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A No-Purification Theorem

Matthias Kleinmann, Hermann Kampermann, Theresa Meyer, Dagmar Bruß

Published Sep 14, 2005• 0 citations
OpenAlex

Abstract

We prove that no physical map can provide a perfect purification of an unknown quantum state. Here, a purification of a mixed quantum state is a pure state in a higher-dimensional Hilbert space, for which the reduced density matrix is identical to the original state. We investigate the possibility of performing an imperfect purification, by either admitting a non-pure output state, or a non-zero distance of its reduced state to the original. Surprisingly, the restriction of the input set to two states leads to non-trivial purifying maps.

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We prove that no physical map can provide a perfect purification of an unknown quantum state. Here, a purification of a mixed quantum state is a pure state in a higher-dimensional Hilbert space, for which the reduced density matrix is identical to the original state. We investigate the possibility of performing an imperfect purification, by either admitting a non-pure output state, or a non-zero distance of its reduced state to the original. Surprisingly, the restriction of the input set to two states leads to non-trivial purifying maps.
We introduce the concept of a physical process that purifies a mixed quantum state, taken from a set of states, and investigate the conditions under which such a purification map exists. Here, a purification of a mixed quantum state is a pure state in a higher-dimensional Hilbert space, the reduced density matrix of which is identical to the original state. We characterize all sets of mixed quantum states, for which perfect purification is possible. Surprisingly, some sets of two noncommuting states are among them. Furthermore, we investigate the possibility of performing an imperfect purification.
We study the task of lifting arbitrary quantum states and channels to purifications and Stinespring dilations, respectively, in both the probabilistic exact and deterministic approximate settings. We formalize this task through a general framework of quantum purification machines that, given a finite number of copies or uses of a black-box input, aim to output a corresponding purification or Stinespring dilation. In the probabilistic exact setting, we show that universality is not necessary to rule out such transformations: the simple requirement that a machine purifies two inputs of different rank with non-zero probability already implies that it cannot be described by a linear positive map. This simple argument captures a fundamental obstruction of quantum theory and recovers the impossibility of universal probabilistic purification from finitely many copies. In the approximate setting, we allow for general machines that are not required, in general, to produce a pure output. Using the minimum average error as our figure of merit, we derive analytical expressions for the performance of several physically motivated strategies as well as a general upper bound on the achievable error, which is tight in a specific regime. Our analysis reveals a trade-off: strategies that produce a pure output - among which we prove the optimal to be a strategy that produces as a fixed output a maximally entangled purification of the fully depolarizing channel - perform optimally between those considered for large environment dimension, while append-environment strategies that generally produce non-pure outputs perform better at small environment dimension.
Many central tasks in fundamental physics and quantum information processing are possible only insofar as mixed quantum states can be made purer. In this work, we prove that the linearity and positivity of quantum mechanics impose general restrictions on quantum purification, unveiling a new fundamental principle of quantum information processing. We first establish that no quantum operation can transform a finite number of copies of an unknown quantum state or channel into an exactly pure output that depends non-trivially on the input, thereby ruling out an important form of universal purification in both static and dynamical settings. Building on this, we show that, upon relaxing the requirement of exact purity, one can establish quantitative sample-complexity lower bounds for approximate purification that hold for arbitrary physically allowed strategies, whose scaling matches the performance of purification-related tasks across several different areas of quantum information processing. Moreover, this lower bound leads to a generalized standard quantum limit for learning arbitrary functions of a quantum state, greatly extending earlier results based on quantum Fisher information and revealing a deep connection between purification and quantum learning. Extending this principle to other important settings, we establish, for the first time, an exponential sample-complexity lower bound for approximate pure dilation state preparation and a no-go theorem for approximate bosonic Gaussian state purification with passive Gaussian operations, establishing much more stringent limitations under practical operational constraints.
Purification is a well-known tool to represent mixed quantum states as pure states on an enlarged Hilbert space. A purification of a given state is not unique and its spatial entanglement strongly depends on the particular choice. This entanglement is linked to how efficiently the purified state can be represented using matrix-product states (MPS). Here, the authors introduce an MPS-based method that allows the user to efficiently find the minimally entangled purification of a given mixed state and demonstrate that it can significantly reduce the entanglement in certain cases.
Pure states are fundamental for the implementation of quantum technologies. Several methods for the purification of the state of a quantum system S have been developed in the past years. Considering two-level systems and assuming a particular interaction between the system S and probe P, we propose a control strategy that purifies an arbitrary mixed state to an eigenstate. An interaction control is designed to improve the quality of the purification of mixed state. The Lyapunov-based control method is used to design the part of interaction control. Compared numerical simulation experiments demonstrate superiority of the purification strategy proposed: the purification of an arbitrary mixed state can be realized in the energy difference between S and P being not only zero case, but also in the nonzero case.
Abstract We generalize the Fubini-Study method for pure-state complexity to generic quantum states by taking Bures metric or quantum Fisher information metric (QFIM) on the space of density matrices as the complexity measure. Due to Uhlmann’s theorem, we show that the mixed-state complexity exactly equals the purification complexity measured by the Fubini-Study metric for purified states but without explicitly applying any purification. We also find the purification complexity is non-increasing under any trace-preserving quantum operations. We also study the mixed Gaussian states as an example to explicitly illustrate our conclusions for purification complexity.