Quantum direct communication with continuous variables

We show how continuous-variable systems can allow the direct communication of messages with an acceptable degree of privacy. This is possible by combining a suitable phasespace encoding of the plain message with real-time checks of the quantum communication channel. The resulting protocol works properly when a small amount of noise affects the quantum channel. If this noise is non-tolerable, the protocol stops leaving a limited amount of information to a potential eavesdropper. Copyright c © EPLA, 2008 Introduction. – In recent years, quantum communication protocols have been extended to the domain of continuous-variable (CV) systems, i.e., quantum systems, like the bosonic modes of the radiation field, which are characterized by infinite dimensional Hilbert spaces [1]. In particular, it has been understood how a sender (Alice) can exploit bosonic modes in order to send analog signals to a receiver (Bob) and then extract a secret binary key from these signals [2,3]. Beyond the possibility of such a continuous-variable quantum key distribution (QKD), here we show how to use these systems in order to perform a (quasi)confidential quantum direct communication (QDC) [4], i.e., the (quasi)private communication of a message from Alice to Bob which is directly encoded in CV systems. The ideal situation for QDC trivially occurs when Alice and Bob are connected by a noiseless channel. However, in general, this is not the case and the honest users must randomly switch their confidential communication with real-time checks on the channel. As soon as they detect the presence of a non-tolerable noise, they promptly stop the communication. The maximum noise that can be tolerated is connected to the maximum amount of information that they are willing to give up to an eavesdropper. In other words, a good QDC protocol should enable Alice and Bob to communicate all the message when the noise is suitably low, while losing a small amount of information when it is not. (a)E-mail: [email protected] Let us consider a bosonic mode described by quadrature operators q̂ and p̂, satisfying [q̂, p̂] = i. An arbitrary state of the system (density operator ρ) must fulfill the uncertainty principle V (q̂)V (p̂) 1/4, where V (x̂) =Tr(ρx̂2)− [Tr(ρx̂)] denotes the variance of the arbitrary quadrature x̂= q̂ or p̂. In particular, coherent states satisfy V (q̂) = V (p̂) :=∆, where ∆= 1/2 represents the quantum shot-noise. This is the fundamental noise that affects disjoint measurements of the quadratures q̂ and p̂ (homodyne detection), and it is doubled to ∆= 1 when the two quadratures are jointly measured (heterodyne detection). A density operator ρ may be faithfully represented by the Wigner quasi-probability distribution W (q, p), whose continuous variables q and p are the eigenvalues of the quadratures. In this phase-space representation, states with Gaussian Wigner functions are called Gaussian states. This is the case of a coherent state |ᾱ〉, whose Gaussian Wigner function is centered at ᾱ= 2(q̄+ ip̄). For coherent states the detection of an arbitrary quadrature x̂ provides outcomes x following the marginal distribution G∆(x− x̄) = 1 √ 2π∆ exp [ − (x− x̄) 2