ua nt - p h / 01 04 06 5 v 1 1 2 A pr 2 00 1 Are all noisy quantum states obtained from pure ones ?

Density matrices and the question of their entanglement have been studied very intensively during the last few years [1]. However the question of how these density matrices arise in the first place has received much less attention. It has been tacitly assumed that density matrices arise when a number of parties, separated in space, start by sharing a pure state, and then this state gets contaminated by noise, due to interaction with the environment. Is it the case that any density matrix can be obtained in this way? Surprisingly we will show that the answer is no. The point is that, in the above scenario, the very fact that we can talk about separated parties, means that the noise is local. This imposes constraints as to how entangled states may be degraded by the environment. We study here the effects of these constraints. We will consider two different situations. The first is where we consider a pure state shared by the parties which is contaminated by local noise; this is a typical situation of obvious physical significance. We will refer to states which can be produced from pure states by local contamination as LC states. The second situation is one in which the parties actively try to degrade the pure state: in addition to local noise we allow local measurements and classical communication between the parties. We refer to states which can be produced from pure states by local contamination and classical communication as LCCC states. We note that in both situations, when we wish to get a density matrix of a given number of parties each with a given dimension of local Hilbert space, we demand that the precursor pure states lie in a system Hilbert space of the same local dimensions [2]. The specific question we address is: given a general “target” density matrix, can we find a pure state from which this density matrix can be obtained by local contamination, in either situation. We will show that generically, in both situations, the answer is no. In a sense, density matrices which can be obtained by local contamination are simple in that their entanglement properties are simply related to those of their pure state precursor. The density matrices which cannot be obtained in this way have a more subtle and complex structure. Let us first discuss the situation in which no classical communication is allowed. We consider n separated parties each of which has a d level system. The number of real parameters describing pure states is 2dn−2, and the number describing general mixed states is d − 1. The fact that it is not possible to reach an arbitrary “target” density matrix by degrading any pure state follows simply from the fact that the number of parameters describing local degrading is linear in n. Thus for sufficiently large n, the number of parameters describing density matrices will be larger that the number describing pure states and local contamination. In order to see how large n needs to be we need to calculate the number of parameters describing the set of general local transformations. The most general interaction of a d-level system with its environment is as follows [3]: