Local Broadcasting of Quantum Correlations

Abstract

Operations that are trivial in the classical world, like accessing information without introducing any change or disturbance, or like copying information, become non-trivial in the quantum world. In this note we discuss several limitations in the local redistributing correlations, when it comes to dealing with bipartite quantum states. In particular, we focus on the task of local broadcasting, by discussing relevant no-go theorems, and by quantifying the non-classicality of correlations in terms of the degree to which local broadcasting is possible in an approximate fashion. I. CLONING, BROADCASTING, AND LOCAL BROADCASTING A. CloningA key aspect of quantum information, which strongly differentiates it from classical information, is the inability to freely copy quantum states. Suppose we are given a quantum system S 1 in an unknown state |ψ , and that we want to put -equivalently, prepare -another system S 2 (let us say, of similar physical nature, and initially in some fiducial state |0 ) in the same state, without changing the state of the system that was given to us. Thinking in terms of a unitary evolution U , what we want to accomplish is the following:where E is an ancillary system that we may want to use in the process, initially prepared in some fiducial state |0 E independent of |ψ and ending up in a state |ξ ψ , which potentially depends on |ψ . Consider now two known states |ψ and |ψ ′ , for both of which we assume (1) to hold for the same unitary U . By taking the inner product of the left-hand and of the right-hand sides of the two occurrencies (one for ψ, and one for ψ ′ ) of (1), and using the fact that U preserves inner products, we arrive at the relationGiven that the modulus of the inner product of two normalized vector states is always less or equal to 1, the latter relation can be satisfied only if | ψ ′ |ψ | is either 0 (ψ and ψ ′ are orthogonal) or 1 (ψ and ψ ′ are the same). This is the content of the no-cloning theorem [1, 2], which says that, within the quantum formalism, there is no physical process able to clone pure quantum states that are not orthogonal.In the general case where one adopts the formalism of density matrices and channels, the cloning of a state ρ of a system S by means of an S → S 1 S 2 channel Λ corresponds to the requestwhere the introduction of a system onto which to copy the state and the possibility of using an ancillary system in the process are already taken into account by the quantum channel formalism. It is useful to recall that every quantum channel Λ S→S ′ from a system S to a system S ′ (of potentially different dimensionality) can be seen as the result of an isometry V S→S ′ E from S to a combined system S ′ E, followed by the tracing out of E:

Approximate broadcasting of quantum correlations

“…The goal of Alice is to broadcast the mutual information between them to many, say n, recipients, using local operation only. If the state is not classical on A, she cannot perform the task perfectly [5][6][7], and the mutual information between each recipient and Bob would decrease in general. Now her task naturally becomes to design a broadcasting operation in order to minimize the average loss of mutual information.…”

“…This is called the no-local-broadcasting theorem [4]. Furthermore, when the local operations ar only allowed for one party (e.g., Alice), the task can be done if and only if ρ AB is classical on A [5][6][7].…”

Hadamard quantum broadcast channels

“…Quantum broadcast channels were introduced in [5] and take on a particular relevance in quantum information theory, due to the no-cloning theorem [6,7] and associated "no-go" results [8,9,10,11]. A variety of information-theoretic results are now known for quantum broadcast channels.…”