Abstract
Hereunder, we study the class of irreducible private states that are private states from which all the secret content is accessible via measuring their key part. We provide the first protocol which distills key not only from the key part, but also from the shield if only the state is reducible. We prove also a tighter upper bound on the performance of that protocol, given in terms of regularized relative entropy of entanglement instead of relative entropy of entanglement previously known. This implies in particular that the irreducible private states are all strictly irreducible if and only if the entangled but key-undistillable states (‘entangled key-undistilable states’) exist. In turn, all the irreducible private states of the dimension 4 ⨂ 4 are strictly irreducible, that is, after an attack on the key part they become separable. Provided the bound key states exist, we consider different subclasses of the irreducible private states and their properties. Finally we provide a lower bound on the trace norm distance between key-undistillable states and private states, in sufficiently high dimensions.
Highlights
Hereunder, we study the class of irreducible private states that are private states from which all the secret content is accessible via measuring their key part
It has been shown that the classical secure key, is in, fact an entanglement measure denoted as KD [5, 6]
This led to upper bounds on the secure content of quantum states via relative entropy of entanglement [5,6,7] and squashed entanglement [8,9,10,11] and further generalizations for quantum channels via squashed entanglement of a quantum channel [12, 13] and relative entropy of entanglement extended to quantum channel [14]
Summary
We introduce rigorous definitions of private states as well as irreducible and strictly irreducible private states which are crucial in further parts of this manuscript. One example of a private state is a basic private state ρABA B , acting on a Hilbert space CdA ⊗ CdB ⊗ CdA ⊗ CdB. AB of a pdit is called the key part, while the subsystem A B its shield. D − 1 we denote states which appear on the shield of the pdit, after obtaining an outcome |ii AB in the measurement performed in the standard basis on its key part. The set of irreducible private states with local dimension of key part d and total dimension of the shield part d , will be denoted as IRd,d. 1 d i |ii ii| ⊗ UiσUi† if consists of separable states with respect to the systems on the shield part on the diagonal, i.e. UiσUi ∈ SEP We denote such states as γ or γ d.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
