state as the local face of a pure whole.

\rho_A=\mathrm{Tr}_R\!\big(|\psi\rangle\langle\psi|_{AR}\big) \qquad (1)

where: $\rho_A$ — the density operator on $\mathcal{H}_A$ to be purified; $|\psi\rangle_{AR}$ — a pure purifying state on $\mathcal{H}_A\otimes\mathcal{H}_R$ ; $\mathrm{Tr}_R$ — the partial trace over the reference; $\mathcal{H}_R$ — the reference (purifying, or ancilla) Hilbert space. Read it as: $\rho_A$ is what remains of a pure $|\psi\rangle_{AR}$ once $R$ is thrown away. The canonical construction turns the eigendecomposition of $\rho_A$ directly into a pure state: diagonalize $\rho_A=\sum_i p_i|i\rangle\langle i|$ , pick any orthonormal set $\{|i\rangle_R\}$ in the reference, and pair them up with matching amplitudes. This is a Schmidt decomposition by construction, exhibiting the eigenvalues of $\rho_A$ as squared Schmidt coefficients.

|\psi\rangle_{AR}=\sum_i\sqrt{p_i}\;|i\rangle_A\otimes|i\rangle_R \qquad (2)

where: $p_i$ — the eigenvalues of $\rho_A$ , with $p_i\ge0$ and $\sum_i p_i=1$ ; $|i\rangle_A$ — the orthonormal eigenbasis of $\rho_A$ ; $|i\rangle_R$ — any chosen orthonormal set in the reference; $\sqrt{p_i}$ — the Schmidt coefficients of the global pure state. Verifying $\mathrm{Tr}_R$ of this returns $\rho_A$ is the first derivation below. Tracing out the system instead of the reference reveals the symmetry at the heart of purification: the reference carries the identical nonzero spectrum, so the two halves are equally mixed and their von Neumann entropies coincide exactly.

\rho_R=\mathrm{Tr}_A\!\big(|\psi\rangle\langle\psi|\big)=\sum_i p_i\,|i\rangle_R\langle i|,\qquad S(\rho_A)=S(\rho_R)=-\sum_i p_i\log p_i \qquad (3)

where: $\rho_R$ — the reduced state of the reference; $S(\cdot)$ — the von Neumann entropy, in bits when $\log=\log_2$ ; $p_i$ — the shared eigenvalues. The entropy measures the entanglement of the purification across the $A\,{:}\,R$ cut, and is the same seen from either side. Purifications are not unique, and the freedom is completely tamed: two purifications of the same $\rho_A$ differ only by an inner-product-preserving operation on the reference — an isometry, or a unitary if the reference dimension is fixed.

|\psi'\rangle_{AR'}=(\mathbb{1}_A\otimes V)\,|\psi\rangle_{AR},\qquad V^\dagger V=\mathbb{1}_R \qquad (4)

where: $V$ — an isometry $\mathcal{H}_R\to\mathcal{H}_{R'}$ (a unitary when $\dim\mathcal{H}_{R'}=\dim\mathcal{H}_R$ ); $\mathbb{1}_A$ — the identity on the system, left untouched; $|\psi'\rangle$ — another purification of the very same $\rho_A$ . This is the operational GHJW freedom, proved in the second derivation. How large must the reference be? Exactly large enough to carry the nonzero Schmidt terms; any larger reference simply adds decoupled dimensions of zero weight.

\dim\mathcal{H}_R\;\ge\;\mathrm{rank}(\rho_A) \qquad (5)

where: $\mathrm{rank}(\rho_A)$ — the number of nonzero eigenvalues of $\rho_A$ ; equality gives the minimal purification. A pure state has rank one and needs a one-dimensional (trivial) reference. A basis-free way to write every purification uses the operator–vector correspondence: with the unnormalized maximally entangled "identity vector" $|\Gamma\rangle$ , any matrix $M$ on the system acting on $|\Gamma\rangle$ produces a purification of $MM^\dagger$ .

|\Gamma\rangle=\sum_i|i\rangle_A\otimes|i\rangle_R,\qquad |\psi\rangle=(M\otimes\mathbb{1})\,|\Gamma\rangle,\qquad \mathrm{Tr}_R\big(|\psi\rangle\langle\psi|\big)=MM^\dagger \qquad (6)

where: $|\Gamma\rangle$ — the vectorized identity (unnormalized maximally entangled state); $M$ — any operator on $\mathcal{H}_A$ ; choosing $M=\sqrt{\rho_A}$ gives the canonical purification, and $M=\sqrt{\rho_A}\,U$ with $U$ unitary gives every other one — the isometric freedom again. Uhlmann's theorem uses exactly this freedom to define the fidelity of mixed states as the maximal overlap of their purifications, $F(\rho,\sigma)=\max|\langle\psi_\rho|\psi_\sigma\rangle|=\|\sqrt{\rho}\sqrt{\sigma}\|_1$ .

Derivation AExistence and uniqueness of the purification

The goal is to establish the two structural facts that make purification usable: that a purification always exists — built directly from the eigendecomposition of $\rho_A$ — and that any two purifications on references of equal dimension are related by a unitary acting on the reference alone. Together they say the purification is essentially canonical, and its only freedom is a harmless rotation of the part you never touch; ensemble equivalence and no-signalling are corollaries.

Step 1. Diagonalize $\rho_A=\sum_i p_i|i\rangle_A\langle i|$ and define $|\psi\rangle_{AR}=\sum_i\sqrt{p_i}\,|i\rangle_A|i\rangle_R$ on any reference with an orthonormal set $\{|i\rangle_R\}$ . Expanding the projector, $|\psi\rangle\langle\psi|=\sum_{i,k}\sqrt{p_ip_k}\,|i\rangle_A\langle k|\otimes|i\rangle_R\langle k|$ . Why: writing the outer product over both indices exposes the reference factors $|i\rangle_R\langle k|$ that the partial trace is about to contract.

Step 2. Take the partial trace over $R$ in the basis $\{|j\rangle_R\}$ , using $\mathrm{Tr}_R(|i\rangle_R\langle k|)=\langle k|i\rangle_R=\delta_{ik}$ . The double sum collapses to $\mathrm{Tr}_R|\psi\rangle\langle\psi|=\sum_i p_i\,|i\rangle_A\langle i|=\rho_A$ . Existence is proved: the canonical Schmidt state is a purification. Why: orthonormality of the reference kills every cross term $i\ne k$ , leaving exactly the eigendecomposition we started from — the construction is self-verifying.

Step 3. For uniqueness, let $|\psi\rangle_{AR}$ and $|\phi\rangle_{AR}$ both purify $\rho_A$ on a reference of dimension $d_R\ge\mathrm{rank}(\rho_A)$ . Schmidt-decompose both: $|\psi\rangle=\sum_i\sqrt{p_i}\,|i\rangle_A|e_i\rangle_R$ and $|\phi\rangle=\sum_i\sqrt{p_i}\,|i\rangle_A|f_i\rangle_R$ . They share the same coefficients $\sqrt{p_i}$ and the same system Schmidt basis $|i\rangle_A$ because both reduce to the identical $\rho_A=\sum_i p_i|i\rangle_A\langle i|$ ; only the reference vectors $|e_i\rangle_R$ and $|f_i\rangle_R$ may differ. Why: the reduced state fixes the coefficients and the system basis uniquely (up to degeneracy), so all the remaining freedom is quarantined on the reference.

Step 4. Define $U=\sum_i|f_i\rangle_R\langle e_i|$ . On the rank- $r$ support both $\{|e_i\rangle\}$ and $\{|f_i\rangle\}$ are orthonormal, so $U$ is a partial isometry; since $d_R$ is equal on both sides, extend it to a full unitary by mapping the orthogonal complements onto each other. Then $(\mathbb{1}_A\otimes U)|\psi\rangle=\sum_i\sqrt{p_i}\,|i\rangle_A\,U|e_i\rangle=\sum_i\sqrt{p_i}\,|i\rangle_A|f_i\rangle=|\phi\rangle$ . Why: $U$ acts only on the reference and turns one purification into the other, so nothing observable at $A$ can distinguish them — the concrete meaning of "unique up to a reference unitary." *Verified: the partial-trace identity $\mathrm{Tr}_R|\psi\rangle\langle\psi|=\rho_A$ and the unitary relation reproduce Nielsen and Chuang §2.5; degenerate eigenvalues only enlarge the freedom (the system Schmidt basis is then non-unique) without changing the conclusion.*

Go deeper · Derivation B · 1 minSteering a remote ensemble

Derivation BSteering a remote ensemble

The aim is to show what a measurement on the reference actually accomplishes: by choosing a measurement basis on $R$ , one can prepare at a distance any pure-state ensemble whose average is $\rho_A$ , and every ensemble decomposition of $\rho_A$ arises this way — yet no such choice alters $\rho_A$ itself. This is the Hughston–Jozsa–Wootters theorem, the physical content of the reference-side freedom and the mechanism behind ensemble interchangeability and no-signalling.

Step 1. Fix a purification $|\psi\rangle_{AR}$ of $\rho_A$ and measure $R$ projectively in a basis $\{|\mu\rangle_R\}$ . The unnormalized conditional state left on $A$ when outcome $\mu$ occurs is $|\tilde\chi_\mu\rangle_A=\langle\mu|_R\,|\psi\rangle_{AR}$ , arising with probability $q_\mu=\langle\tilde\chi_\mu|\tilde\chi_\mu\rangle$ by the Born rule. Why: projecting the reference onto $|\mu\rangle$ slices the global state into a system piece; its norm is exactly the outcome probability.

Step 2. Sum the conditional projectors over a complete basis: $\sum_\mu|\tilde\chi_\mu\rangle\langle\tilde\chi_\mu|=\sum_\mu\langle\mu|\psi\rangle\langle\psi|\mu\rangle=\mathrm{Tr}_R|\psi\rangle\langle\psi|=\rho_A$ . So whatever basis is chosen on $R$ , the unconditioned state on $A$ is $\rho_A$ . Why: summing over a complete measurement basis is literally the partial trace, so the average state at $A$ is basis-independent — this is the no-signalling guarantee in one line.

Step 3. Change the reference basis, $\{|\mu\rangle_R\}\to\{|\nu\rangle_R\}$ . The conditional states and probabilities change, $\{q_\mu,|\chi_\mu\rangle\}\to\{r_\nu,|\omega_\nu\rangle\}$ , giving a genuinely different ensemble decomposition of the same $\rho_A$ . By choosing the reference basis appropriately, one realizes any ensemble consistent with $\rho_A$ . Why: the reference stores the "which pure state" information, and reading it in different bases sorts $A$ into different pure ensembles — all averaging back to the one $\rho_A$ . *Verified: the unconditioned $\rho_A$ is invariant under the choice of reference measurement, so a remote party cannot signal by choosing what to measure; cross-checked against Preskill Ch. 2 and the GHJW classification.*

Go deeper · Derivation C · 1 minFrom a purification to a quantum channel

Derivation CFrom a purification to a quantum channel

The purpose here is to connect state purification to its dynamical sibling, the Stinespring dilation of channels. Just as a mixed state is a pure state with a discarded reference, a noisy channel is a unitary evolution on a dilated space followed by a discard. Making this precise turns "adjoin an ancilla, evolve unitarily, trace it out" into the operator-sum (Kraus) form of every completely positive trace-preserving map.

Step 1. Dilate the input by adjoining a reference in a fixed pure state, $\rho_A\mapsto\rho_A\otimes|0\rangle\langle0|_R$ , then act with a global unitary $U_{AR}$ and discard $R$ : $\mathcal{N}(\rho_A)=\mathrm{Tr}_R\!\big(U_{AR}(\rho_A\otimes|0\rangle\langle0|_R)U_{AR}^\dagger\big)$ . Why: a fresh ancilla in a definite state is the trivial purification of the environment's "blank" input; the unitary then entangles system and ancilla exactly as system–environment evolution would.

Step 2. Expand the discarded reference in a basis $\{|k\rangle_R\}$ and define $K_k=\langle k|_R\,U_{AR}\,|0\rangle_R$ , operators on $\mathcal{H}_A$ alone. Unitarity of $U_{AR}$ gives $\sum_k K_k^\dagger K_k=\langle0|U^\dagger\big(\sum_k|k\rangle\langle k|\big)U|0\rangle=\mathbb{1}$ . Why: tracing out the reference in a chosen basis peels the global unitary into a family of system operators, and the completeness of the basis enforces the trace-preserving condition automatically.

Step 3. Assemble the operator-sum representation $\mathcal{N}(\rho_A)=\sum_k K_k\,\rho_A\,K_k^\dagger$ . This is the Kraus form of a CPTP map, so every physical channel is unitary evolution on a purified/dilated space followed by a partial trace. Why: the same "enlarge, evolve, discard" recipe that reads a mixed state as a reduced pure state reads a channel as a restricted isometry — one idea, applied to states and to dynamics. *Verified: equivalent to the Stinespring dilation theorem; the freedom in the discarded basis $\{|k\rangle\}$ reproduces the unitary freedom among Kraus sets, matching Watrous §2.2 and Wilde Ch. 5.*

Limiting cases

Reading the purification at the extremes of mixedness exposes its whole structure at a glance.

Case	Result	Meaning
Pure state	$\rho_A=\vert\phi\rangle\langle\phi\vert$ , rank 1	The minimal purification is the product state $\vert\phi\rangle_A\otimes\vert0\rangle_R$ : no entanglement, $S=0$ , the reference is trivial.
Maximally mixed qubit	$\rho_A=\tfrac12 I$	Rank 2 with equal weights; minimal purification is the Bell state $\tfrac{1}{\sqrt2}(\vert0\rangle_A\vert0\rangle_R+\vert1\rangle_A\vert1\rangle_R)$ , maximally entangled with $S=\log_2 2=1$ bit.
Maximally mixed qudit	$\rho_A=\tfrac1d I$	Minimal reference dimension $d$ ; canonical purification $\tfrac{1}{\sqrt d}\sum_{i}\vert i\rangle_A\vert i\rangle_R$ , entropy $\log_2 d$ — the maximum for dimension $d$ .
Generic rank- $r$ state	$\mathrm{rank}(\rho_A)=r$	Minimal reference dimension is exactly $r$ ; a larger reference works but the extra dimensions decouple, carrying zero Schmidt weight.
Thermal (Gibbs) state	$\rho_A=e^{-\beta H}/Z$	Purified by the thermofield double $\propto\sum_n e^{-\beta E_n/2}\vert n\rangle_A\vert n\rangle_R$ , with Schmidt weights $e^{-\beta E_n}/Z$ and $S=$ thermal entropy.

Common trapThe purifying system is a real environment, so measuring it disturbs the state at A

It does not. The reference $R$ is a mathematical device: for any $\rho_A$ you may invent it, choose its dimension (anything $\ge\mathrm{rank}\,\rho_A$ ), and choose the purification (any two differ by a reference-side isometry). Every physically accessible prediction at $A$ depends only on $\rho_A$ , never on which purification you imagined. The tempting error is to treat a chosen purification — an "environment," or an eavesdropper "holding the rest of the wavefunction" — as an objective object whose manipulation reaches back to $A$ . It cannot: any operation confined to $R$ , including a projective measurement in any basis, leaves $\mathrm{Tr}_R|\psi\rangle\langle\psi|=\rho_A$ exactly invariant, so it can neither change $A$ 's statistics nor transmit a signal. Steering only sorts $A$ into ensembles that already averaged to $\rho_A$ ; it never alters $\rho_A$ . The purification is a lens for reasoning, not a hidden extra fact.

Worked examples

1 · Purifying a qubit, and building a second purification (fully worked)

Purify $\rho_A=0.7\,|0\rangle\langle0|+0.3\,|1\rangle\langle1|$ . Give the canonical purification, verify the partial trace, find the entropy and minimal reference dimension, then exhibit a physically distinct purification of the same $\rho_A$ .

Step 1. The state is already diagonal, with eigenvalues $p_0=0.7$ , $p_1=0.3$ and eigenbasis $\{|0\rangle,|1\rangle\}$ . Equation (2) gives the canonical purification on a 2-dimensional reference, $|\psi\rangle_{AR}=\sqrt{0.7}\,|0\rangle_A|0\rangle_R+\sqrt{0.3}\,|1\rangle_A|1\rangle_R$ , with Schmidt coefficients $\sqrt{0.7}=\mathbf{0.837}$ and $\sqrt{0.3}=\mathbf{0.548}$ .

Step 2. Verify by partial trace. $\mathrm{Tr}_R|\psi\rangle\langle\psi|=0.7\,|0\rangle\langle0|\,\langle0|0\rangle+0.3\,|1\rangle\langle1|\,\langle1|1\rangle+(\text{cross terms})\,\langle1|0\rangle=0.7\,|0\rangle\langle0|+0.3\,|1\rangle\langle1|=\rho_A$ . The cross terms die because $\langle1|0\rangle_R=0$ . The construction returns $\rho_A$ exactly.

Step 3. Entropy and dimension. Since $\mathrm{rank}(\rho_A)=2$ , the minimal reference dimension is $\mathbf{2}$ . The entropy is $S(\rho_A)=-0.7\log_2 0.7-0.3\log_2 0.3=\mathbf{0.881\ bits}$ , and by equation (3) the reference carries the identical value $S(\rho_R)=0.881$ bits — the marked point on the second figure.

Step 4. A distinct purification. Apply a Hadamard on the reference via equation (4): $|\psi'\rangle=(\mathbb{1}_A\otimes H)|\psi\rangle=\sqrt{0.7}\,|0\rangle_A|+\rangle_R+\sqrt{0.3}\,|1\rangle_A|-\rangle_R$ . Its reference vectors differ from $|0\rangle,|1\rangle$ , yet $\mathrm{Tr}_R|\psi'\rangle\langle\psi'|=\rho_A$ still, because $\{|+\rangle,|-\rangle\}$ is orthonormal. $|\psi\rangle$ and $|\psi'\rangle$ are different global states with the same reduced $\rho_A$ — the reference-side freedom made concrete.

2 · The thermofield double of a two-level system

A qubit with Hamiltonian $H=\varepsilon\,|1\rangle\langle1|$ is in thermal equilibrium at inverse temperature $\beta$ with $e^{-\beta\varepsilon}=\tfrac14$ . Write its Gibbs state and its thermofield-double purification, and compute the entanglement entropy.

Step 1. The partition function is $Z=1+e^{-\beta\varepsilon}=1+\tfrac14=\tfrac54$ , so the Gibbs state is $\rho=\mathrm{diag}(1/Z,\ e^{-\beta\varepsilon}/Z)=\mathrm{diag}(0.8,\ 0.2)$ . Eigenvalues $p_0=0.8$ , $p_1=0.2$ .

Step 2. The thermofield double is the canonical purification with amplitudes $\sqrt{p_n}$ : $|\mathrm{TFD}\rangle=\sqrt{0.8}\,|0\rangle_A|0\rangle_R+\sqrt{0.2}\,|1\rangle_A|1\rangle_R$ , i.e. Schmidt coefficients $\mathbf{0.894}$ and $\mathbf{0.447}$ . Tracing out the reference copy returns the Gibbs state, and tracing out the system returns the same thermal spectrum on $R$ .

Step 3. The entanglement entropy equals the thermal entropy, $S=-0.8\log_2 0.8-0.2\log_2 0.2=\mathbf{0.722\ bits}$ . The doubled entangled pure state encodes the finite-temperature physics of the single copy — the basis of thermofield dynamics and of two-sided eternal black holes in holography.

Physics of the portrait

Purification began not as a computational tool but as Erwin Schrödinger's attempt, in the wake of the 1935 EPR paper, to understand what entanglement does to the parts of a whole.

Origins · 1 minErwin Schrödinger and how this was discovered

Erwin Schrödinger

1887–1961 · entanglement and the steering of remote states

Erwin Schrödinger, who a decade earlier had given quantum mechanics its wave equation, turned in 1935 to the strangeness of composite systems. Reacting to the Einstein–Podolsky–Rosen argument, he wrote a pair of papers on the "probability relations between separated systems" and coined the word entanglement (Verschränkung), calling it "the characteristic trait of quantum mechanics." His decisive technical result — the ancestor of purification and the GHJW theorem — was the steering theorem: given one half of an entangled pure state, a distant experimenter can, by choosing what to measure on the other half, collapse the near half into any ensemble compatible with its reduced state, and into no ensemble that is not. In modern language, Schrödinger had shown that every mixed state is the restriction of a pure entangled state, and that the many ensembles hiding behind a single density operator are exactly the ones a remote measurement can prepare. His steering became the operational content of the density operator, sharpened by Jaynes in 1957 and classified definitively by Hughston, Jozsa, and Wootters in 1993. The "thermofield double" that doubles a thermal state into a pure entangled state, central to modern finite-temperature and black-hole physics, is his construction wearing a statistical-mechanics hat.

History

1935

— Schrödinger names entanglement and analyzes the EPR state's separated subsystems

1936

— Schrödinger proves the steering theorem: a remote measurement prepares any ensemble consistent with the reduced state

1955

— Stinespring's dilation theorem casts every completely positive map as an isometry followed by restriction

1975

— Uhlmann defines the fidelity of mixed states as a maximal overlap of purifications

1993

— Hughston, Jozsa, and Wootters give the operational classification of purifications and ensembles

Going further

Texts. Nielsen and Chuang, Quantum Computation and Quantum Information, §2.5 (Schmidt decomposition and purifications) · Preskill, Quantum Information Lecture Notes, Ch. 2 (density operators, GHJW, and the HJW theorem) · Watrous, The Theory of Quantum Information, §2.2 (purifications and isometric equivalence) · Wilde, Quantum Information Theory, Ch. 5 (purification and isometric extension) · Kitaev, Shen, and Vyalyi, Classical and Quantum Computation, §11 (mixed states and purification).

Provenance. A verifier should recompute: (1) the partial-trace identity $\mathrm{Tr}_R|\psi\rangle\langle\psi|=\sum_i p_i|i\rangle\langle i|=\rho_A$ for the canonical purification; (2) equality of the spectra of $\rho_A$ and $\rho_R$ , hence $S(\rho_A)=S(\rho_R)$ ; (3) the Bell-state purification of $\tfrac12 I$ with entropy $1$ bit, and $\log_2 d$ for $I/d$ ; (4) the uniqueness-up-to-reference-unitary argument via shared Schmidt coefficients; (5) the Kraus/Stinespring identities $K_k=\langle k|U|0\rangle$ with $\sum_k K_k^\dagger K_k=\mathbb{1}$ . The worked-example numbers — $\sqrt{0.7}=0.837$ , $\sqrt{0.3}=0.548$ , $S(0.7,0.3)=0.881$ bits; the thermofield double $p=(0.8,0.2)$ with $S=0.722$ bits — were checked independently. All are standard results from Nielsen and Chuang §2.5, Preskill Ch. 2, Watrous §2.2, and Wilde Ch. 5; no non-standard constants are claimed.

Prerequisites. Density Operators · Dirac Bra-Ket Notation · Inner Product, Norm, and Normalization · Projective Measurements. Leads to. Stinespring Dilation · Von Neumann Entropy · Schmidt Decomposition · Quantum Key Distribution.

Ask the tutor: Ask me to purify a given density matrix, steer a remote ensemble two different ways, or turn a noisy channel into a Stinespring dilation.

Why can any mixed state be purified?

Because a density operator is Hermitian and positive with unit trace, it has a spectral decomposition $\rho_A=\sum_i p_i|i\rangle\langle i|$ with non-negative eigenvalues summing to one. That is all you need: pair each eigenvector $|i\rangle_A$ with a fresh orthonormal reference vector $|i\rangle_R$ , weight by $\sqrt{p_i}$ , and the state $\sum_i\sqrt{p_i}\,|i\rangle_A|i\rangle_R$ is pure while tracing out the reference returns $\rho_A$ . The only ingredients are a valid probability distribution and enough room in the reference to hold the orthonormal labels — and the minimal room is the rank of $\rho_A$ .

Is the reference system a real physical object?

Not necessarily. The reference is a mathematical construct you are free to invent; its dimension is any number at least the rank of $\rho_A$ , and the purification is fixed only up to a unitary on it. Sometimes a genuine environment plays the role — a photon that carried away which-path information, or the partner of an entangled pair — and then the purification is realized in the lab. But nothing about $A$ depends on whether the reference is real or imagined: all of $A$ 's statistics come from $\rho_A$ alone. Purification embeds a mixed state in a pure one, and the embedding is a choice, not a fact.

Why must the reference dimension be at least the rank of ρ_A?

Because the canonical purification has exactly $\mathrm{rank}(\rho_A)$ nonzero Schmidt terms, and the reference must supply that many orthonormal vectors to hold them — fewer cannot reproduce the spectrum. A larger reference is allowed and changes nothing observable: the extra dimensions carry zero Schmidt weight and decouple, so the reduced state is still $\rho_A$ ; the only effect is that the family of isometries relating purifications grows. The minimal choice, dimension equal to the rank, gives the tightest purification.

Does measuring the reference let me signal to A?

No. Any operation confined to the reference — a unitary, a measurement in any basis, even discarding it — leaves $\mathrm{Tr}_R|\psi\rangle\langle\psi|=\rho_A$ unchanged, because summing a projective measurement over its complete basis is precisely the partial trace. A remote party can steer $A$ into different ensembles by choosing what to measure, but only ensembles that already average to $\rho_A$ , and only if they announce the outcome; without that classical message $A$ 's state is untouched. This no-signalling property makes shared entanglement a safe cryptographic resource.

How is purification related to the Schmidt decomposition?

They are two views of one object. Any pure bipartite state has a Schmidt decomposition $\sum_i\sqrt{\lambda_i}\,|i\rangle_A|i\rangle_R$ with orthonormal bases and non-negative coefficients. Read left to right it is a purification of $\rho_A=\sum_i\lambda_i|i\rangle_A\langle i|$ ; read whole it is the Schmidt form of a pure state. So the squared Schmidt coefficients are the eigenvalues of the reduced state, the Schmidt rank equals $\mathrm{rank}\,\rho_A$ , and $S(\rho_A)=S(\rho_R)$ is immediate because both reductions share the coefficients. Purification is the statement that every mixed state is the reduction of a Schmidt-decomposed pure state.

What does purification have to do with quantum channels?

Everything, through the Stinespring dilation. A purification reads a mixed state as a pure state with a discarded reference; Stinespring reads a noisy channel as a unitary (isometric) evolution with a discarded reference. Concretely, adjoin an ancilla in a fixed state, apply a global unitary, and trace the ancilla out — the result is a completely positive trace-preserving map with Kraus operators $K_k=\langle k|U|0\rangle$ . Every physical channel arises this way, and the freedom in the discarded basis is the same reference-side freedom that relates purifications of a state. So purification and dilation are one principle applied to statics and to dynamics: enlarge the space until everything is pure and unitary, then restrict. The same freedom lets Uhlmann's theorem define the fidelity of mixed states as the maximal purification overlap $\|\sqrt{\rho}\sqrt{\sigma}\|_1$ .