Uniqueness of Purifications in Quantum Systems

• Uniqueness of purifications is defined as the equivalence of distinct pure-state representations sharing the same mixed-state marginal, connected by ancillary unitaries or isometries.
• The concept extends from finite-dimensional tensor-product formulations to operator-algebraic frameworks where Haag duality and the Uhlmann property provide rigorous foundations.
• Operational constructions and tensor-network approaches reveal that purification non-uniqueness impacts computational efficiency and informs resource-theoretic no-go theorems.

Uniqueness of purifications is the principle that a mixed quantum state can be represented as the marginal of a pure state on a larger system, and that different such pure-state representations are equivalent under transformations acting only on the purifying degrees of freedom. In standard finite-dimensional quantum information theory this appears as uniqueness up to an isometry or unitary on the ancillary system, while in more general operator-algebraic settings the statement becomes nontrivial: for commuting von Neumann algebras it is equivalent to Haag duality, and can fail in infinite systems even when local tomography holds (Luijk et al., 16 Sep 2025).

1. Standard finite-dimensional formulation

For a mixed state $\rho_A$ on a Hilbert space $\mathcal H_A$, a purification is a pure state $|\Psi_{AB}\rangle$ on an enlarged Hilbert space $\mathcal H_A\otimes\mathcal H_B$ such that

$$\rho_A=\operatorname{Tr}_B\!\left(|\Psi_{AB}\rangle\langle\Psi_{AB}|\right).$$

In finite-dimensional quantum mechanics, purifications always exist, and any two purifications of the same state are related by an isometry acting on the purifying system; in the usual bipartite tensor-product setting this is commonly stated as uniqueness up to local unitaries on the purifying subsystem (Liu et al., 25 Sep 2025).

This uniqueness is an equivalence statement rather than literal singleness. A purification of a given state is not unique, and its entanglement strongly depends on the particular choice made. That dependence is operationally important in tensor-network algorithms, entanglement of purification, and related optimization problems, because different representatives of the same purification class can have sharply different computational cost (Hauschild et al., 2017).

In the standard tensor-product model $\mathcal H=\mathcal H_A\otimes\mathcal H_B$, with

$$M_A=B(\mathcal H_A)\otimes 1,\qquad M_B=1\otimes B(\mathcal H_B),$$

Haag duality holds automatically, and the standard uniqueness statement is recovered as a special case of a more general operator-algebraic theorem (Luijk et al., 16 Sep 2025).

2. Operator-algebraic characterization and Haag duality

Let $M_A$ and $M_B$ be commuting von Neumann algebras on a Hilbert space $\mathcal H$. In this setting, Haag duality is the identity

$\mathcal H_A$0

and the Uhlmann property is the requirement that for any $\mathcal H_A$1 satisfying

$\mathcal H_A$2

and for any $\mathcal H_A$3, there exists a unitary $\mathcal H_A$4 such that

$\mathcal H_A$5

The main theorem of "Uniqueness of purifications is equivalent to Haag duality" states that a pair of commuting von Neumann algebras has the Uhlmann property if and only if Haag duality holds, and that in this case there exists a partial isometry $\mathcal H_A$6 with $\mathcal H_A$7 whenever two pure states have identical marginals on $\mathcal H_A$8 (Luijk et al., 16 Sep 2025).

The theorem identifies the precise operator-algebraic condition behind purification uniqueness. In the direction $\mathcal H_A$9 Uhlmann property, the argument uses the uniqueness of the GNS representation to obtain an intertwiner commuting with $|\Psi_{AB}\rangle$0, hence lying in $|\Psi_{AB}\rangle$1. In the converse direction, the Uhlmann property forces projections in $|\Psi_{AB}\rangle$2 into $|\Psi_{AB}\rangle$3, yielding $|\Psi_{AB}\rangle$4. The result replaces the finite-dimensional intuition based on explicit tensor factorization by a purely algebraic criterion.

A common misconception is that local tomography suffices. The operator-algebraic analysis shows otherwise. If $|\Psi_{AB}\rangle$5 and $|\Psi_{AB}\rangle$6 are commuting factors that jointly generate $|\Psi_{AB}\rangle$7, then local tomography holds, but uniqueness of purification can still fail when $|\Psi_{AB}\rangle$8. In particular,

$|\Psi_{AB}\rangle$9

and the implication is strict. The paper gives counterexamples, including an irreducible subfactor inclusion and the infinite surface code, where orthogonal purifications with the same marginal on one side cannot be connected by any local unitary on the other side (Luijk et al., 16 Sep 2025).

3. Generalized probabilistic and indefinite-causal-order frameworks

In general probabilistic theories with purification, the uniqueness statement is elevated to a postulate: every mixed state has a purification, unique up to reversible transformations on the purifying system. If $\mathcal H_A\otimes\mathcal H_B$0 and $\mathcal H_A\otimes\mathcal H_B$1 are two purifications of the same state, then

$\mathcal H_A\otimes\mathcal H_B$2

for some reversible transformation $\mathcal H_A\otimes\mathcal H_B$3. Within that framework, the purification principle is equivalent to reversible realization of every physical process: every channel arises from a reversible interaction with an environment in a pure state followed by discarding, and such reversible dilations are themselves unique up to reversible transformations on the environment (Chiribella et al., 2009).

This GPT formulation shows that uniqueness of purification is not merely a kinematic statement about states. It also constrains the structure of processes, supports a states-transformations isomorphism analogous to Choi-Jamiolkowski, and underlies structural consequences such as no cloning, teleportation, and the characterization of entanglement-breaking channels as measure-and-prepare channels (Chiribella et al., 2009).

In the process-matrix formalism for indefinite causal order, purification becomes selective rather than universal. A process $\mathcal H_A\otimes\mathcal H_B$4 is purifiable if there exists a pure process $\mathcal H_A\otimes\mathcal H_B$5 such that

$\mathcal H_A\otimes\mathcal H_B$6

with purity defined by mapping local unitaries to unitary global evolutions. Theorem 2 gives necessary and sufficient conditions for purifiability in terms of vectors $\mathcal H_A\otimes\mathcal H_B$7 obeying linear and quadratic constraints, while the purification itself is unique up to isometries on the purifying environment. Unlike standard mixed states, however, not all process matrices are purifiable, and the purification postulate is proposed only as a necessary, not sufficient, condition for physicality (Araújo et al., 2016).

4. Constraints and failures in extended quantum systems

When the purifying subsystem is a real interacting environment rather than a formal ancilla, the freedom in choosing a purification can become sharply constrained. In the study of purification reproducing the time evolution of an open quantum system, the purified state

$\mathcal H_A\otimes\mathcal H_B$8

must reproduce the reduced dynamics generated by the actual product initial state. For a macroscopic bath, if $\mathcal H_A\otimes\mathcal H_B$9, then almost any purification built from a typical orthonormal system of bath vectors yields reduced dynamics practically identical to the mixed-state evolution. The successful purifications are therefore not unique, but the class of valid choices is not arbitrary (Inoue et al., 2018).

In free bosonic and Ising conformal field theories, reduced vacuum states are Gaussian mixed states, and Gaussian purifications are organized by a continuous orbit on the purifying subsystem. For a mixed Gaussian state with complex structure $$\rho_A=\operatorname{Tr}_B\!\left(|\Psi_{AB}\rangle\langle\Psi_{AB}|\right).$$0, a minimal Gaussian purification exists on $$\rho_A=\operatorname{Tr}_B\!\left(|\Psi_{AB}\rangle\langle\Psi_{AB}|\right).$$1, and all Gaussian purifications are generated by

$$\rho_A=\operatorname{Tr}_B\!\left(|\Psi_{AB}\rangle\langle\Psi_{AB}|\right).$$2

with $$\rho_A=\operatorname{Tr}_B\!\left(|\Psi_{AB}\rangle\langle\Psi_{AB}|\right).$$3 for bosons or $$\rho_A=\operatorname{Tr}_B\!\left(|\Psi_{AB}\rangle\langle\Psi_{AB}|\right).$$4 for fermions. Entanglement of purification and complexity of purification are then defined by minimizing over this orbit of ancillary Gaussian unitaries (Camargo et al., 2020).

Relativistic quantum field theory furnishes a different type of non-uniqueness. In the analysis of null-shifted Rindler wedges, distinct global pure states lead to identical local thermal occupation numbers. The standard Unruh construction produces mixed Gibbsian thermality through entanglement across a horizon, whereas null-shifted wedge constructions can yield pure tensor-product states with selective and non-Gibbsian thermality. The paper interprets these as inequivalent purifications of thermal spectra and argues that thermal behavior can arise from Bogoliubov mixing and modular time evolution rather than entanglement-induced mixedness (Jha, 28 Jan 2026).

These examples delimit the range of the textbook uniqueness theorem. They show that once one leaves the finite-dimensional tensor-product setting, the relevant question is often not whether purifications exist, but which ancillary transformations remain admissible, which dynamical or algebraic constraints restrict them, and whether the usual equivalence class survives at all.

5. Tensor-network and many-body formulations

In tensor-network descriptions of mixed many-body states, uniqueness up to ancilla unitaries coexists with strong representational asymmetries. For one-dimensional systems, the amount of entanglement in a purification controls the efficiency of an MPS representation. "Finding purifications with minimal entanglement" introduces an MPS-based iterative procedure that minimizes the second Renyi entropy by applying local unitaries on the ancilla. The method exploits the non-uniqueness of purification to reduce entanglement substantially; for the thermofield double at large $$\rho_A=\operatorname{Tr}_B\!\left(|\Psi_{AB}\rangle\langle\Psi_{AB}|\right).$$5, the entropy after disentangling matches that of the ground state, i.e. exactly half the TFD value (Hauschild et al., 2017).

A more structural limitation appears in the comparison between matrix product density operators and local purifications. For mixed states described as MPDOs of bond dimension $$\rho_A=\operatorname{Tr}_B\!\left(|\Psi_{AB}\rangle\langle\Psi_{AB}|\right).$$6, there is in general no function $$\rho_A=\operatorname{Tr}_B\!\left(|\Psi_{AB}\rangle\langle\Psi_{AB}|\right).$$7 such that the bond dimension $$\rho_A=\operatorname{Tr}_B\!\left(|\Psi_{AB}\rangle\langle\Psi_{AB}|\right).$$8 of a purification MPS satisfies

$$\rho_A=\operatorname{Tr}_B\!\left(|\Psi_{AB}\rangle\langle\Psi_{AB}|\right).$$9

The paper proves this inequivalence even for classical states. It nevertheless provides constructive methods: the sum-of-squares approach gives exact purifications with

$\mathcal H=\mathcal H_A\otimes\mathcal H_B$0

while the eigenbasis method gives

$\mathcal H=\mathcal H_A\otimes\mathcal H_B$1

in the exact case and $\mathcal H=\mathcal H_A\otimes\mathcal H_B$2 for truncated approximations (Cuevas et al., 2013).

An even stronger obstruction is known for translational invariance. There exist translationally invariant MPDOs valid for all system sizes that are positive for all $\mathcal H=\mathcal H_A\otimes\mathcal H_B$3, yet admit no translationally invariant purification valid for all $\mathcal H=\mathcal H_A\otimes\mathcal H_B$4. The proof combines undecidability of the positivity problem with the uniqueness of canonical forms of matrix product states: existence of a translationally invariant purification is decidable in principle, positivity for all sizes is not, and this mismatch forces failure of universal TI purification representability (Cuevas et al., 2015).

These results separate two distinct notions of uniqueness. The ordinary statement that purifications of a fixed state are related by ancilla transformations does not imply that there is a canonical, efficient, or translationally invariant purification ansatz for many-body representations.

6. Operational constructions, resource value, and no-go theorems

The non-uniqueness of purification can be operationally useful. The random purification channel maps $\mathcal H=\mathcal H_A\otimes\mathcal H_B$5 i.i.d. copies of a mixed state to a uniform convex combination of $\mathcal H=\mathcal H_A\otimes\mathcal H_B$6 i.i.d. copies of its purifications. More generally, for any permutationally symmetric state, it outputs a uniform convex combination of permutationally symmetric purifications, each differing only by a tensor-product unitary acting on the purifying system. This construction makes explicit that the output is independent of the initial choice of reference purification and gives a one-line proof of a stronger version of Uhlmann's theorem for quantum divergences (Girardi et al., 28 Nov 2025).

Access to a purification can itself be a resource in learning theory. For low-rank mixed states, a constant number of ancilla qubits in a purification suffices for constant-sample estimation of quantities related to purity, cooled form, principal component, and quantum Fisher information. Without access to a purification, the same tasks require exponentially many copies of the target mixed state for strategies using a bounded number of ancilla qubits, even when the rank is known (Liu et al., 2024).

At the same time, purification cannot be made universal as a physical transformation on unknown inputs. "No Universal Purification in Quantum Mechanics" proves that if a nonzero positive trace-non-increasing map $\mathcal H=\mathcal H_A\otimes\mathcal H_B$7 sends $\mathcal H=\mathcal H_A\otimes\mathcal H_B$8 copies of every state to a pure output, then that output must be independent of the input: $\mathcal H=\mathcal H_A\otimes\mathcal H_B$9 Approximate purification also obeys task-independent bounds, and approximate preparation of a pure dilation requires

$$M_A=B(\mathcal H_A)\otimes 1,\qquad M_B=1\otimes B(\mathcal H_B),$$0

Thus, purification is always possible for a fixed state, but there is no universal input-dependent purification machine for arbitrary unknown states or channels (Liu et al., 25 Sep 2025).

For qubit mixed states, recent geometric work reformulates purification freedom in Fano variables. A purification is specified by the system Bloch vector, the ancilla Bloch vector, and a real correlation matrix, while all purifications of a fixed state are generated by proper rotations $$M_A=B(\mathcal H_A)\otimes 1,\qquad M_B=1\otimes B(\mathcal H_B),$$1 acting on the ancillary degrees of freedom. Uhlmann overlap optimization then reduces to an orthogonal Procrustes problem on $$M_A=B(\mathcal H_A)\otimes 1,\qquad M_B=1\otimes B(\mathcal H_B),$$2, and the resulting misalignment angle

$$M_A=B(\mathcal H_A)\otimes 1,\qquad M_B=1\otimes B(\mathcal H_B),$$3

captures geometric information beyond scalar fidelity-based measures. Because the optimal Procrustes rotation lifts to a local unitary on the ancilla, the standard uniqueness class acquires an explicit geometric and operational realization (Osán, 18 May 2026).

Across these settings, uniqueness of purifications is best understood as a symmetry principle with sharply context-dependent scope. In finite-dimensional quantum mechanics it is a unitary equivalence theorem; in operator algebras it is equivalent to Haag duality; in generalized theories it can be postulated as a structural axiom; in many-body and field-theoretic settings it may be constrained, weakened, or fail; and in operational tasks it functions both as a resource and as a source of fundamental no-go theorems.