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arXiv:quant-ph/9511010 · Physical Review Letters

Noncommuting Mixed States Cannot Be Broadcast

Howard Barnum, Carlton M. Caves, Christopher A. Fuchs, Richard Jozsa, Benjamin W. Schumacher

Published Apr 8, 1996• 654 citations
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Abstract

We show that, given a general mixed state for a quantum system, there are no physical means for broadcasting that state onto two separate quantum systems, even when the state need only be reproduced marginally on the separate systems. This result extends the standard no-cloning theorem for pure states.
DOI
10.1103/physrevlett.76.2818

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It is well known that the non-broadcasting theorem proved by Barnum et al is a fundamental principle of quantum communication. As we are aware, optimal broadcasting (OB) is the only method to broadcast noncommuting mixed states approximately. In this paper, motivated by the probabilistic cloning of quantum states proposed by Duan and Guo, we propose a new way for broadcasting noncommuting mixed states—probabilistic broadcasting (PB), and we present a sufficient condition for PB of mixed states. To a certain extent, we generalize the probabilistic cloning theorem from pure states to mixed states, and in particular, we generalize the non-broadcasting theorem, since the case that commuting mixed states can be exactly broadcast can be thought of as a special instance of PB where the success ratio is 1. Moreover, we discuss probabilistic local broadcasting (PLB) of separable bipartite states.
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The NNN to MMM $(M\ensuremath{\geqslant}N)$ universal quantum broadcasting of mixed states ${\ensuremath{\rho}}^{\ensuremath{\bigotimes}N}$ is proposed for a qubit system. The broadcasting of mixed states is universal and optimal in the sense that the shrinking factor is independent of the input state and achieves the upper bound. The quantum broadcasting of mixed qubits is a generalization of the universal quantum cloning machine for identical pure input states. A pure state decomposition of the identical mixed qubits ${\ensuremath{\rho}}^{\ensuremath{\bigotimes}N}$ is obtained.
In this work, we extensively study the problem of broadcasting of quantum correlations (QCs). This includes broadcasting of quantum entanglement as well as correlations that go beyond the notion of entanglement (QCsbE). It is quite well known from the ``no-broadcasting theorem'' that perfect broadcasting of QCs is not possible. However, it does not rule out the possibility of partial broadcasting of QCs where we can get lesser correlated states from a given correlated state. In order to have a holistic view of broadcasting, we investigate this problem by starting with a most general representation of two qubit mixed states in terms of the Bloch vectors. As a cloning transformation we have used universal symmetric optimal Buzek-Hillery (BH) cloner both locally and nonlocally. Unlike entanglement, we find that it is impossible to broadcast QCsbE optimally. Lastly, we generalize these results for any symmetric or asymmetric cloning machines as well. This result brings out a fundamental difference between the correlations defined from the perspective of entanglement and the correlations measure which claims to go beyond entanglement.