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arXiv:2003.01885

No-go Theorem of Purification

Sadegh Raeisi

Published Mar 4, 2020• 0 citations
arXivHTMLPDF

Abstract

The Shannon's bound for compression is one of the key restrictions for the compression of quantum information. Here we show that the unitarity of the compression operation imposes new bounds on the compression that are more limiting than Shannon's compression bound. This translates to a no-go theorem for the purification of quantum states. For a specific case of a two-qubit system, our results indicate that it is not possible to distill purity beyond the maximum of the individual purities. We show that this restriction results in the cooling limit of the heat-bath algorithmic cooling techniques. We formalize the limitations imposed by the unitarity of the compression operation in two theorems and use the theorems to show that the limitations of unitarity lead to the cooling limit of heat-bath algorithmic cooling. To this end, we introduce a new optimal cooling technique and show that without the limitations of the unitary operations, the new cooling technique would have exceeded the limit of Heat-bath algorithmic cooling. This work opens up new avenues to understanding the limits of dynamic cooling.

DOI
10.48550/arxiv.2003.01885

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The Shannon's bound for compression is one of the key restrictions for the compression of quantum information. Here we show that the unitarity of the compression operation imposes new bounds on the compression that are more limiting than Shannon's compression bound. This translates to a no-go theorem for the purification of quantum states. For a specific case of a two-qubit system, our results indicate that it is not possible to distill purity beyond the maximum of the individual purities. We show that this restriction results in the cooling limit of the heat-bath algorithmic cooling techniques. We formalize the limitations imposed by the unitarity of the compression operation in two theorems and use the theorems to show that the limitations of unitarity lead to the cooling limit of heat-bath algorithmic cooling. To this end, we introduce a new optimal cooling technique and show that without the limitations of the unitary operations, the new cooling technique would have exceeded the limit of Heat-bath algorithmic cooling. This work opens up new avenues to understanding the limits of dynamic cooling.
Algorithmic cooling techniques provide tools to increase the purity of quantum states. It is known that the cooling with these techniques is limited. However, the physical root of the limit is still unclear. Here we show that the unitarity of the compression operation imposes the cooling limit of heat-bath algorithmic cooling. Specifically, we prove that the unitarity of the compression operation does not allow increasing the purity beyond the maximum of the individual purities. We formalize the limitations imposed by the unitarity of the compression operation in two theorems. We then introduce an optimal cooling technique and show that without the limitations of the unitary operations the cooling technique would exceed the limit of heat-bath algorithmic cooling. We also show that the limitations of unitarity lead to the exact cooling limit of algorithmic cooling. This work opens up new avenues to understanding the limits of dynamic cooling.
Abstract Quantum technologies require pure states, which are often generated by extreme refrigeration. Heat-bath algorithmic cooling is the theoretically optimal refrigeration technique: it shuttles entropy from a multiparticle system to a thermal bath, thereby generating a quantum state with a high degree of purity. Here, we show how to surpass this hitherto-optimal technique by taking advantage of a single binary-outcome measurement. Our protocols can create arbitrary numbers of pure quantum states without any residual mixedness by using a recently discovered device known as a quantum switch to put two operations in superposition, with postselection certifying the complete purification.
Global unitary transformations (OPTSWAPS) that optimally increase the bias of any mixed computation qubit in a quantum system -- represented by a diagonal density matrix -- towards a particular state of the computational basis which, in effect, increases its purity are presented. Quantum circuits that achieve this by implementing the above data compression technique -- a generalization of the 3B-Comp used before -- are described. These circuits enable purity increment in the computation qubit by maximally transferring part of its von Neumann or Shannon entropy to any number of surrounding qubits and are valid for the complete range of initial biases. Using the optswaps, a practicable new method that algorithmically achieves hierarchy-dependent cooling of qubits to their respective limits in an engineered quantum register opened to the heat-bath is delineated. In addition to multi-qubit purification and satisfying two of DiVincenzo's criteria for quantum computation in some architectures, the implications of this work for quantum data compression and quantum thermodynamics are discussed.
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.
The purity of quantum states is a key requirement for many quantum applications. Improving the purity is limited by fundamental laws of thermodynamics. Here, we are probing the fundamental limits for a natural approach to this problem, namely, heat-bath algorithmic cooling (HBAC). The existence of the cooling limit for HBAC techniques was proved by Schulman, Mor, and Weinstein. A bound for this value was found by Elias et al. and numerical testing supported the hypothesis that their bound may be the actual limit. A proof or disproof of whether their bound was the actual limit remained open for the past decade. Here, for the first time, we prove this limit. In the context of quantum thermodynamics, this corresponds to the maximum extractable work from the quantum system. We also establish, in the case of higher dimensional reset systems, how the performance of HBAC depends on the energy spectrum of the reset system.
Preparing a quantum system in a pure state is ultimately limited by the nature of the system's evolution in the presence of its environment and by the initial state of the environment itself. We show that, when the system and environment are initially uncorrelated and arbitrary joint unitary dynamics is allowed, the system may be purified up to a certain (possibly arbitrarily small) threshold if and only if its environment, either natural or engineered, contains a "virtual subsystem" which has the same dimension and is in a state with the desired purity. Beside providing a unified understanding of quantum purification dynamics in terms of a "generalized swap process," our results shed light on the significance of a no-go theorem for exact ground-state cooling, as well as on the quantum resources needed for achieving an intended purification task.