dense key distribution ”

In his Comment [1], Wójcik proposes two simple schemes for performing eavesdropping attacks to the Quantum Dense Key Distribution protocol (QDKD) in Ref. [2]. These schemes enable Eve to achieve a larger mutual information than Alice’s and Bob’s one, but maintaining inviolated the security condition (Eq. (6) in Ref. [2]). Eve’s attack in Wójcik’s schemes relies on the possibility of subtraction of photons from the quantum channel without being disclosed by the Anticorrelation Check. In fact the security proof given in [2] is incomplete because it includes only the effect that Eve’s presence induces a bit flip on the travelling qubit. In this reply we complete the security proof of QDKD even against individual eavesdropping attack with injection or subtraction of photons in the quantum channels. Furthermore we discuss the security of the experimental realization of the QDKD protocol performed in [2], according to the arguments raised by the modified proof. A dedicated formalism is introduced to account for these attacks. Let |mX〉 be the state with n photons on the quantum channel X, where m photons have horizontal polarization and (n −m) vertical polarization. The orthonormal base BX : {|m n X〉} spans the Hilbert space HX of photons in channel X; |0 0 X〉 is the vacuum state. In the actual formalism, referring to Fig. 1 in Ref. [2], Alice produces pairs of photons in the singlet state |ψ− AB〉 = 1 √ 2 (|0A1 1 B〉− |1 1 A0 1 B〉). Photon A is stored in her laboratory while on photon B she performs either the operation 1B (identity operator) or ẐB before sending it to Bob. A generalized gate ẐB acts in the Hilbert space HB as ẐB|m n B〉 = (−1) |mB〉. ẐB is hermitian and unitary, and corresponds to the Pauli matrix σ̂Z acting individually on each photon in the channel B. Alice’s selection of gate 1B or ẐB is encoded according to