Purification in Infinite Dimensions

Every trace-class density operator admits a purification in a Hilbert-space tensor product, even when its rank is infinite. The construction is a norm-convergent countable series. Its ancilla must contain enough orthogonal vectors to accommodate the support rank; the familiar claim that all purifications differ by an ancillary unitary needs a qualification when infinite-dimensional unused subspaces have different dimensions.

Helpful background. Purification supplies the physical construction and finite-dimensional interpretation; trace and partial trace supplies the trace-class extension.

Constructing the purifying vector

Let $\rho$ be a density operator on a separable Hilbert space $\mathcal H_A$, with positive spectral expansion

$$\rho=\sum_np_n|u_n\rangle\langle u_n|, \qquad\sum_np_n=1.$$

Choose an ancillary Hilbert space $\mathcal H_B$ with an orthonormal family $(b_n)$ of the same cardinality as the nonzero spectrum. Define

$$|\Psi\rangle =\sum_n\sqrt{p_n}\,|u_n\rangle\otimes|b_n\rangle.$$

Orthogonality gives a tail norm squared of $\sum_{n>N}p_n$, so the series is Cauchy and converges to a unit vector. The rank-one projector is trace class, and its partial trace is $\rho$.

One rigorous way to verify the last assertion is to pair with every bounded local observable:

$$\langle\Psi,(A\otimes I)\Psi\rangle =\sum_np_n\langle u_n,Au_n\rangle =\operatorname{Tr}(\rho A).$$

The sums converge absolutely because $|\langle u_n,Au_n\rangle|\leq\|A\|$. Uniqueness of the trace-class partial trace identifies the reduced operator. No generalized eigenvectors or nonnormalizable “maximally entangled vector” are required.

Minimal ancillas and support

The reduced state on $B$ is $\rho_B=\sum_np_n|b_n\rangle\langle b_n|$. Hence its nonzero spectrum agrees with that of $\rho_A$, including multiplicity.

The smallest possible ancillary dimension is the rank of $\rho$, understood as the Hilbert-space dimension of its support. Necessity follows from the Schmidt decomposition: a vector in $\mathcal H_A\otimes\mathcal H_B$ cannot have more nonzero Schmidt coefficients than $\dim\mathcal H_B$. The construction proves sufficiency.

An infinite-rank density operator therefore requires an infinite-dimensional ancilla for an exact purification. A finite-rank approximation admits a finite ancilla, with an error controlled by the omitted spectral weight. Even on a nonseparable ambient space, a trace-class density operator has separable support, so its minimal purification uses at most a countably infinite ancilla.

Isometries and the unitary qualification

Fix a minimal ancilla $\mathcal K$ for a purification $\Psi_0$. Any other purification of the same $\rho$ can be expressed as

$$|\Psi\rangle=(I_A\otimes V)|\Psi_0\rangle,$$

where $V:\mathcal K\to\mathcal H_B$ is an isometry. This follows by expanding the other vector in the eigenbasis of $\rho$: the corresponding ancillary coefficient vectors have the inner products needed to define the isometry, including within degenerate eigenspaces.

For two arbitrary purifications, the relation on their occupied ancillary support spaces is unitary between those supports. On full ancillary spaces it is naturally represented by a partial isometry. It extends to a unitary between the full ancillary spaces only when the orthogonal complements can also be matched.

This distinction can matter even when both ancillary spaces are the same $\ell^2$. Let every $p_n$ be positive and compare

$$\Psi=\sum_{n\geq1}\sqrt{p_n}\,u_n\otimes b_n, \qquad \Phi=\sum_{n\geq1}\sqrt{p_n}\,u_n\otimes b_{n+1}.$$

Both reduce to $\rho$. The shift $Vb_n=b_{n+1}$ relates them, but $V$ is an isometry with a one-dimensional missing range. No unitary on this same ancilla maps $\Psi$ to $\Phi$: coefficient comparison would require $Ub_n=b_{n+1}$ for every $n$, contradicting surjectivity. Enlarging ancillary spaces can remove the mismatch.

Thermal purification and moment conditions

For $p_n=(1-q)q^n$, $n\geq0$, $0<q<1$, the purification is

$$\Psi_q=\sqrt{1-q}\sum_{n=0}^{\infty}q^{n/2}|n\rangle_A|n\rangle_B.$$

It is normalized by the geometric series. Each marginal has mean occupation $q/(1-q)$ and the same entropy. The full vector state has zero entropy, while either subsystem is mixed.

Normalization alone would not guarantee these finite moments for a different spectral tail. A purification of a state with infinite mean energy is still a valid unit vector. Its membership in the domain of a local Hamiltonian, or in its form domain, is a separate weighted-summability question. The same warning applies to infinite entropy.

Holevo gives the tensor-product state framework. The infinite-dimensional fidelity treatment by Hou and Qi makes the corresponding purification and isometry issues explicit.

Exercises

1. Let $t_N=\sum_{n>N}p_n$, and normalize the first $N$ terms of $\Psi$. Compute the overlap with $\Psi$ and the trace distance of their pure projectors.

Solution

The overlap is $\sqrt{1-t_N}$. The pure-state trace distance is therefore $\sqrt{t_N}$. The reduced density operators have trace distance $t_N$, consistent with contractivity of partial trace.

2. Can a faithful infinite-rank density operator be purified with a two-dimensional ancilla?

Solution

No. A vector with a two-dimensional ancillary factor has Schmidt rank at most two, hence a reduced operator of rank at most two. Approximation by a rank-two state is possible only with the corresponding nonzero truncation error.

References

• Holevo, Alexander S., Quantum Systems, Channels, Information: A Mathematical Introduction, 2nd revised and expanded ed., De Gruyter, 2019, Chapters 3 and 11 — purifications and infinite-system states. Publisher.
• Hou, Jinchuan, and Xiaofei Qi, “Fidelity of states in infinite-dimensional quantum systems,” Science China Physics, Mechanics & Astronomy 55, 1820–1827, 2012 — infinite-dimensional purification and fidelity arguments. Preprint; DOI.