Noncommuting mixed states cannot be broadcast.

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 generalizes and extends the standard no-cloning theorem for pure states. 1995 PACS numbers: 03.65.Bz, 89.70.+c, 02.50.-r Typeset using REVTEX 1 The fledgling field of quantum information theory [1] draws attention to fundamental questions about what is physically possible and what is not. An example is the theorem [2,3] that there are no physical means by which an unknown pure quantum state can be reproduced or copied—a result summarized by the phrase “quantum states cannot be cloned.” In this paper we formulate and prove an impossibility theorem that extends and generalizes the pure-state no-cloning theorem to mixed quantum states. The theorem answers the question: are there any physical means for broadcasting an unknown quantum state, pure or mixed, onto two separate quantum systems? By broadcasting we mean that the marginal density operator of each of the separate systems is the same as the state to be broadcast. The pure-state “no-cloning” theorem [2,3] prohibits broadcasting pure states, for the only way to broadcast a pure state |ψ〉 is to put the two systems in the product state |ψ〉 ⊗ |ψ〉, i.e., to clone |ψ〉. Things are more complicated when the states are mixed. A mixed-state no-cloning theorem is not sufficient to demonstrate no-broadcasting, for there are many conceivable ways to broadcast a mixed state ρ without the joint state being in the product form ρ⊗ ρ, the mixed-state analog of cloning; the systems might be correlated or entangled in such a way as to give the right marginal density operators. For instance, if the density operator has the spectral decomposition ρ = ∑ b λb|b〉〈b|, a potential broadcasting state is the highly correlated joint state ρ̃ = ∑ b λb|b〉|b〉〈b|〈b|, which, though not of the product form ρ⊗ ρ, reproduces the correct marginal probability distributions. The general problem, posed formally, is this. A quantum system AB is composed of two parts, A and B, each having an N -dimensional Hilbert space. System A is secretly prepared in one state from a set A={ρ0, ρ1} of two quantum states. System B, slated to receive the unknown state, is in a standard quantum state Σ. The initial state of the composite system AB is the product state ρs ⊗ Σ, where s = 0 or 1 specifies which state is to be broadcast. We ask whether there is any physical process E , consistent with the laws of quantum theory, that leads to an evolution of the form ρs ⊗ Σ → E(ρs ⊗ Σ) = ρ̃s, where ρ̃s is any state on the N-dimensional Hilbert space AB such that