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redpajama_set_name: RedPajamaGithub


text:
\section{Introduction}
Quantum key distribution (QKD) protocols are designed to allow two users, Alice ($A$) and Bob ($B$), to establish a shared secret key, secure against even an all-powerful adversary Eve ($E$).  Since the creation, in 1984, of the BB84 protocol \cite{QKD-BB84}, there have been several protocols developed which achieve this end.  For a general survey, the reader is referred to \cite{QKD-survey}.

Semi-quantum key distribution (SQKD) protocols, first introduced in 2007 by Boyer et al., \cite{SQKD-first}, have the same goal: the establishment of a secret key, secure against an all-powerful adversary.  However now, instead of allowing both $A$ and $B$ to manipulate quantum resources (e.g., prepare and measure qubits in a variety of bases) as is permissible in a typical QKD protocol, only $A$ is allowed such liberties while $B$ is limited to performing certain ``classical'' or ``semi-quantum'' operations (what operations $B$ is limited to are discussed shortly).  In this scenario, $A$ is called the \emph{quantum user} while $B$ is called the \emph{classical user} (in a fully quantum protocol, such as BB84\cite{QKD-BB84}, both $A$ and $B$ are fully quantum).  Such protocols are theoretically interesting as they attempt to answer the question ``how quantum does a protocol need to be in order to gain an advantage over a classical one?'' \cite{SQKD-first,SQKD-second}

These SQKD protocols, by necessity, rely on a two-way quantum communication channel - one which permits a qubit to travel from the quantum user $A$, to the classical user $B$, then back to $A$.  The limited classical $B$, upon receiving a qubit, is able to do one of two things: he may \emph{measure and resend} the qubit or \emph{reflect} the qubit.  Measuring and resending involves taking the qubit received from $A$ and subjecting it to a $Z$ basis measurement (the $Z$ basis being $\{\ket{0}, \ket{1}\}$).  His result $\ket{r}$, for $r \in \{0,1\}$ is then resent to $A$.  Reflecting the qubit involves $B$ allowing the qubit to simply pass through his lab undistributed in which case he learns nothing about its state.  Thus, the classical user is only able to work directly with the computational $Z$ basis - he cannot, for example, perform a measurement in the Hadamard $X$ basis (denoted $\{\ket{\pm} = 1/\sqrt{2}(\ket{0}\pm\ket{1})\}$).

This two-way quantum channel, of course, greatly complicates the security analysis of these protocols as the attacker $E$ is allowed two opportunities to interact with the traveling qubit.  While several SQKD protocols have been proposed \cite{SQKD-first,SQKD-second,SQKD-3,SQKD-4,SQKD-Single-Security,SQKD-cl-A,SQKD-random-prepare}, up until recently, the bulk of security proofs for such semi-quantum protocols have focused on the notion of \emph{robustness}.  This property, introduced in \cite{SQKD-first,SQKD-second}, requires that any attack which causes $E$ to potentially gain information, by necessity causes a disturbance which $A$ and $B$, with non-zero probability, may detect.  Note that robustness says nothing about the amount of information gained (which may be high) nor the probability of detection (which may be low).

Recently, some work has been accomplished moving beyond robustness.  In particular \cite{SQKD-information} derived expressions relating the disturbance of $E$'s attack to her information gain for the SQKD protocol of \cite{SQKD-first}, assuming she is limited to performing \emph{individual attacks} (those attacks where $E$ is limited to performing the same attack operation each iteration of the protocol and is forced to measure her quantum memory before $A$ and $B$ utilize their key for any purpose - these are weaker attacks than \emph{collective attacks} - where $E$ performs the same operation each iteration but can postpone her measurement until any future time of her choice - and \emph{general attacks} - those attacks where $E$ is allowed to do anything within the laws of physics; the reader is referred to \cite{QKD-survey} for more information on these possible attack models).  Also, \cite{SQKD-cl-A} described a new SQKD protocol and computed a similar relation between the disturbance and information gain of $E$'s attack, though again assuming individual attacks.

Our work recently has been in the proof of unconditional security (making no assumptions on the type of attack $E$ uses) of several SQKD protocols.  In particular, in \cite{SQKD-Krawec-SecurityProof}, we proved the security of Boyer et al.,'s original SQKD protocol \cite{SQKD-first} showing that $A$ and $B$ are able to distill a secure secret key so long as the noise in the channel is less than $5.34\%$.  We have also, in \cite{SQKD-Single-Security}, derived a series of security results for single state SQKD protocols (single state protocols were introduced in \cite{SQKD-lessthan4}, though without proofs of unconditional security).  While that particular work stopped short of unconditional security proofs, the security results we prove there can be applied toward that goal \cite{SQKD-Krawec-dissertation}.

Finally, and also the topic of this current paper, in \cite{SQKD-MultiUser}, we designed a new \emph{mediated SQKD} protocol.  Such a protocol allows two limited classical users $A$ and $B$ to establish a secure secret key with the help of a quantum server (denoted $C$ for ``Center'').  With such a system, one could envision a QKD infrastructure consisting of several limited ``classical'' users, utilizing the services of this quantum server in order to distill secret keys (each key known only to a pair of classical users, not the server).  If the quantum server $C$ were honest, such a goal would be trivial.  Instead, we assumed the server is untrusted - indeed, we can assume the server is the all-powerful quantum adversary.

In our original work in \cite{SQKD-MultiUser}, we proved our mediated protocol's unconditional security by computing a lower bound on its key rate (to be defined shortly, roughly speaking, the key rate is the ratio of the number of secret key bits to the number of qubits sent) in the asymptotic scenario (as the number of qubits sent approaches infinity).  We showed that, if the server is ``semi-honest,'' that is the server follows our protocol correctly but afterwards tries to learn something about the key, then $A$ and $B$ may distill a secret key so long as the noise in the channel is less than $19.9\%$.  If the server is fully adversarial (that is, the server does not follow the protocol and instead performs any operations he likes within the laws of quantum physics), then $A$ and $B$ may distill a secret key so long as the noise in the channel is less than $10.65\%$ (a number that is close to BB84's $11\%$ \cite{QKD-renner-keyrate}).

In this paper, we reconsider our security proof and improve on these tolerated noise levels by computing a new bound on the protocol's key rate using an alternative method of proof.  In our original proof, we utilized a bound on the Jensen-Shannon Divergence from \cite{QC-info-trace-bound} to find bounds on the quantum mutual information between $A$ and $C$'s system.  Here, we will use a technique adapted from \cite{QKD-keyrate-general} (which we also successfully applied in our proof of security for a different SQKD protocol in \cite{SQKD-Krawec-SecurityProof}) which allows us to bound the conditional von Neumann entropy between $A$ and $C$'s system.  This new technique provides a far more optimistic bound.  Indeed, as we will see, if the server is semi-honest, using our new key rate bound in this paper, we will see that $A$ and $B$ may distill a secret key so long as the noise in the quantum channel is less than $22.05\%$.  If the server $C$ is adversarial, then we can withstand up to $12.5\%$.  Furthermore, our new proof of security does not make any assumptions concerning the symmetry of $E$'s attack as was done in our original proof.

Thus the contributions of this paper are two-fold.  First, we improve our original security proof finding a more optimistic bound on our mediated SQKD protocol's tolerated noise level.  Our new proof in this paper also requires fewer assumptions.  Secondly, we describe an alternative proof method, which as we've already shown in \cite{SQKD-Krawec-SecurityProof}, can be applied to other semi-quantum protocols (and perhaps other QKD protocols utilizing a two-way channel).  We also show how this alternative method can produce more optimistic bounds than produced by bounding the Jensen-Shannon divergence as done in our original proof \cite{SQKD-MultiUser}.  Thus, this observation can be useful in other QKD protocol proofs and, perhaps, may lead to insight into various quantum information theoretic bounds.  Furthermore, it may be possible to adapt the techniques we use in this proof to work with other QKD protocols which utilize a two-way quantum communication channel.

\section{The Protocol}

We now review the mediated protocol of \cite{SQKD-MultiUser}.  This protocol is designed to allow two classical users $A$ and $B$ (users who can only measure and resend in the $Z$ basis or reflect qubits) to distill a secret key, known only to themselves, with the help of a quantum server $C$.  Note that, since $A$ and $B$ can only make $Z$ basis measurements, they must rely on the server to perform measurements in alternative bases.  However, neither $A$ nor $B$ trust $C$; indeed we will prove the security of this protocol assuming $C$ is adversarial.  Since this paper is concerned only with devising an improved key rate bound, we do not alter the protocol in any way.

This mediated protocol assumes the existence of a quantum communication channel connecting Alice ($A$) to the server $C$ and Bob ($B$) to $C$.  We do not require a quantum channel directly connecting $A$ and $B$.  Besides this, we assume an authenticated classical channel connects $A$ and $B$.  The server $C$, and indeed any other third party eavesdropper, may listen to the messages sent on this channel, but they may not send messages of their own.  Finally, we assume a classical channel connects the server to either $A$ or $B$ (or both).  See Figure \ref{fig:diagram}.  Using similar arguments as in \cite{SQKD-MultiUser}, this channel connecting $C$ to $A$ or $B$ need not be authenticated; though, indeed, better security bounds may be achieved if it is authenticated as we discuss later.  We will assume that any message sent from $C$ is received by both parties.  That is, $C$ cannot send two different messages $m_A \ne m_B$ to $A$, respectively $B$, without being caught.  Such a mechanism is easy to achieve: any message sent from $C$ to $A$ (or $B$) is then forwarded by $A$ (or $B$) to the other honest party $B$ (or $A$) using the authenticated channel.

\begin{figure}\center
\includegraphics[width=\linewidth]{figure1.png}
\caption{A diagram of the scenario we consider.  Here, $A$ and $B$ are the honest, classical users who wish to establish a secret key; $E$ is a third party eavesdropper; and $C$ is the untrusted, potentially adversarial quantum server.  A quantum channel (the parallel lines in the center of the diagram) connects $C$ to $A$ and also $C$ to $B$; these channels pass through the third party eavesdropper (which may be a single entity or two separate entities).  An authenticated classical channel connects $A$ to $B$ (the dotted line); only $A$ and $B$ may write to this channel, however all parties can read from it.  Finally a classical channel connects the server $C$ to $A$ and $B$; this channel is not necessarily authenticated and the eavesdroppers may write their own messages on it.}\label{fig:diagram}
\end{figure}

Our protocol will utilize the Bell basis, the states of which we denote:
\begin{align*}
\ket{\Phi^+} &= \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})\\
\ket{\Phi^-} &= \frac{1}{\sqrt{2}}(\ket{00} - \ket{11})\\
\ket{\Psi^+} &= \frac{1}{\sqrt{2}}(\ket{01} + \ket{10})\\
\ket{\Psi^-} &= \frac{1}{\sqrt{2}}(\ket{01} - \ket{10})
\end{align*}

We first describe the protocol assuming an honest server $C$.  The quantum communication stage of our protocol repeats the following process:

\begin{enumerate}
  \item $C$ prepares $\ket{\Phi^+}$, sending one qubit to $A$, the other to $B$.
  \item $A$ and $B$ (the classical users) choose, independently of each other, to either reflect the qubit back to $C$ or to measure and resend it.  If they measure in the $Z$ basis and resend, they save their measurement results as their potential raw key bit.  However, $A$ and $B$ do not yet reveal their choice of operation.
  \item Regardless of $A$ and $B$'s choice, $C$ will receive two qubits back from them.  $C$ then performs a Bell measurement.  If this measurement produces the result $\ket{\Phi^-}$, $C$ sends the message ``$-1$'' to both $A$ and $B$ using the classical channel.  Otherwise, for all other measurement results, $C$ sends the message ``$+1$''.
  \item $A$ and $B$ now divulge, using the authenticated classical channel, their choice in step 2 (but not their measurement results).  If $A$ and $B$ both measure and resent, and if $C$ sends the message ``$-1$'', they will \emph{accept} this iteration; that is to say, they will both use this iteration's measurement results to contribute towards their raw key.  They will not accept this iteration (i.e., it will not be used to contribute to the raw key) if $C$ sent the message ``$+1$''.
  Otherwise, if $A$ and $B$ both reflected, it should be the case that $C$ sends ``$+1$''; any other message by $C$ is counted as an error.
  All other cases are discarded.
\end{enumerate}

Assuming an honest $C$ and the lack of any channel noise or third-party eavesdropper, it is clear that the above protocol is correct (i.e., $A$ and $B$ will agree on the same raw key).  Indeed, if $A$ and $B$ both reflect, the state arriving back to $C$ on step 3 is $\ket{\Phi^+}$; thus $C$ should always respond with the message ``$+1$''.  Alternatively, if $A$ and $B$ both measure and resend, the state arriving at $C$ is $\ket{i,i}$ (where $i \in \{0,1\}$ is $A$ and $B$'s measurement result and potential raw key bit for this iteration).  $C$ will then perform a Bell basis measurement (projecting the state into one of the Bell basis states).  This measurement results in outcome $\ket{\Phi^+}$ or $\ket{\Phi^-}$ each with probability $1/2$.  Only if $C$ reports ``$-1$'' (i.e., he measures $\ket{\Phi^-}$) will $A$ and $B$ use their measurement results ``$i$'' as their raw key bit. (Note that if they also use those iterations where $C$ sends ``$+1$'', this opens a potential easy attack strategy for $C$: he can always measure in the computational basis and send ``$+1$'' each iteration.)

Let $p_M$ be the probability that $A$ or $B$ measure and resend.  Note that, without noise and assuming an honest $C$, only $p_Mp_M/2 = p_M^2/2$ iterations are expected to be accepted.  However, to improve efficiency, we may use a technique from \cite{QKD-BB84-Modification} (which was meant to improve the key rate of the BB84 protocol).  Namely, we may set $p_M$ arbitrarily close to 1.  Thus, in the asymptotic scenario, we can expect $1/2$ of the qubits sent to contribute to the raw key.  It is an open question as to whether or not a mediated SQKD protocol can be designed which improves this rate to something larger than $1/2$.

\section{Security Proof}
While this protocol's unconditional security was proven in \cite{SQKD-MultiUser}, we will now provide an alternative proof of security which, as we demonstrate later, provides a more optimistic lower bound on the protocol's key rate (we will define the key rate in the asymptotic scenario shortly).  Note that, to prove the security of this protocol, we must not only consider an adversarial $C$, but also the existence of third-party, all-powerful eavesdroppers (as shown in Figure \ref{fig:diagram}).

Our proof will work in stages: we will first prove security against an adversarial $C$, assuming $C$ is limited to collective attacks (those where he performs the same attack operation each iteration, but, unlike with an independent attack discussed earlier, he is free to postpone the measurement of his quantum memory until any future time of his choice) and assuming there are no third-party eavesdroppers.  Following this, we will prove security against general attacks (those attacks where no assumptions are made other than $C$ follows the laws of physics) and assuming the existence of third-party eavesdroppers.

\subsection{Notation}
We denote by $H(\cdot)$ the Shannon entropy function.  Given a set $\{p_1, p_2, \cdots, p_n\}$ where $\sum_ip_i = 1$ and $p_i \ge 0$, then:
\[
H(p_1, p_2, \cdots, p_n) = -\sum_{i=1}^np_i\log p_i,
\]
where all logarithms in this paper are base two unless otherwise specified.  We will occasionally use the notational shortcut $H(\{p_i\}_i)$ to mean $H(p_1, p_2, \cdots, p_n)$.  Furthermore, when $n=2$ (which forces $p_2 = 1-p_1$), we will occasionally write $h(p_1)$ to mean $H(p_1, p_2)$.

Given a density operator $\rho$, we write $S(\rho)$ to be its von Neumann entropy.  Let $\{\lambda_1, \lambda_2, \cdots, \lambda_n\}$ be the eigenvalues of finite dimensional $\rho$.  Then $S(\rho) = -\sum_i\lambda_i\log\lambda_i$.

When $\rho$ acts on a bipartite system $\mathcal{H}_A\otimes\mathcal{H}_B$, we will often write $\rho_{AB}$.  Then, later, if we write $\rho_A$ we take that to mean the result of tracing out the $\mathcal{H}_B$ portion of $\rho_{AB}$ (i.e., $\rho_A = tr_B\rho_{AB}$).  Similarly for $\rho_B$.  Similarly, for multi-partite systems (e.g., $\rho_{ABC}$ acts on a tripartite system and $\rho_{BC} = tr_A\rho_{ABC}$).

Given $\rho_{AB}$, we write $S(AB)$ to mean $S(\rho_{AB})$ and $S(B)$ to mean $S(\rho_B)$.  Finally, we write $S(A|B)$ to be the conditional von Neumann entropy defined: $S(A|B) = S(AB) - S(B) = S(\rho_{AB}) - S(\rho_B)$.

\subsection{Modeling the Protocol}
For now, we will consider only an adversarial center $C$ and not third-party eavesdroppers.  That is, the system in question consists only of $A$, $B$, and $C$ and no one else (the case when there are third-party attackers will be considered later).  We will also assume $C$ employs collective attacks.  Assuming this, a single iteration of our protocol may be described as a closed system in the Hilbert space:
\[
\mathcal{H} = \mathcal{H}_A\otimes\mathcal{H}_B\otimes\mathcal{H}_{T_A}\otimes\mathcal{H}_{T_B}\otimes\mathcal{H}_C\otimes\mathcal{H}_{cl},
\]
where:
\begin{itemize}
\item $\mathcal{H}_A$ and $\mathcal{H}_B$ are $A$ and $B$'s private registers storing their raw key bit.
\item $\mathcal{H}_{T_A}$ and $\mathcal{H}_{T_B}$ are two dimensional spaces modeling the qubit channel connecting $C$ to $A$ and $C$ to $B$ respectively (they are the \emph{transit} space).
\item $\mathcal{H}_C$ is $C$'s private quantum system.
\item $\mathcal{H}_{cl}$ is a two-dimensional subspace, spanned by orthonormal basis $\{\ket{+1}, \ket{-1}\}$ used to model the classical message $C$ sends on this iteration.
\end{itemize}

We model the protocol in the same way as in \cite{SQKD-MultiUser}, the details of which we now quickly review.  At start, $C$, who we now assume is fully adversarial, prepares, not necessarily $\ket{\Phi^+}$ as prescribed by the protocol, but instead the state $\ket{\phi_0} = \sum_{i,j}\alpha_{i,j}\ket{i,j}_{T_A,T_B}\otimes\ket{c_{i,j}}$ where the $\ket{c_{i,j}}$ are arbitrary normalized, though not necessarily orthogonal, states in $\mathcal{H}_C$.

As shown in our original proof, there is no advantage to $C$ in sending the above state, versus sending the far simpler state: $\ket{{\psi}_0} = \sum_{i,j}\alpha_{i,j}\ket{i,j}_{T_A,T_B}$.  That is, there is no advantage to $C$ in preparing a state, on step (1) of the protocol, that is entangled with his private quantum memory.  The proof of this, which may be found in \cite{SQKD-MultiUser}, uses a technique introduced in \cite{SQKD-Single-Security}.  Thus, when analyzing the security of this protocol, we may assume the state $C$ sends is unentangled with $\mathcal{H}_C$.

Following $A$ and $B$'s operation (either measuring and resending, or reflecting), the qubits return to $C$.  The server, $C$, is now permitted to perform any operation of his choice, potentially entangling the qubits with his private quantum memory.  Since he must also send a single classical bit to both $A$ and $B$ (from our earlier discussion, it is impossible for $C$ to send different messages to $A$ and $B$ thus we may assume he sends only a single message and it is received by both), his attack is modeled as a \emph{quantum instrument} \cite{QC-Instrument} $\mathcal{I}$ which acts on density operator $\rho = \rho_{T_AT_BC}$ as follows:
\begin{equation}\label{eq:instrument}
\mathcal{I}(\rho) = \ket{+1}\bra{+1}_{cl}\otimes\sum_{i=1}^{N_0}E_{i,0}\rho E_{i,0}^* + \ket{-1}\bra{-1}_{cl}\otimes\sum_{i=1}^{N_1}E_{i,1}\rho E_{i,1}^*,
\end{equation}
where the $E_{i,j}$ satisfy:
\[
\sum_{i=1}^{N_0} E_{i,0}^*E_{i,0} + \sum_{i=1}^{N_1}E_{i,1}^*E_{i,1} = I.
\]
(Above, $I$ is the identity operator.)

We may assume, without loss of generality that $N_0$ and $N_1$ are both finite.  Finally, it can be shown (see \cite{SQKD-MultiUser} for details) that this attack may be represented, without loss of power to $C$, by a unitary operator $U = U_\mathcal{I}$ acting on a larger, though still finite dimensional, Hilbert space.  Providing $C$ with this larger space potentially increases his power, thus providing us with a lower bound on the security of our protocol.  Working with unitary attack operators turns out to be far simpler than working with attacks of the form in Equation \ref{eq:instrument}.

This unitary attack operator, $U$, acts on $\mathcal{H}_{T_A}\otimes\mathcal{H}_{T_B}\otimes\mathcal{H}_C\otimes\mathcal{H}_{cl}$, where $\mathcal{H}_C$ has been suitably expanded.  After $A$ and $B$'s operation, $C$ will apply unitary $U$.  $C$ will then perform a projective measurement on the $\mathcal{H}_{cl}$ system in the $\{\ket{+1}, \ket{-1}\}$ basis.  This measurement determines the message he sends.  The post-measurement state represents the state of his ancilla in the event he sends that particular message.  Such a procedure is mathematically equivalent to his use of $\mathcal{I}$ as proven in \cite{SQKD-MultiUser}.

Clearly, a single iteration of the protocol, conditioning on the event the iteration is accepted (i.e., $A$ and $B$ both measure and resend, and $C$ sends ``$-1$'') may be described by a density operator of the form:
\[
\rho_{ABC} = \sum_{i,j}p_{i,j}\ketbra{i,j}_{AB}\otimes\sigma_C^{(i,j)},
\]
where $p_{i,j}$ is the probability that $A$ and $B$'s raw key bit is $i$ and $j$ respectively and $\sigma_C^{(i,j)}$ is the state of $C$'s quantum memory in that event.

Following $N$ iterations of the quantum communication stage, assuming collective attacks, the overall system is in the state $\rho_{ABC}^{\otimes N}$.  $A$ and $B$ will then perform parameter estimation, error correction, and privacy amplification protocols (see \cite{QKD-survey} for more information on these, now standard, processes) resulting in a secret key of size $\ell(N) \le N$ (possibly $\ell(N) = 0$ if $C$ has too much information on the raw key).  We are interested in the ratio of secret key bits to raw key bits as the latter approaches infinity; that is, we are interested in the \emph{key rate}, denoted $r$, in the \emph{asymptotic scenario} \cite{QKD-survey}:
\[
r = \lim_{N \rightarrow \infty} \frac{\ell(N)}{N}.
\]

For a state of the form $\rho^{\otimes N}$, it was shown in \cite{QKD-Winter-Keyrate,QKD-renner-keyrate,QKD-renner-keyrate2} that the key rate is:
\begin{equation}
r = \inf(I(A:B) - I(A:C)) = \inf(S(A|C) - S(A|B)).
\end{equation}
Here $I(A:B)$ is the (classical) mutual information held between $A$ and $B$'s system; $I(A:C)$ is the quantum mutual information between $A$ and $C$; the conditional entropy $S(A|C)$ was defined earlier in the notation section; $H(A|B)$ is the conditional Shannon entropy defined: $H(A|B) = H(AB) - H(B)$.  Finally, the infimum is over all attack operators which induce the observed statistics (e.g., the observed error rate).

Note that the first equation, involving mutual information, is due to \cite{QKD-Winter-Keyrate}, the second, equivalent version, is from \cite{QKD-renner-keyrate,QKD-renner-keyrate2} (showing the two are equal is trivial).  While in our previous paper \cite{SQKD-MultiUser}, we used the key rate equation based on mutual information, in this paper we will bound $r$ by bounding the conditional entropy.  Though these two equations produce equal results, the technique we use to bound the latter, as we show in this paper, allows us to provide a more optimistic lower bound on $r$.

\subsection{The New Key Rate Bound}
We must now describe the state of $A$, $B$, and $C$'s system after a single iteration of the protocol, conditioning on the event that this iteration is used to contribute towards the raw key (i.e., both $A$ and $B$ measure and resend, and $C$ sends ``$-1$'').  Let $\ket{\psi_0} = \sum_{i,j}\alpha_{i,j}\ket{i,j}_{T_A,T_B}$ be the initial state prepared and sent by $C$ (from our earlier discussion this is without loss of generality), where the qubit $T_A$ is sent to $A$ and the qubit $T_B$ is sent to $B$.  Now, assume both $A$ and $B$ measure and resend (other cases, though potentially useful for parameter estimation, do not contribute towards the raw key and thus are not considered for the time being).  After $A$ and $B$'s measure and resend operation, the state is clearly:

\[
\rho_1 = \sum_{i,j\in\{0,1\}} |\alpha_{i,j}|^2\ketbra{i,j}_{A,B}\otimes\ketbra{i,j}_{T_A,T_B}.
\]

At this point, the transit system returns to $C$'s control where he will perform an arbitrary, without loss of generality unitary, operator acting on $\mathcal{H}_{T_A}\otimes\mathcal{H}_{T_B}\otimes\mathcal{H}_C\otimes\mathcal{H}_{cl}$ (since we are working with collective attacks, the latter two subspaces are assumed to be cleared to some ``zero'' state).  We write $U$'s action on basis states as follows:

\begin{equation}\label{eq:U-action}
\ket{i,j} \overset{U}\mapsto \ket{+1, e_{i,j}} + \ket{-1, f_{i,j}},
\end{equation}
where $\ket{e_{i,j}}$ and $\ket{f_{i,j}}$ are arbitrary, not necessarily normalized nor orthogonal, states in $\mathcal{H}_{T_A}\otimes\mathcal{H}_{T_B} \otimes \mathcal{H}_C$.  Unitarity of $U$ imposes various restrictions on these states which will become important later in our analysis.  Note that, unlike our analysis in our original paper, we do not make any symmetry assumptions at this point.

Following $C$'s operation, and conditioning on the event that $C$ sends ``$-1$'' (if $C$ sends ``$+1$'' the iteration is discarded and thus does not contribute to the raw key - though such iterations will be important later for parameter estimation as we discuss shortly), the final state is:

\begin{equation}\label{eq:final-state-ABC}
\rho_{ABC} = \frac{1}{p_a} \sum_{i,j}|\alpha_{i,j}|^2 \ketbra{i,j}_{AB} \otimes \ketbra{f_{i,j}}_C,
\end{equation}
where:
\begin{equation}\label{eq:pa}
p_a = \sum_{i,j}|\alpha_{i,j}|^2 \braket{f_{i,j}|f_{i,j}}.
\end{equation}
(Note that we have disregarded the $\mathcal{H}_{cl}$ portion of the above state as it is projected to $\ketbra{-1}_{cl}$ and thus not needed; we have also abused notation slightly by ``absorbing'' the $\mathcal{H}_{T_A}\otimes\mathcal{H}_{T_B}$ subspaces into $\mathcal{H}_C$.

Our goal is to compute a bound on $S(A|C) - H(A|B)$.  We will first bound $S(A|C)$.  However, the high dimensionality of the system proves a hinderance.  To overcome this, we will employ a technique first proposed in \cite{QKD-keyrate-general}, and later adapted successfully by us, to the key rate computation of a different (not a mediated) SQKD protocol in \cite{SQKD-Krawec-SecurityProof}.  This technique requires us to condition on a new random variable of our choice.  By appending a carefully chosen auxiliary system, we can simplify the entropy computations.  Due to the strong sub additivity of von Neumann entropy, it holds that, for any tripartite system $\mathcal{H}_{A}\otimes\mathcal{H}_{C}\otimes\mathcal{H}_X$, we have:
\[
S(A|C) \ge S(A|CX) \Longrightarrow S(A|C) - H(A|B) \ge S(A|CX) - H(A|B),
\]
thus allowing us to find a lower bound on the key rate of this mediated SQKD protocol.  Appending this system $\mathcal{H}_X$ and providing $C$ access to it, though unrealistic, does allow us to compute a lower bound on the key rate; the realistic case, then, where $C$ does not have this additional information $\mathcal{H}_X$ can only be better for $A$ and $B$.

The system we append, denoted $\mathcal{H}_X$, is two-dimensional and spanned by orthonormal basis states $\{\ket{C}, \ket{W}\}$ where $\ket{C}$ will be used to denote the event that $A$ and $B$'s raw key bit is correct, while $\ket{W}$ will be used to describe the event that their raw key bits are wrong.

Incorporating this system into $\rho_{ABC}$ yields the state:
\begin{align}
\rho_{ABCX} &= &&\frac{1}{p_a}\ketbra{C}\otimes\left(|\alpha_{0,0}|^2\ketbra{0,0}_{AB}\otimes\ketbra{f_{0,0}}\right.\label{eq:final-state-ABCX}\\
&&&\left.+ |\alpha_{1,1}|^2\ketbra{1,1}_{AB}\otimes\ketbra{f_{1,1}}\right) \notag\\ \notag\\
&+&&\frac{1}{p_a}\ketbra{W}\otimes\left(|\alpha_{0,1}|^2\ketbra{0,1}_{AB}\otimes\ketbra{f_{0,1}}\right.\notag\\
&&&\left.+|\alpha_{1,0}|^2\ketbra{1,0}_{AB}\otimes\ketbra{f_{1,0}}\right).\notag
\end{align}

Given such a state, we may more readily compute $S(A|CX) = S(ACX) - S(CX)$.  Indeed, it is not difficult to see that:
\begin{equation}\label{eq:entropy-ACX}
S(ACX) = H\left( \left\{ \frac{1}{p_a}|\alpha_{i,j}|^2 \braket{f_{i,j}|f_{i,j}}\right\}_{i,j}\right).
\end{equation}
(Choosing a suitable basis, we may write $\rho_{ACX}$ - which is the result of tracing out $B$ from Equation \ref{eq:final-state-ABCX} - as a diagonal matrix with diagonal elements equal to $\frac{1}{p_a} |\alpha_{i,j}|^2\braket{f_{i,j}|f_{i,j}}$.)

We must now compute an upper bound on $S(CX)$ which will provide us with a lower bound on the key rate equation.  Indeed, if $S(CX) \le \eta$ then $S(A|C) \ge S(A|CX) \ge S(ACX) - \eta$.

Define the following:
\begin{align}
p_C &= \frac{1}{p_a}(|\alpha_{0,0}|^2\braket{f_{0,0}|f_{0,0}} + |\alpha_{1,1}|^2\braket{f_{1,1}|f_{1,1}})\label{eq:pc}\\ \notag\\
p_W &= \frac{1}{p_a}(|\alpha_{0,1}|^2\braket{f_{0,1}|f_{0,1}} + |\alpha_{1,0}|^2\braket{f_{1,0}|f_{1,0}})\label{eq:pw}.
\end{align}
Clearly, $p_C$ is the probability that $A$ and $B$'s raw key bit is correct (they match) while $p_W$ is the probability that they are wrong.  For the following, assume that both $p_C$ and $p_W$ are both strictly positive.  The case when $p_W$ is zero is similar as we will comment later.  Of course if $p_C = 0$ then there is far too much noise and $A$ and $B$ should abort. (If $p_C = 0$ all their key bits are wrong!)  Note that $p_C$ and $p_W$ are parameters that may be estimated by $A$ and $B$.  Also note that $A$ and $B$ may estimate the quantities $\braket{f_{i,j}|f_{i,j}}$ which are simply the probabilities that $C$ sends ``$-1$'' in the event $A$ and $B$ both measure and resend $\ket{i,j}$.

Tracing out $A$ and $B$ from Equation \ref{eq:final-state-ABCX} yields:
\begin{align}
\rho_{CX} &= \frac{1}{p_a}\ketbra{C}\otimes\left( |\alpha_{0,0}|^2 \ketbra{f_{0,0}} + |\alpha_{1,1}|^2 \ketbra{f_{1,1}} \right)\label{eq:final-state-CX}\\
&+\frac{1}{p_a}\ketbra{W} \otimes \left( |\alpha_{0,1}|^2 \ketbra{f_{0,1}} + |\alpha_{1,0}|^2 \ketbra{f_{1,0}} \right).\notag \\ \notag\\
&= p_C\ketbra{C} \otimes \sigma_C + p_W\ketbra{W} \otimes \sigma_W,\notag
\end{align}
where $\sigma_x$ are the following (unit trace - recall we are assuming for now that both $p_C$ and $p_W$ are non-zero) density operators:
\begin{align}
\sigma_C &= \frac{|\alpha_{0,0}|^2 \ketbra{f_{0,0}} + |\alpha_{1,1}|^2\ketbra{f_{1,1}}}{|\alpha_{0,0}|^2\braket{f_{0,0}|f_{0,0}} + |\alpha_{1,1}|^2\braket{f_{1,1}|f_{1,1}}}.\\ \notag\\
\sigma_W &= \frac{|\alpha_{0,1}|^2 \ketbra{f_{0,1}} + |\alpha_{1,0}|^2\ketbra{f_{1,0}}}{|\alpha_{0,1}|^2\braket{f_{0,1}|f_{0,1}} + |\alpha_{1,0}|^2\braket{f_{1,0}|f_{1,0}}}.
\end{align}

It is not difficult to show (see, for instance, \cite{SQKD-Krawec-SecurityProof} for a proof) that the entropy of such a system is simply:
\begin{align}
S(CX) &= H(p_C, p_W) + p_WS(\sigma_W) + p_CS(\sigma_C)\notag\\
&\le H(p_C, p_W) + p_W + p_CS(\sigma_C).\label{eq:entropy-bound}
\end{align}
The inequality above follows from the fact that $S(\sigma_W) \le \log \dim \sigma_W$.  Since $\sigma_C$ and $\sigma_W$ are both two-dimensional, $S(\sigma_W) \le 1$.  Note that it is not difficult to show that if $p_W = 0$, then $|\alpha_{0,1}|^2\ketbra{f_{0,1}} + |\alpha_{1,0}|^2\ketbra{f_{1,0}} \equiv 0$.  Thus, this term does not appear in Equation \ref{eq:final-state-CX}, and so the bound in Equation \ref{eq:entropy-bound} holds even in this case.

Obviously if the noise is small, than $p_W$ should also be small.  All that remains, therefore, is to upper bound $S(\sigma_C)$.

We may write, without loss of generality, $\ket{f_{0,0}} = x\ket{f}$ and $\ket{f_{1,1}} = y\ket{f} + z\ket{\zeta}$, where $\braket{f|f} = \braket{\zeta|\zeta} = 1$, $\braket{f|\zeta} = 0$, and $x,y,z \in \mathbb{C}$.  This of course implies:
\begin{align}
&|x|^2 = \braket{f_{0,0}|f_{0,0}}\label{eq:identity-ev-1}\\
&|y|^2+|z|^2 = \braket{f_{1,1}|f_{1,1}}\label{eq:identity-ev-2}\\
&x^*y = \braket{f_{0,0}|f_{1,1}} \Longrightarrow |y|^2 ={|\braket{f_{0,0}|f_{1,1}}|^2}/{|x|^2}.\label{eq:identity-ev-3}
\end{align}

Using this $\{\ket{f}, \ket{\zeta}\}$ basis, we may write $\sigma_C$ as:

\[
\sigma_C = q_0 \left( \begin{array}{ccc}
|\alpha_{0,0}|^2 |x|^2 + |\alpha_{1,1}|^2|y|^2 &,& |\alpha_{1,1}|^2y^*z \\\\
|\alpha_{1,1}|^2yz^* &,& |\alpha_{1,1}|^2|z|^2\end{array}\right),
\]
where:
\[
q_0 = (|\alpha_{0,0}|^2\braket{f_{0,0}|f_{0,0}} + |\alpha_{1,1}|^2\braket{f_{1,1}|f_{1,1}})^{-1} = [|\alpha_{0,0}|^2|x|^2 + |\alpha_{1,1}|^2(|y|^2+|z|^2)]^{-1}.
\]

The eigenvalues of this matrix are easily computed to be:
\begin{align*}
\lambda_{\pm} &= \frac{1}{2} \pm \frac{q_0}{2}\sqrt{\left(|\alpha_{0,0}|^2|x|^2 + |\alpha_{1,1}|^2|y|^2 - |\alpha_{1,1}|^2|z|^2\right)^2 + 4 |\alpha_{1,1}|^4|y|^2|z|^2}\\
&=\frac{1}{2} \pm \frac{q_0}{2}\sqrt{ \left( |\alpha_{0,0}|^2F_{0,0} + |\alpha_{1,1}|^2[2|y|^2 - F_{1,1}] \right)^2 + 4 |\alpha_{1,1}|^4|y|^2(F_{1,1} - |y|^2)},
\end{align*}
where above we have defined $F_{i,i} = \braket{f_{i,i}|f_{i,i}}$ and have used the identities \ref{eq:identity-ev-1} and \ref{eq:identity-ev-2}.  Now, define $\Delta = |\alpha_{0,0}|^2F_{0,0} - |\alpha_{1,1}|^2F_{1,1}$ and, continuing, we have:
\begin{align*}
\lambda_{\pm} &= \frac{1}{2} \pm \frac{q_0}{2}\sqrt{\left(\Delta + 2 |\alpha_{1,1}|^2|y|^2\right)^2 + 4|\alpha_{1,1}|^4|y|^2F_{1,1} - 4|\alpha_{1,1}|^4|y|^4}\\
&= \frac{1}{2} \pm \frac{q_0}{2}\sqrt{\Delta^2 + 4|\alpha_{1,1}|^2|y|^2\left(\Delta + |\alpha_{1,1}|^2F_{1,1}\right)}\\
&= \frac{1}{2} \pm \frac{q_0}{2}\sqrt{\Delta^2 + 4|\alpha_{1,1}|^2|y|^2|\alpha_{0,0}|^2|x|^2}.
\end{align*}
Finally, using identity \ref{eq:identity-ev-3} (and that $|x|^2 = F_{0,0}$), we have:
\begin{equation}
\lambda_{\pm} = \frac{1}{2} \pm \frac{q_0}{2}\sqrt{\Delta^2 + 4|\alpha_{0,0}|^2|\alpha_{1,1}|^2|\braket{f_{0,0}|f_{1,1}}|^2}
\end{equation}

Note that $\Delta, q_0, |\alpha_{0,0}|^2$, and $|\alpha_{1,1}|^2$ are all parameters that $A$ and $B$ may estimate.  Computing a bound on $|\braket{f_{0,0}|f_{1,1}}|^2$ may be achieved using the error rate when both $A$ and $B$ reflect as we demonstrate later.

Combining everything together yields the bound:
\begin{align*}
S(A|C) \ge H\left( \left\{ \frac{1}{p_a}|\alpha_{i,j}|^2 \braket{f_{i,j}|f_{i,j}}\right\}_{i,j}\right) - H(p_C, p_W) - p_W - p_CH(\lambda_+, \lambda_-).
\end{align*}

Note that $H(\lambda_+, \lambda_-) = h(\lambda_+)$ takes its maximum when $\lambda_+ = \frac{1}{2}$.  It is not difficult to see that, as $|\braket{f_{0,0}|f_{1,1}}|^2 \ge 0$ increases, $\lambda_+$ increases.  Furthermore, when $|\braket{f_{0,0}|f_{1,1}}|^2 = 0$, $\lambda_+ \ge \frac{1}{2}$.  Thus, as $|\braket{f_{0,0}|f_{1,1}}|^2$ increases, $h(\lambda_+)$ necessarily decreases.  By finding a lower-bound $\mathcal{F}$ (which will be a function of certain observed parameters as we will soon discuss) such that $|\braket{f_{0,0}|f_{1,1}}|^2 \ge \mathcal{F}$, we will have an upper-bound on $h(\lambda_+)$ and thus a lower bound on $S(A|CX)$.  That is to say, if we define:
\begin{equation}
\tilde{\lambda} = \frac{1}{2} + \frac{q_0}{2}\sqrt{\Delta^2 + 4|\alpha_{0,0}|^2|\alpha_{1,1}|^2\mathcal{F}},
\end{equation}
then:
\[
\frac{1}{2} \le \tilde{\lambda} \le \lambda_+ \Longrightarrow h(\lambda_+) \le h(\tilde{\lambda}),
\]
This implies:
\begin{equation}\label{eq:entropy-bound-final}
S(A|C) \ge H\left( \left\{ \frac{1}{p_a}|\alpha_{i,j}|^2 \braket{f_{i,j}|f_{i,j}}\right\}_{i,j}\right) - H(p_C, p_W) - p_W - p_Ch(\tilde{\lambda}).
\end{equation}

This concludes our bound on $S(A|C)$.  Determining a bound on $\mathcal{F}$ (needed to bound $h(\lambda_+)$) can be easily determined from the probability that $C$ sends ``$-1$'' if both $A$ and $B$ reflect (note that $C$ should send ``$+1$'' if they both reflect - $C$ sending ``$-1$'' in this event is counted as an error).  In the next section, we will demonstrate this in two specific cases comparing our new result with our older one from \cite{SQKD-MultiUser}.

Of course, computing $H(A|B)$, the last remaining term from the key rate equation, is easy after parameter estimation.  Indeed, let $p(a,b)$ be the probability of $A$'s raw key bit being $a$ and $B$'s raw key bit being $b$.  From Equation \ref{eq:final-state-ABC}, we see these values are:
\begin{equation}\label{eq:pab}
p(i,j) = \frac{1}{p_a}|\alpha_{i,j}|^2 \braket{f_{i,j}|f_{i,j}}
\end{equation}
Also, if we let $p(b)$ be the probability that $B$'s raw key bit is $b$ (i.e., $p(b) = p(0,b) + p(1,b)$) then our final key rate bound is found to be:
\begin{align*}
r &\ge H\left( \left\{ \frac{1}{p_a}|\alpha_{i,j}|^2 \braket{f_{i,j}|f_{i,j}}\right\}_{i,j}\right) - H(p_C, p_W) - p_W - p_Ch(\tilde{\lambda})\\
& - H\left(\{p(a,b)\}_{a,b}\right) + h(p(0)).
\end{align*}
Using Equation \ref{eq:pab}, this can be simplified to:
\begin{equation}\label{eq:key-rate-bound}
r \ge h(p(0)) - H(p_C, p_W) - p_W - p_Ch(\tilde{\lambda})
\end{equation}

Note that all terms in the above expression, with the exception of $\tilde{\lambda}$, are directly observable by $A$ and $B$.  Indeed, $|\alpha_{i,j}|^2$ is simply the probability that $A$ and $B$ measure $\ket{i,j}$.  The quantity $\braket{f_{i,j}|f_{i,j}}$ is the probability that $C$ sends ``$-1$'' in the event $A$ and $B$ measure and resend $\ket{i,j}$.  If $A$ and $B$ divulge complete information about certain randomly chosen iterations (in particular their choices and measurement results if applicable), this information is easily estimated.  The only parameter that cannot be directly observed is $\tilde{\lambda}$; in particular the quantity $|\braket{f_{0,0}|f_{1,1}}|^2$ upon which that eigenvalue depends.  They may, however, estimate it using the probability that $C$ sends ``$-1$'' if $A$ and $B$ both reflect (the probability of which $A$ and $B$ may estimate).  We will show how this is done in the next section when we actually evaluate the above key rate expression for certain scenarios.

\subsection{General Attacks and Third Party Eavesdroppers}
We considered only collective attacks above.  However, if $A$ and $B$ permute their raw key bits, using a randomly chosen (and publicly disclosed) permutation, the protocol becomes permutation invariant \cite{QKD-renner-keyrate}.  Thus the results in \cite{QKD-general-attack,QKD-general-attack2} apply showing that, to prove security against general attacks (where $C$ is allowed to perform any operation of his choice - perhaps altering his attack operator each iteration) it is sufficient to prove security against collective attacks.  Furthermore, since we are considering the asymptotic scenario in this paper, our key rate bound is equivalent in both cases.

Finally, it is clear that any attack by a third-party eavesdropper (including attacks whereby the eavesdropper alters $C$'s classical messages - recall that channel is not authenticated) can simply be ``absorbed'' into $C$'s attack operator $U$.  Thus our bound holds even in this case.

\section{Evaluation}
Our key rate bound applies in the most general of cases.  $A$ and $B$ must simply observe certain parameters and, based on these, they may determine a lower bound on their secret key fraction (bounding $|\braket{f_{0,0}|f_{1,1}}|^2$ may also be achieved using these parameters as we discuss shortly).  Of course, due to its reliance on many parameters, it is difficult to visualize this bound here in this paper.  However, we may consider certain scenarios which $A$ and $B$ may encounter and evaluate our bound based on these particular scenarios.  First, we will consider the case of a semi-honest server and a noisy quantum channel.  Later we will consider an adversarial server whose attack is ``symmetric'' (a common assumption in QKD security proofs).  These two scenarios were considered in our earlier work \cite{SQKD-MultiUser} and so will allow us to compare our new results with our old showing the superiority of our new bound in both of these scenarios.

\subsection{Bounding $|\braket{f_{0,0}|f_{1,1}}|^2$}
We now show how to bound the quantity $|\braket{f_{0,0}|f_{1,1}}|^2$ based on the probability that $C$ sends ``$-1$'' if $A$ and $B$ had sent the Bell state $\ket{\Phi^+}$ to $C$.  Obviously this is not a parameter that is directly observable ($A$ and $B$ are classical and so cannot prepare $\ket{\Phi^+}$).  However, it may be bounded, as we show later, using the error rate in those iterations where $A$ and $B$ reflected (i.e., the probability that $C$ sends ``$-1$'' if $A$ and $B$ both reflect).

Consider the action of $C$'s attack operator $U$ on Bell basis states.  This is:
\begin{align*}
U\ket{\Phi^+} &= \ket{+1, g_0} + \ket{-1, h_0}\\
U\ket{\Phi^-} &= \ket{+1, g_1} + \ket{-1, h_1}\\
U\ket{\Psi^+} &= \ket{+1, g_2} + \ket{-1, h_2}\\
U\ket{\Psi^-} &= \ket{+1, g_3} + \ket{-1, h_3},
\end{align*}
where each $\ket{g_i}$ and $\ket{h_i}$ are linear functions of $\ket{e_{k,l}}$ and $\ket{f_{k,l}}$ respectively (see Equation \ref{eq:U-action}).  In particular, we have $\ket{h_0} = \frac{1}{\sqrt{2}}(\ket{f_{0,0}} + \ket{f_{1,1}})$.  Imagine, for the time, that, in addition to the other parameters mentioned in the last section, $A$ and $B$ are also able to estimate the parameter $\braket{h_0|h_0}$:
\begin{equation}\label{eq:h0}
\braket{h_0|h_0} = \frac{1}{2}(\braket{f_{0,0}|f_{0,0}} + \braket{f_{1,1}|f_{1,1}} + 2Re\braket{f_{0,0}|f_{1,1}}).
\end{equation}
This then provides an estimate of $|\braket{f_{0,0}|f_{1,1}}|^2$.  Indeed, if $Re\braket{f_{0,0}|f_{1,1}} = x$, then of course $|\braket{f_{0,0}|f_{1,1}}|^2 = Re^2\braket{f_{0,0}|f_{1,1}} + Im^2\braket{f_{0,0}|f_{1,1}} \ge Re^2\braket{f_{0,0}|f_{1,1}} = |x|^2$.  Thus:
\begin{align}
\braket{h_0|h_0} &= \eta\notag\\
\Rightarrow |\braket{f_{0,0}|f_{1,1}}|^2 &\ge \left(\frac{1}{2}\braket{f_{0,0}|f_{0,0}} + \frac{1}{2}\braket{f_{1,1}|f_{1,1}} - \eta\right)^2\label{eq:bound-equal-f00f11}.
\end{align}

This quantity $\braket{h_0|h_0}$ is simply the probability that $C$ sends ``$-1$'' if $A$ and $B$ jointly send the state $\ket{\Phi^+}$ (this should be small).  Obviously they cannot estimate this parameter directly in reality (they are neither of them quantum).  However, as we will show later, they are able to compute an upper bound on it.  In particular, if they bound $\braket{h_0|h_0} \le \eta$, then, for $\eta$ small enough (and recall it should be small since $C$ is supposed to send ``$+1$'' if he receives $\ket{\Phi^+}$), it holds that:
\begin{align}
\braket{h_0|h_0} &\le \eta\notag\\
\Rightarrow Re\braket{f_{0,0}|f_{1,1}} &\le \eta - \frac{1}{2}\braket{f_{0,0}|f_{0,0}} - \frac{1}{2}\braket{f_{1,1}|f_{1,1}} \le 0\notag\\
\Rightarrow Re^2\braket{f_{0,0}|f_{1,1}} &\ge \left(\frac{1}{2}\braket{f_{0,0}|f_{0,0}} + \frac{1}{2}\braket{f_{1,1}|f_{1,1}} - \eta\right)^2\notag\\
\Rightarrow |\braket{f_{0,0}|f_{1,1}}|^2 &\ge \left(\frac{1}{2}\braket{f_{0,0}|f_{0,0}} + \frac{1}{2}\braket{f_{1,1}|f_{1,1}} - \eta\right)^2.\label{eq:bound-general-f00f11}
\end{align}
(We see that $\eta$ must be small enough so that the right hand side of the second inequality is negative.)

Both of these two identities will be useful in the subsequent examples where we bound $\eta$ as a function of $Q$ - the probability that $A$ and $B$'s measurement results differ - and $p_w$ - the probability that $C$ sends the wrong message when $A$ and $B$ both reflect (i.e., he sends ``$-1$''); with $Q = p_w = 0$ implying $\eta = 0$.

\subsection{Semi-Honest Center}

In this first example, we will consider the case where $C$ is \emph{semi-honest} - that is, he always follows the protocol exactly, preparing the correct state in step one of the protocol, performing the correct measurement in step three, and reporting, honestly, his measurement result.  Beyond that, however, he is free to do whatever he likes, for instance, he can listen in on the public communication channel to try and gain additional information on the key.  This is not only a practically relevant scenario to analyze, but it also allows us to compare our new key rate bound with our old from \cite{SQKD-MultiUser}.

Besides the semi-honest sever, we will also assume a noisy quantum channeled modeled using two independent depolarization channels, one for the forward direction (qubits traveling from $C$ to $A$ and $B$) with parameter $p$, the other for the reverse (qubits returning from $A$ and $B$ to $C$) with parameter $q$.  That is, if the joint qubit state is $\rho$ (a density operator acting on a four dimensional Hilbert space), then the depolarization channel with parameter $p$ is:
\[
\mathcal{E}_p(\rho) = (1-p)\rho + \frac{p}{4}I,
\]
where $I$ is the identity operator.  Furthermore, for the remainder of this sub-section, we will relabel the Bell basis states as follows: $\ket{\Phi^+} = \ket{\phi_0}$, $\ket{\Phi^-} = \ket{\phi_1}$, $\ket{\Psi^+} = \ket{\phi_2}$, and $\ket{\Psi^-} = \ket{\phi_3}$.

To compute a key rate bound, we need only compute a few parameters.  First are the probabilities that $C$ sends ``$-1$'' if $A$ and $B$ measure and resend the value $\ket{i,j}$ (i.e., we estimate the value $\braket{f_{i,j}|f_{i,j}}$).  These are easy to compute given our assumptions of a semi-honest server and depolarization channels.  For instance, if $A$ and $B$ measure and resend $\ket{0,0}$, then the state arriving at $C$'s lab is:
\[
\mathcal{E}_q(\ketbra{0,0}) = (1-q)\ketbra{0,0} + \frac{q}{4}\sum_{i,j\in\{0,1\}}\ketbra{i,j}.
\]

$C$ now performs a Bell measurement of this system, with the message ``$-1$'' being sent only if he measures $\ketbra{\phi_1}$ (i.e., $\ketbra{\Phi^-}$).  It is easy to see that the probability of him sending this message, given state $\mathcal{E}_q(\ketbra{0,0})$ is:
\[
\frac{1-q}{2} + \frac{q}{4} = \frac{2-q}{4}.
\]
Thus $\braket{f_{0,0}|f_{0,0}} = (2-q)/4$.  Using a similar process, the remaining $\braket{f_{i,j}|f_{i,j}}$ may be computed:
\begin{align}
\braket{f_{0,0}|f_{0,0}} &= \frac{2-q}{4} = \braket{f_{1,1}|f_{1,1}}\label{eq:dc:f00}\\
\braket{f_{0,1}|f_{0,1}} &= \frac{q}{4} = \braket{f_{1,0}|f_{1,0}}.\label{eq:dc:f01}
\end{align}

Next, we compute $|\alpha_{i,j}|^2$ (the probability that $A$ and $B$ measure $\ket{i,j}$).  Since we are assuming $C$ is semi-honest and thus he prepared $\ket{\phi_0}$ initially, these are computed using the state:
\[
\mathcal{E}_p(\ketbra{\phi_0}) = (1-p)\ketbra{\phi_0} + \frac{p}{4}\sum_{i=0}^3\ketbra{\phi_i}.
\]
Clearly:
\begin{align*}
|\alpha_{0,0}|^2 &= \frac{2-p}{4} = |\alpha_{1,1}|^2\\
|\alpha_{0,1}|^2 &= \frac{p}{4} = |\alpha_{1,0}|^2.
\end{align*}

Thus, the value of $p$ may be estimated by $A$ and $B$ using the observable parameters $|\alpha_{i,j}|^2$, while $q$ can be estimated using the observable parameters $\braket{f_{i,j}|f_{i,j}}$.  Furthermore, we have that $Q$, which is the probability that $A$ and $B$'s measurement results are different, is $Q = |\alpha_{0,1}|^2 + |\alpha_{1,0}|^2 = p/2$.

$A$ and $B$ are now able to compute $p_a$, $p_C$, and $p_W$ in terms of $p$ and $q$.  All that remains is to bound $|\braket{f_{0,0}|f_{1,1}}|^2$.  We will use our above discussion and instead determine a value $\eta$ such that $\braket{h_0|h_0}$ (which is the probability that $C$ sends ``$-1$'' if the joint state leaving $A$ and $B$'s lab is $\ket{\phi_0}$) is upper bounded by $\eta$.

In the general case, which we consider next, this value $\eta$ can only be estimated (since $A$ and $B$ are classical users, they cannot prepare a Bell state to directly observe $\braket{h_0|h_0}$).  However, given the assumptions in this sub-section, they may in fact compute a value for $\eta$ based on $q$.  Indeed, if the state leaving $A$ and $B$ is $\ket{\phi_0}$, then the system, when it reaches $C$, has evolved via:
\[
\mathcal{E}_q(\ketbra{\phi_0}) = (1-q)\ketbra{\phi_0} + \frac{q}{4}\sum_{i=0}^3\ketbra{\phi_i}.
\]
Since $C$ is semi-honest, he will only send ``$-1$'' if he measures $\ket{\phi_1}$.  This probability is simply $\braket{h_0|h_0} = q/4 = \eta$.  We may now use Equation \ref{eq:bound-equal-f00f11} to lower bound the quantity $|\braket{f_{0,0}|f_{1,1}}|^2$.

Using these values and Equation \ref{eq:key-rate-bound}, we see that the key rate of this mediated protocol, in the event $p=q=2Q$ (i.e., the noise in the forward and reverse channels is equivalent and the probability that $A$ or $B$'s measurement results are wrong is $Q$), remains positive for all $Q \le 22.05\%$ as shown in Figure \ref{fig:keyrate-semihonest}.  This is an improvement over our original bound of $Q \le 19.9\%$.

Note that, to achieve this high tolerance level in the presence of third-party eavesdroppers, the classical channel connecting $C$ to $A$ or $B$ needs to be authenticated.  This requirement is not necessary in the next scenario we consider where $C$ is fully adversarial.

\begin{figure}\center
\includegraphics[width=250pt]{semihonest.png}
\caption{A graph of our key rate lower bound when the server $C$ is semi-honest and there is a noisy quantum channel, modeled by two independent depolarization channels, connecting the users.  This graph assumes the noise in the forward direction (when the qubits travel from $C$ to $A$ and $B$) is equal to the noise in the reverse (when qubits return from $A$ and $B$ to $C$).  Observe that the key rate remains positive for all $Q \le 22.05\%$, where $Q$ is the probability that $A$ and $B$'s measurement results differ.}\label{fig:keyrate-semihonest}
\end{figure}

\subsection{Adversarial Server, Symmetric Attack}
Of course the above section assumed $C$ was semi-honest - that is he followed the protocol exactly.  Now, we assume the server $C$ is adversarial.  He is allowed to prepare any state he likes on step (1) of the protocol, and he may perform any arbitrary operation (allowed by the laws of quantum physics) when the qubits return to him, sending any message of his choice based on his operation.  The only assumption we make in this section is that $C$'s attack is symmetric in that it can be parameterized by only a few parameters (to be discussed).  We make this assumption first, so that we can compare our new key rate bound with our old one from \cite{SQKD-MultiUser} (which assumed a symmetric attack); and second, so that we can better visualize the key rate bound by reducing the many parameters to only a few.  Note that in our original proof of security in \cite{SQKD-MultiUser}, we assumed throughout that $C$'s attack was symmetric (the key rate bound we derived there was based on this assumption).  In our new bound in this paper, however, we made no such assumptions - Equation \ref{eq:key-rate-bound} works even in the most general case.  So, while assuming a symmetric attack is not required for our new proof in this paper, it is easier to visualize our key rate bound and it also allows us to compare our new results with our old.

The first assumption we make is that we may parameterize the values $|\alpha_{i,j}|^2$ using only a single parameter $Q$.  Namely, we have:
\begin{align*}
|\alpha_{0,0}|^2 &= \frac{1-Q}{2} = |\alpha_{1,1}|^2\\
|\alpha_{0,1}|^2 &= \frac{Q}{2} = |\alpha_{1,0}|^2.
\end{align*}

Next, we assume that the probability that $C$ sends ``$-1$'' in the event $A$ and $B$ measure $\ket{0,0}$ is equal to the probability he sends that same message if they both measure $\ket{1,1}$.  Similarly for the case $\ket{0,1}$ and $\ket{1,0}$.  That is:
\begin{align*}
\braket{f_{0,0}|f_{0,0}} &= \braket{f_{1,1}|f_{1,1}} = \mathcal{F}_{=}\\
\braket{f_{0,1}|f_{0,1}} &= \braket{f_{1,0}|f_{1,0}} = \mathcal{F}_{\ne}
\end{align*}
We make no assumptions regarding the comparative relationship between $\mathcal{F}_=$ and $\mathcal{F}_{\ne}$.

Using similar arguments as in \cite{SQKD-MultiUser} (in particular see Equation 25 from that source, along with its derivation), we may find the following upper-bound on $\braket{h_0|h_0}$:
\[
\braket{h_0|h_0} \le \left(\frac{ \sqrt{1-Q}\left(\sqrt{Q\mathcal{F}_{\ne}} + \sqrt{p_w} \right) } {1-Q}\right)^2 = \eta,
\]
where $p_w$ is the probability that $C$ sends the wrong message (namely ``$-1$'') in the event both $A$ and $B$ reflect.

This is our value for $\eta$ allowing us to bound $|\braket{f_{0,0}|f_{1,1}}|^2$ using Equation \ref{eq:bound-general-f00f11} from the previous section.  However, to evaluate this bound, we must determine what to set $\mathcal{F}_=$ and $\mathcal{F}_{\ne}$ to.  Naturally, in practice, these values are simply observed by $A$ and $B$.  However, for this paper, to evaluate our bound, we will consider two examples.

First, in order to compare with our previous bound from our original paper, we will use $\mathcal{F}_{\ne} = Q$ and:
\[
\mathcal{F}_= = \frac{\tilde{p}_a - Q^2}{1-Q},
\]
where $\tilde{p}_a$ is the desired probability of acceptance.  It is trivial to check that, when setting these values thusly, we have $p_a = \tilde{p}_a$ (see Equation \ref{eq:pa}).  We have to set this parameter this way since, in our original proof, we evaluated our key rate equation assuming $p_a$ was $0.5$, then $0.4$, and finally $0.3$ (our original proof did not establish a clear relationship between the value of $p_a$ and the other observed parameters as we did in this new proof).

Using these values, we see that, when $Q = p_w$ and $\tilde{p}_a = .5$ the key rate expression is positive for all $Q \le 12.5\%$; when $\tilde{p}_a = .4$ it is positive for $Q \le 10.8\%$; and for $\tilde{p}_a = .3$ it remains positive for $Q \le 8.86\%$.  See Figure \ref{fig:keyrate-adv-1}.  Compare this with our old bound from \cite{SQKD-MultiUser} - there, when $\tilde{p}_a = .5$ our old bound remained positive for $Q \le 10.65\%$, while for $\tilde{p}_a = .3$, the old bound was positive only for $Q \le 5.25\%$.

\begin{figure}\center
\includegraphics[width=250pt]{advcompare.png}
\caption{Showing a graph of our key rate lower bound when the server is fully adversarial and when $C$'s attack is such that $p_a$, the probability that any particular iteration is accepted by $A$ and $B$, is equal to $0.5$, $0.4$, and $0.3$.  Here, $Q$ represents the probability that $A$ and $B$'s measurement results are different.  This is done to compare with our old bound from \cite{SQKD-MultiUser} (see text).  Indeed, we see that in all three cases, our new bound is superior.  For instance, when $p_a = .5$, our new bound remains positive for all $Q \le 12.5\%$ whereas our old bound from the previous paper remained positive only for $Q \le 10.65\%$.  The difference is even more noticeable for smaller values of $p_a$ as mentioned in the text.}\label{fig:keyrate-adv-1}
\end{figure}

The second example we consider is based on values determined from the depolarization channel example in the previous section.  We will assume the noise in both channels is equal.  In that case, we have $\mathcal{F}_{\ne} = Q/2$ while $\mathcal{F}_= = 1/2 - Q/2$ (see Equations \ref{eq:dc:f00} and \ref{eq:dc:f01} and recall that $Q = 2p = 2q$).

Using these values, we see that, when $Q = p_w$, the key rate remains positive for all $Q \le 13.04\%$.  See Figure \ref{fig:keyrate-adv-2}.

\begin{figure}\center
\includegraphics[width=250pt]{advsecond.png}
\caption{Showing a graph of our key rate lower bound when the server is fully adversarial and when $C$'s attack is such that certain parameters agree with the depolarization example (see text).  Here, $Q$ represents the probability that $A$ and $B$'s measurement results are different.  Note that the key rate remains positive for all $Q \le 13.04\%$.}\label{fig:keyrate-adv-2}
\end{figure}

For comparison, BB84 can tolerate up to $11\%$ error while the six-state BB84 can tolerate up to $12.6\%$ (both bounds without pre-processing since we have not considered pre-processing in our proof) \cite{QKD-renner-keyrate,QKD-renner-keyrate2}.  B92 \cite{QKD-B92} was recently shown to tolerate up to $6.5\%$ error assuming a depolarization channel \cite{QKD-B92-Improved}.

We stress that, in practice, there is no need to make such assumptions to determine $\mathcal{F}_=$ and $\mathcal{F}_{\ne}$: these are parameters that are observed directly.  We made assumptions only to visualize our key rate bound and to compare it with our previous work.

\section{Conclusion}
We have provided a new proof of security for the mediated semi-quantum key distribution protocol presented in \cite{SQKD-MultiUser}.  While a proof of security was provided in that original source, we have demonstrated that our new key rate bound provides a more optimistic rate.  Indeed we see that in every scenario considered, our new proof demonstrates that the tolerated noise level of the protocol (the maximal amount of noise before $A$ and $B$ should abort) is strictly larger than the old key rate bound.  Our new proof also does not make any assumptions about the attack used by $C$.  Note that, while we only evaluated our key rate bound in two specific examples - the semi-honest case and the symmetric adversarial case - our proof works in even the most general of scenarios (i.e., a non-symmetric adversarial attack).  Our work in this paper has shown that this mediated SQKD protocol can tolerate noise levels surpassing the maximal tolerated noise thresholds of many other fully quantum protocols.  Finally, the proof technique we used may hold utility in security proofs of other protocols (quantum or semi-quantum) which rely on a two-way quantum communication channel.

Many open problems remain.  Most important, perhaps, is that we considered only the perfect qubit scenario; dealing with problems such as multi-photon attacks is an interesting area of research (in any two-way quantum protocol).  Also, can a mediated SQKD protocol be designed that is more efficient - recall that, in the absence of noise and with an honest server, only half of the sent qubits can contribute to the raw key.  Finally, can a mediated SQKD protocol be designed without the need for $C$ to prepare and measure in the Bell basis.  As mentioned in \cite{SQKD-MultiUser}, the answer to this seems to be positive, however the security proof is more involved.  Perhaps the techniques we developed and used in this paper can be adapted towards this goal.

redpajama_set_name: RedPajamaArXiv


text:
Displaying items by tag: budva
WPD AG and Vjetroelektrana Budva to Invest €101.3m in Brajici Wind Farm
November 26, 2019 - A consortium of the German WPD AG company and Vjetroelektrana Budva [in English: Budva Wind Farm] is the only bidder that met all the requirements for the construction of the Wind Farm Brajići on state-owned land in the municipalities of Budva and Bar, suggests the Decision on the shortlist of bidders.
The document drew up by the Ministry of Economy says the German-Montenegrin consortium was the only bidder which made a bid in a timely manner and offered an investment of EUR 101,3 million for the construction of a wind farm.
The bid encompasses the construction of a wind farm with the potential of producing 100,8 MW, while the consortium also offered to pay €2,50 per square meter for the lease of state-owned land.
The bid also envisages the participation of national companies in the amount of €21,6 million and the construction of a new wind power plant at an area of 235,462 square meters.
Text by CDM on November 26, 2019, read more here.
To read more about business in Montenegro, follow TMN's dedicated page.
Budvanska Rivijera HG's Revenue Amounts to €4,88 Million
November 11, 2019 - Budvanska Rivijera Hotel Group generated revenue of €4,88 million at the end of September this year, the report on the operating performance of the company shows.
Limited liability company, which encompasses Palas and Castellastva hotels in Petrovac, Aleksandar and Mogren in Budva, as well as tourist settlement Slovenska plaža, made great results during the main season despite a portfolio of investments in internal restructuring.
Operating income amounted to €20,05 million in the first nine months. Around €18,44 million were used for the sale of capacities and services in five hotels.
In the same period, expenses amounted to €15,33 million. Around €5,69 million were related to employees' salaries, €3,76 million went for the material, amortization required €1,83 million and more than €4 million related to other operational expenses.
Since other business activities brought another €174.165, Budvanska Riviera finished the third quarter with the final €4,88 million.
The company's total assets amounted to €140,78 million in this period.
The company earmarked €1,65 million for the short-term financial arrangements – €12,54 million for the cash and €1,41 million for stocks.
The current initial capital is worth €55,04 million and is divided into more than 8 million shares.
Deferred tax liabilities amount to €7,96 million whereas retained earnings reached €9,76 million.
On 25 June, the majority shareholders of Budvanska Riviera HG gave green light to restructuring project. A new limited liability company was established – Sveti Stefan hotels, encompassing Sveti Stefan and Miločer hotels.
To read more about business in Croatia, follow TCN's dedicated page.
29 Oct 2019, 12:31 PM
Sea Property Working on Tender for Budva Promenade Design Solutions
October 29, 2019 - Director of the Public Company Sea Property Predrag Jelusic expects the announcement of an international tender for the development of an architectural solution for the promenade in Budva, which will cover the hinterland of Slovenska Plaza and Bečići beaches, including their connection. The Ministry of Sustainable Development and Tourism, Jelusic said in an interview with Dnevne Novine, has sent requests to all competent institutions for issuing approvals for urban technical conditions.
"Planning documentation is being prepared. It is extensive material, as several plans treat this area - Detailed Urban Plan Center Budva, Detailed Urban Plan Center Becici, as well as basic studies. The team formed within the Ministry of Sustainable Development and Tourism is currently working to unify the studies with the preparation of all the elements that are relevant to the upcoming international competition. The construction of a five-kilometer-long promenade from the Old Town to the location of Sole Mio and the "blue stone" in Rafailovići is an emancipation project of Budva. It is an urgent project in the metropolis of tourism for Montenegro as well since today in Budva; we still have about 45 percent of the total traffic of Montenegro", emphasizes Jelusic.
The director of the Sea Property says that the company has earmarked 300,000 euros for announcing an international competition and drafting the preliminary and general design, for which there is already a great interest in renowned design and architectural companies. There is also an active investor interest in building high-class hotels along the current and future promenade.
"It will give extra quality. This content project will have squares, green areas. It will have contents that will be different from the current ones that can function for all 12 months. It is the most beautiful part of Budva, the most beautiful promenade in Montenegro, and it deserves to be in the function of both recreationists and tourists and citizens. I must point out that we have the full attention of the Government of Montenegro and local government that together, as we did the Sutomore project, Pine in Tivat, Dobrota in Kotor, the Pet Danica promenade in Herceg Novi…. provide Budva with a new quality. It is a must for this city. Budva has to have this project if it thinks it does not lose the tourist match in Montenegro, let alone in the region. That is the mission and goal of this whole initiative," says Jelusic, recalling that the construction of the promenade will cost between 10 and 12 million euros and that this investment should not be a problem for Budva, the state, and the Sea Property.
Jelusic points out that it was crucial for the implementation of the project that the Detailed Urban Plan Budva Center and the Detailed Urban Plan Becici Center were not put out of force.
"In the local parliament, the reason and arguments put forward by DPS councilors prevailed, and two key projects of public interest in the Lungo Mare and the promenade were given a chance to materialize. Otherwise, we would not be in a position to realize these two projects that are crucial for Budva, for tourism, for urban planning of Budva, for its citizens," says the Sea Property director.
Commencement of work on the promenade is expected during autumn of next year. In the meantime, Budva will receive landscaped space at a location below the main post office building, where discos were during the season.
"After the tenants removed their objects, by agreement, we began the realization of a project of horticultural and park landscaping of the newly created area of over 3,000 m2. It is planned to connect the footpath from the 'upper' promenade to Slovenska Plaza, set up park benches, greenery, and mark the 'five friendly' park area. All these works will be completed by the beginning of the next tourist season. This solution should be in the function of the locals and tourists until the complete reconstruction of the promenade Budva-Becici, which will cover the whole area of the most attractive zone on the Budva Riviera," Jelusic points out.
Source: Dnevne Novine
Wind Farm Brajići Project in Budva Estimated at 100 Million EUR
28 October 2019 - A consortium of the German WPD AG company and Vjetroelektrana from Budva is the only bidder for the construction of Brajići wind farm, offering 100 million EUR.
As Cafe del Montenegro reported, five companies have purchased tender documentation in the public tender invited for the lease state-owned land. Only one bid was submitted within the established deadline.
The bid encompasses the construction of a wind warm with a capacity of 100,8 MW, a 100 million EUR investment. The lease of the land will be 2,50 EUR per square meter for a total of 220.770 square meters of land.
WPD AG, a renowned German company, is experienced in the implementation of similar projects in many world countries, with a 4.400 MW portfolio of installed capacity.
The bid also envisages the participation of national companies for 21,6 million EUR, which will further enhance the multiple effects of the project.
Research and measurements of wind potential in Brajići were carried on the grounds of licenses issued by the competent Ministry, between May 2013 and May 2015 and between November 2016 and November 2017.
German WPD expressed its interest in the construction of the Brajići wind farm at the beginning of this year. Andreas Scholl, WPD Director for Europe, met in February with Montenegro's Prime Minister Duško Marković and said that they wanted such investment in Montenegro.
Similarly, the wind farm Gvozd project was recently estimated at 58 million EUR, rated at 55 megawatts and, according to the estimates, it will have 13 turbines. Shareholders in the state energy company supported the decision on the construction of the Gvozd wind farm. Executive manager of the Directorate for Development and Engineering in the Electric Power Company of Montenegro Ivan Mrvaljević said that development of the Study on the potential of wind and energy production assessment, and the Study on the market and business model of Gvozd wind farm had been carried out before the investment decision.
Read more news about business in Montenegro at TMN's dedicated page.
UK-Based Jet2 to Launch Flights to Tivat in May 2020
11 October 2019 - On October 3rd, Jet2.com and Jet2holidays announced new direct services to Tivat in Montenegro, with flights and holidays going on sale from London Stansted and Manchester Airports.
The announcement marks the first time that Jet2.com has operated flights direct to Montenegro, a destination that features everything from golden beaches and cool bars to whitewashed villages and rugged national parks. Currently accessible from Dubrovnik, the launch of this brand new route now gives holidaymakers direct access to Montenegro with Jet2.com and Jet2holidays.
The airline will operate up to two weekly services (Thursday and Sunday) to Tivat from each base, with services running from May through to October.
Located on the west coast of the Balkan Peninsula, Tivat opens up an array of dazzling options for holidaymakers, including resorts such as stylish Budva, chilled-out Bečići and the stunning Herceg Novi. The classy coastal town of Tivat itself is nearby, as is the jaw-dropping Kotor Bay.
This is Jet2.com's third brand new route for Summer 20, in addition to Preveza (Greece) and Zadar (Croatia).
Steve Heapy, CEO of Jet2.com and Jet2holidays said: "The introduction of services to Tivat means that we are operating direct flights to Montenegro for the first time. With stunning scenery, a wonderful climate, and fantastic resorts such as Budva and Bečići on offer, we are expecting this destination to be hugely popular with holidaymakers. Yet again, Jet2.com and Jet2holidays are giving holidaymakers and independent travel agents even more choice and flexibility when it comes to booking our award-winning flights and holidays."
Low-cost carrier Jet2 will enter the Montenegrin market in 2020 with the launch of two new routes to Tivat. The airline will inaugurate two weekly flights from Manchester and London Stansted starting May 21. Both will be operated with a 189-seat Boeing 737-800 aircraft. The announcement marks the first time the leisure airline and package holiday specialist has operated flights to Montenegro.
"The Balkan beauty of Montenegro sits beside neighbours Croatia and Bosnia and Herzegovina. Majestic mountains stand proud against brilliant beaches, with terracotta-topped towns in between. You'll find laid-back rural areas alongside upmarket marinas too. Sound up your street? Let us show you more…," says the open invitation of the Jet2 company to all interested travellers to visit Montenegro and its beauty.
Tickets are already available for purchase on the airline's website.
Get more information on when and how to visit Montenegro at TMN's dedicated page.
Budva City Theatre Hosts TOURNEE Festival
22 September 2019 - The Budva City Theatre will host the TOURNEE Festival from 22 to 25 September this year in Budva. During these four festival nights, Budva will host partners from Italy and Albania with their programs.
The Public Institution City Theatre, together with the Municipality of Budva, is a partner in the project "Theatres for tOURism developmeNt in Europe, TOURNEE", funded through the EU trilateral cooperation program CBC Italy-Albania- Montenegro. This program aims to enhance the attractiveness of the cross-border area for all visitors, through the creation of new community festivals, the City Theatre representatives announced.
The overall objective of TOURNEE is to establish closer cooperation among the key actors in the theatrical sector, by creating a durable Cluster of Theatre Institutions in the cross border area. TOURNEE involves cross border experts in theatre productions, territorial marketing, cultural heritage management and tourism strategic policies, in creating innovative cultural products to strengthen the exploitation of tourist destinations. As a result, an increasing role of theatres in supporting tourist economic growth is expected, as well as an improvement of public policies in cultural and tourism sectors.
As reported by RTV Budva, on Sunday, September 22, the TOURNEE Festival Opening Ceremony will take place between the churches starting at 7.30 pm.
After the opening ceremony, at 8 pm, the audience will have the opportunity to enjoy the opera "The Servant turned Mistress" (La Serva Padrona) by Giovanni Pergolesi, directed by Vladan Đurković, as part of the TOURNEE project. This opera is a co-production of the City of Budva Theatre, Tirana Opera and the Pula Public Theatre, under the direction of the Operosa Foundation in Podgorica.
On the second night of the festival, on September 23, there will be a performance by partners from Tirana "Johan Padan and the discovery of the Americas".
The third night is scheduled for the performance of "Farce of the Crown" produced by partners from Fier, Albania.
On Wednesday, September 25, two programs will be presented: the musical "Island" and the theatre performance "Oresteia. Dystopian".
According to the organisers, the tickets will not be charged during the festival, as the project is co-financed by the European Union.
Property of the Week: New First Row Apartments near Budva
September 18, 2019 - A rare chance to buy first row to the sea close to Budva in the latest RE/MAX Property of the Week.
They say that the three most important things in real estate are location, location, location.
There is perhaps a twist on that maxim when it comes to the Adriatic coast - first row to the sea, first row to the sea, first row to the sea.
In addition to having the security of knowing that nobody is going to interrupt that perfect view, nothing quite beats the mesmerising connection you can get with just you and the sea.
And the good news is that there are some new (and increasingly rare) opportunities to get your own slice of first-row real estate heaven in our latest Property of the Week from the RE/MAX catalogue;
For sale two bedroom apartment of 88 m2 in Rafailovici, on the first line to the sea. Rafailovici is a little charming place, located on the beautiful Becici beach, from Budva less than 10 minutes drive. The building is currently under construction, and the completion of works is in May 2020. In addition to the exclusive location, this complex will feature an unusual design as well as modern facilities, which future owners of apartments can enjoy, such as pools, panoramic elevators, reception, video surveillance, etc. This is an opportunity for all hedonists who want to enjoy the peace of a small place on the coast with a perfect sea view.
Sales price - 340,000 euro. For more information, and to book a viewing, visit the original RE/MAX listing.
Total of 204 High-end Hotels in Montenegro Currently Available
16 September 2019 - According to records published by the Ministry of Sustainable Development and Tourism, there are 453 hotels in Montenegro with 43.639 beds. Nearly 55% account for four-star and five-star hotels. This year, 38 high-end hotels have been opened, and there will be even more.
As representatives of the Ministry said, categorization procedure helps achieve one of Government's goals – creating conditions for the improvement and raising the quality level of catering capacities.
One of the key measures defined by the Reforms in the Area of Tourism Agenda is the development of new and improvement of the existing accommodation facilities. The most important activity in that context is defining models for the enhancement of the construction of four and five-star hotels, reports CdM.
"There are currently 25 five-star hotels in Montenegro. One hundred seventy-nine four-star hotels, whereas others refer to one to three stars," said the representatives of the Ministry.
Analysis of the current state of affairs of hotel structure takes into consideration the fact that the Strategy adopted at the end of 2008, when the many-year-long economic crisis started, has severely affected the investment projects in Montenegro.
"However, despite such ambiance, we are very satisfied with the structure of our hotels now, as the number of high-end hotels exceeds the projections envisaged by the Strategy," said the representatives of the Ministry.
Raising the offer quality in hotels and other catering facilities achieves the strategic goal of the development of tourism in Montenegro- high quality, all-year-long tourist offer.
"In that context, Government is focused on creating conditions for enhancing the development of catering services, as the most important segment of the tourism product," said the representatives of the Ministry.
With the aim of raising competitiveness level and creating conditions for the extension of the season, Ministry has been implementing the project of the establishment, improvement and control of the quality level of hotel facilities as well as other accommodation facilities.
Commission of the Ministry of Tourism is in charge of the categorization of catering facilities.
"Commission for categorization of catering facilities conducts the categorization procedure and assigns the category depending on the organization, equipment and maintenance of the facility, type and quality of services and other categorization criteria," explained the representatives of the Ministry.
Read more news about traveling through Montenegro at TMN's dedicated page.
Beach Fest in Budva Tonight: Fešta na Plaži 2019
15 September 2019 - The summer parties on the Budva Riviera are not over, confirm the organisers of the Beach Fest (Montenegrin: Fešta na plaži), which will be held on Sunday, September 15, in Rafailovići, organized by the Tourism Organization of the Municipality of Budva and the Local Community of Bečići.
Budva's Beach Fest was held for the first time two years ago at the initiative of the locals that inhabit Rafailovići. Each year, the festival offers a rich entertainment program to citizens and tourists, and it improves their stay on the Budva Riviera at the end of the summer months. Just as the organisers aimed at the very beginning when they first put great efforts into organising the festival, "Fešta na plaži" has now become a tradition.
This year's program of Budva's Beach Fest begins at 7 pm with the performance of the ensemble Toć, a local band from Tivat and Kotor. From 8.30 pm the audience will be entertained by the acoustic band Trio Gušt, and after them, at 10 pm, the guests will be able to enjoy the performance of the regional star Željko Vasić.
For this occasion, local caterers will prepare delicacies for all visitors, and a particularly interesting part of the program will be the sailing of decorated local boats.
The charm of this small fishing village and the quality program of the event make the Beach Fest perfect end of summer event, so the organisers invite all locals and tourists to join them tonight, on Sunday, September 15th in Rafailovići and have a great time.
Eleanor Lam Falls in Love With Montenegro, Budva Top Destination for Luxurious End Of Summer Trip
01 September 2019 - Hong Kong Tatler, the essential guide to the luxury lifestyle in Hong Kong, recently placed Budva, Montenegro on the top of the list of destinations for a luxurious last-minute trip!
Hong Kong Tatler Magazine that covers everything from the best events and most glamorous people, to fine dining, international fashion and luxury travel wanted to inspire its readers to take advantage of the remaining sunny summer days and organise a last minute trip to a luxurious destination. In fact, Tatler based the destination research on famous people's holiday getaways, and the very first recommendation is Aman Sveti Stefan Resort in Budva!
While exploring Montenegro, one of Hong Kong society's brightest young stars and a social media phenomenon, Eleanor Lam (better known as Elly), fell in love with Aman Sveti Stefan, the charming seaside hideaway resort with lush grounds and stunning sea views.
As Tatle reported, Eleanor Lam is always on the move, and a highlight from her myriad summer jaunts was Budva, a medieval town in Montenegro. She stayed at the Aman Sveti Stefan, an exclusive resort atop a private island that used to be a fishing village in the 15th century.
Pink-sand beaches, crystal-clear waters and rugged, pine-cloaked mountains form the untamed backdrop to Aman Sveti Stefan, in one of Europe's last corners of truly wild beauty – Montenegro, managed to charm the young star. Eleanor loves exploring exotic destinations, and the stroll along the shoreline, Queen's Beach, the scallop-shaped bay considered Montenegro's finest, left her breathless as she managed to take some gorgeous photograph on site.
Lam also couldn't resist revealing more beautiful locations in Montenegro, as she visited a peaceful, authentic place with delicious Montenegrin cuisine, Rijeka Crnojevića. Swimming in a pool with a gorgeous view over the Skadar Lake, Ellie posted a picture on social media, saying that she could get used to the picturesque sites Montenegro has to offer.

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BearWorks
Home > Graduate Theses > 642
MSU Graduate Theses
Thesis Title
The Biosystematics of the Mosquitoes of Greene County, Missouri
Sammie Lee Dickson
Master of Science in Biology
Committee Chair
Steven Jensen
Mosquito larvae, pupae, and adults were collected from April through October 1977, in order to determine the species composition of the fauna of Greene County, Missouri. Twenty species of six genera were collected from a total of 146 localities. These species were: Anopheles punctipennis, A. quadrimaculatus, Uranotaenia sapphirina, Culiseta inornata, Psorophora ciliata, P. confinnis, P. cyanescens, P. discolor, P. howardii, Aedes atropalpus, A. canadensis canadensis, A. thibaulti, A. trivittatus, A. vexans, Culex erraticus, C. pipiens-complex, C. restuans, C. salinarius, C. tarsalis, and C. territans. Habitats most frequently collected were drainage ditches, temporary rain pools, streams, artificial containers, small permanent ponds, and pools in stream beds. The species most often collected were Anopheles punctipennis, Aedes vexans, Culex restuans, and C. salinarius. Psrophora confinnis, Culex pipiens-complex, C. tarsalis, and C. territans were collected in several localities, but were not usually found in large numbers. The remaining species found in this study were collected only in limited numbers. Aedes canadensis canadensis larvae and A. thibaulti were collected for the first time in southwest Missouri.
© Sammie Lee Dickson
Dickson, Sammie Lee, "The Biosystematics of the Mosquitoes of Greene County, Missouri" (1978). MSU Graduate Theses. 642.
https://bearworks.missouristate.edu/theses/642
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Dissertation/Thesis
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© Board of Governors, Missouri State University • Contact Information


Top Corporate Tax Rate was reduced from 35% to 21%. This change is probably the most impactful change of the new legislation. In summary, if you are in the top tax bracket and own a C corporation your profits inside the corporation are taxed at 21%. If you want to distribute profits from your company, you will issue yourself a dividend which would be taxed at the dividend rate. Adding the top dividend rate of 23.8% and corporate tax rate of 21% your total top tax rate would be 44.8%.
20% Deduction for pass-through businesses*. The provisions surrounding how to qualify for the 20% deduction on pass-through entities (S-corporations, LLCs, sole proprietors, and partnerships) are quite complex. To provide an overly simplistic summary: if you own a pass-through entity and your taxable income is less than $157,500 (single) or $315,000 (married, filing jointly) you will be able to take a 20% deduction on your pass-through income. If you are above these limits, there are restrictions depending on the type of company you operate, W-2 wages, and unadjusted basis of qualified property. Translation: talk to your CPA.
Inflation Rate Measure changed to "Chained CPI." The bill changes the IRS inflation index from the Consumer Price Index for urban consumers (CPI-U) to the chained CPI (C-CPI-U). Many economists consider the chained CPI to be a more accurate reflection of cost-of-living changes as it accounts for substitution. Substitution is the theory that holds if the costs of a good increase, then the market will substitute with a cheaper good or reduce the amount used. However; making the adjustment for substitution will lower the growth rate of the index, resulting in smaller adjustments to tax brackets, retirement account contribution limits, and other annual adjustments. This lower growth rate will result in higher taxes over time.
Reductions in individual tax rates*. The bill maintains the current structure of seven investor tax brackets, but changes the threshold amounts and lowers the marginal rate of five brackets. This reduction includes the sunset provision, meaning it's a temporary arrangement from 2018 to 2025.
Standard Deductions increased, and phase-outs eliminated*. For single filers, the standard deduction rises from $6,350 to $12,000; for married couples filing jointly, the standard deduction increases from $12,700 to $24,000. The previous law also limited itemized deductions based on income, which was repealed through 2025. The additional standard deduction for ages (65+) or blind was also maintained. This means single filers meeting this provision will receive an additional $1,600 deduction and each qualified married filer will receive an additional $1,300 deduction.
Personal exemptions eliminated*. Previous law allowed an exemption of $4,050 per person, which was eventually phased out for those with higher adjusted gross income. The new law repeals personal exemptions through 2025.
Child Tax Credit Increased to $2,000*. The child tax credit was expanded from a $1,000 credit to a $2,000 credit for each child under the age of 17 claimed as a dependent. The income phase-out was also increased from $110,000 to $400,000 for married couples filing jointly. $1,400 of the child tax credit is also refundable meaning your tax liability could be less than $0. The expanded child tax credit helps families who would have paid more under the new tax system due to exemptions being eliminated.
State and Local Tax ("SALT") Deduction capped at $10,000*. Under the bill, you may only deduct up to $10,000 in state and local taxes, including sales, income, and property taxes. Previously, this deduction was not subject to limitation.
Mortgage interest deduction limited to $750,000*. The bill keeps the $1 million deductible limit for acquisition debt for current homeowners, but caps the interest deduction at $750,000 in mortgage debt for homes bought after Dec. 31, 2017. Previously $50,000 (single) or $100,000 (married) of home-equity loan interest could be deducted for non-acquisition debt. The bill eliminates this provision. Eliminating the home equity loan provision has created some confusion. Home equity loans used for acquisition debt (purchasing or improving your home) still qualifies for the interest deduction. Home equity loans or cash out refinances for non-acquisition debt will not qualify for the deduction.
Medical Expense Deduction Expanded. Previous law only allowed for deduction of medical expenses above 10% of adjusted gross income (AGI). Taxpayers now (this element of the bill was made retroactive for 2017) only need medical expenses to exceed 7.5% of AGI to qualify for the deduction. However; this provision ends at the end of 2018.
Miscellaneous Itemized Deduction subject to 2% AGI floor eliminated*. All miscellaneous itemized deductions under the 2% AGI floor rule were repealed which includes: various unreimbursed employee expenses, losses on a variable annuity, deduction for advisory fees, and tax preparation costs among others.
Individual Alternative Minimum Tax (AMT) exemption increased*. Both the AMT exemption amounts and phase out limits increased which will dramatically reduce the impact of the AMT tax. AMT exemptions increased from $54,300 to $70,300 for single filers and from $84,500 to $109,400 for married filing jointly. The phase-outs increased from $120,700 to $500,000 (single) and $160,900 to $1m (MFJ). With these changes, only people with extremely large AMT adjustment items will be subject to AMT tax.
Estate Tax Exemption Increased*. The bill maintains the 40% estate tax rate, but increases the exemption from 5.49m to 11.2m. Very high net worth families may want to examine opportunities to move money out of their estate utilizing the higher exemption levels.
Roth recharacterizations eliminated*. Under previous law, if you convert a traditional Individual Retirement Account (IRA) to a Roth IRA, you had the ability to "undo" the conversion and recharacterization back to a traditional IRA. The bill removes this ability for ROTH conversions taking place after Dec. 31st, 2017.
529 plans expanded to include private elementary and secondary school expenses*. Previously 529 plan assets could only be withdrawn tax and penalty free for qualifying college expenses. The new provisions allow $10,000 annually of 529 assets to be used for elementary and secondary school. If you live in a state that has an income tax and provides a state income tax deduction for contributing to a 529 plan, there may be a benefit to using the 529 plan for short term contributions. Otherwise, no federal income tax deduction is provided for contributing to a 529 plan so the benefits are limited to the tax-free growth of the account.
Charitable Contribution Limits Expanded*. Previously cash donations to charities were limited to 50% of your Adjusted Gross Income (AGI). The bill now allows you to include cash gifts to charity of up to 60% of your AGI as a deduction in that calendar year. In general, fewer people will itemizing their deductions so they may not receive a benefit for their charitable contributions. Qualified Charitable Distributions (QCDs) may be used more often as well as bunching gifts into one tax year instead of spreading out to go over the higher itemized deductions.
Taxation of Alimony Reversed*. Previously alimony payments were deductible for the individual paying the alimony and taxable to the individual receiving. Under the new law, alimony payments will no longer be deductible by payors or reportable by recipients. This provision will only apply to divorce agreements filed after December 31st, 2018.
1031 Exchanges limited to real estate transactions*. Previously other types of investment property like airplanes, boats, and classic cars were allowed.
Moving Expense Deduction Repealed*. This means you can no longer deduct moving expenses, and moving expenses reimbursed by your employer will be taxable. An exemption for active military was maintained.
First In First Out (FIFO) in the selling of securities. The senate bill FIFO provision that would have required investors to sell the shares they've held the longest first was eliminated. This is good for investors as it would have impacted your ability to help you realize losses, gains, or gift low basis stock as it fit with your financial plan.
No change to capital gains taxes.
No change to tax-free municipal bonds.
Individual Mandate for Health Insurance repealed beginning 2019, but Affordable Care Act was not repealed*. This means in 2019 there will no longer be penalties for not having an approved health insurance or exemption. However; since the ACA was maintained, so was the 3.8% Net Investment Income Tax (NIIT) and 0.9% Medicare surtax on income above $200k MAGI (single) and $250k MAGI (MFJ).


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Letter of Elise Johnson, R #1, Box 179B, Lobelville TN 37097.
.4 WILLIAM2, 1757; m Mary ---.
.... JAMES2, 1760; was in Anson Co in1820.
.... ELIZABETH3, m Richard Sasser.
.... ANNA3 (?same as Dane M); m Thomas Hendy.
.9a ZIBA DANIEL3, ; m Permelia ---.
.10 ZIBARY3, 1784; m Sarah Dobbs.
.11 ALLEN3, 1795; m Martha Kilpatrick.
6. ABRAHAM3 BARBER, 178?; m Sarah ?Barber; res NC, TN.
.14 JOHN E4, 1809; m/1 Sarah ---; m/2 Sarah Land.
.15 CALVIN4, 1817; m Nancy Dowdy.
.... SUSAN A, 1824; m Simon Campbell.
.... LYDIA ANN4, 1834; m Willis Hertley Mills.
.21 ABRAHAM CALVIN4, b NC 1812; m Bethany Davenport.
.... JAMES C4, 1822; m Catherine ---.
.... NANCY4, (Nicey), 1825; m --- Austin.
.21a ANDREW JACKSON4, c 1853; m/1 Sinthy Ann Grady; m/2 Rebecca Jane Sellers.
.... SARAH L E4, 1835; m William Baucom; res Perry Co; d Johnson Co AR.
.... MARTHA PATRICIA4, 1837; res Perry Co; d in AR.
.... ISABELL JANE4, 1841; m Dennis Baucom.
.... JAMES C4, 1843; m Mary ---.
.... NANCY CAROLINE4, 1846; m Frederick J Clayborn; res. Perry Co; d Johnson Co AR.
.22 ANDREW D J4, 1820; m Tabitha Deliah Smith.
.... MARY4, 1822; m/1 Jordon Dowdy; m/2 Alvin or Allen Warren; m/3 Elijah Warren.
.... GATHASA4 (Gatsey), 1824; m Elijah Warren, Sr.
.23 JOHN4, 1826; m Rhoda S A Beckham.
.... NANCY ANN4, 1833; m Ruffin Sanders Stephens; res. Perry Co.
.... JOSEPH4, 1839; d young.
.24 DANIEL4, 1844; m/1 Mary Ann Kilpatrick; m/2 Martha ---; m/3 Sarah Kilpatrick; m/4 1897 Nora Neumigah Duncan.
.... FRANCES B4, 1825; m Roberson Warren.
.... NANCY B4, 1830; m Elkanah A Land.
.... WILLIAM THOMAS4, 1835; m Ellender P McCullister.
.... WILLIAM W, 1836; m Rachael ---.
.... JOHN CALVIN5, 1835; m Rebecca Stewart/Bennet.
.... SARAH ELIZABETH5, 1836; m/1 James F Skinner; m/2 James Hightower.
.... JAMES MERRILL5, 1838; m Mary Morgan.
.... NANCY CAROLINE5, 1839; m James Bridges.
.... ELIJAH WARREN5, 1840; m Carrie Smith.
.... MARY ETTA5, 1842; m William Riley Kennedy II.
.... LUCINDA CATHERINE5, 1844; m William Parsons Bethell.
.... MELITA5, 1847; m Sidney C Harper.
.... PHILLIP P5, 1848; m Alice Harper.
.... THOMAS JEFFERSON5, 1849; m Sallie Kennedy.
.... WILLIAM WORTH5, 1851; m Mary Dean.
.... OPHELIA B5, 1853; m Leonidas Thurman.
.... GEORGE WALTON5, 1854; d 1883.
.... HENRY ARTHUR5, 1857; d inf.
.... KATURA B5, 1858; m George Alexander Dean.
21A. ANDREW JACKSON4 BARBER, c 1853; m/1 Sinthy Ann Grady; m/2 Rebecca Jane Sellers.
.29a GEORGE LEWIS, 1873; m Sarah Adeline Strother.
.... PARALEE, ; m William Hardy Coker.
.... LUCINDA JANE, 1880; m David Perry Strother.
.29b YANCY JEFFERSON, 1885; m/1 Rebecca Jane Sellers.
.... AMANDA5, 1844; m/1 --- Nuckolls; m/2 William Washington Nuckolls.
.... JAMES5, 1852; m Mary ---.
.30 ALLEN5, 1847; m Amanda P Nuckolls.
.... DAVID WILSON5, 1849; m Elizabeth Nuckolls.
.... SARAH JANE5, 1856; m George Thomas Rogers.
.... ROBERT5, 1858; m Martha Warren.
.... MARTHA5, 1860; m Allen Trull.
.... JOSEPH5, 1865; m Maria Ann Fuller.
.... MARY5, 1868; m Pete Beasley.
.... FANNIE5, 1870; m Rufford Gibbons.
.33 ANDERSON LEVINE5, 1873; m Sarah Elizabeth Morgan.