Device-Independent Quantum Key Distribution

A device-independent quantum key distribution protocol implements the task of Quantum Key Distribution (QKD) without relying on any particular description of the underlying system. The protocol enables two parties, Alice and Bob, to establish a classical secret key by distributing an entangled quantum state and checking for the violation of a Bell inequality in order to certify the security. The output of the protocol is a classical secret key which is completely unknown to any third party, namely an eavesdropper.

Tags: Two Party, Quantum Enhanced Classical Functionality, Specific Task,Quantum Key Distribution, BB84 QKD,

Assumptions

• We assume the existence of an authenticated public classical channel between the two parties
• We assume synchronous network between parties
• We assume security from coherent attacks

Outline

A DIQKD protocol is composed by the following steps:

• Distribution: For each round of the distribution phase:
  • Alice uses the source to prepare a maximally entangled state and send half of the state to Bob.
  • Upon receiving the state, Bob announces that he received it, and they both use their respective devices to measure the quantum systems. They record their output in a string of bits.
• A second phase where Alice and Bob publicly exchange classical information in order to perform error correction, where they correct their strings generating the raw keys, and parameter estimation, where they estimate the parameters of interest. At the end of this phase Alice and Bob are supposed to share the same $n$-bit string and have an estimate of how much knowledge an eavesdropper might have about their raw key.
• In the final phase, Alice and Bob perform privacy amplification, where the not fully secure Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} -bit strings are mapped into smaller strings $K_{A}$ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K_B} , which represents the final keys of Alice and Bob respectively.

Hardware Requirements

• Network Stage: Entanglement Distribution
• Relevant Network Parameters: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon_T, \epsilon_M} (see Entanglement Distribution)
• Distribution of Bell pairs, and measurement in three different bases (two basis on Alice's side and three basis on Bob's side).
• Minimum number of rounds ranging from ${\mathcal {O}}(10^{6})$ to ${\mathcal {O}}(10^{12})$ depending on the network parameters, for commonly used secure parameters.
• $QBER\leq 0.071$, taking a depolarizing model as benchmark. Parameters satisfying $\epsilon _{T}+\epsilon _{M}\leq 0.071$ are sufficient.
• Authenticated classical channel.
• Random number generator.

Notations Used

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} expected number of rounds
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l} final key length
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma} fraction of test rounds
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q} quantum bit error rate
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta} CHSH violation
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_{exp}} expected winning probability on the CHSH game in an honest implementation
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta_{est}} width of the statistical interval for the Bell test
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta_{con}} confidence interval for the Bell test
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon_s} smoothing parameter
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon_{EC},\epsilon'_{EC}} error probabilities of the error correction protocol
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon_{EA}} error probability of Bell violation estimation.
• $\epsilon _{con}$ error probability of Bell violation estimation.
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon_{PA}} error probability of the privacy amplification protocol
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mbox{leak}_{EC}} leakage in the error correction protocol

Properties

Either Protocol (see Pseudo-code) abort with probability higher than Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1-(\epsilon_{EA}+\epsilon_{EC})} , or it generates a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (2\epsilon_{EC}+\epsilon_{PA}+\epsilon_s)} -correct-and-secret key of length
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l\geq& \frac{{n}}{\bar{s}}\eta_{opt} -\frac{{n}}{\bar{s}}h(\omega_{exp}-\delta_{est}) -\sqrt{\frac{{n}}{\bar{s}}}\nu_1 -\mbox{leak}_{EC} }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \quad -3\log\de{1-\sqrt{1-\de{\frac{\epsilon_s}{4(\epsilon_{EA} + \epsilon_{EC})}}^2}}+2\log\de{\frac{1}{2\epsilon_{PA}}}}
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mbox{leak}_{EC}} is the leakage due to error correction step and the functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{s}} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \eta_{opt}} , $\nu _{1}$ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu_2} are specified in Table below. The security parameters of the error correction protocol, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon_{EC}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon'_{EC}} , mean that if the error correction step in Protocol 1 does not abort, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K_A=K_B} with probability at least Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1-\epsilon_{EC}} , and for an honest implementation, the error correction protocol aborts with probability at most Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon'_{EC}+\epsilon_{EC}} .

Pseudo Code

• Input:Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n, \gamma, \epsilon_{\rm PA},\epsilon_{\rm PE},\epsilon_{\rm EC},\epsilon'_{\rm EC},Q_Z}
• Output:$K_{A},K_{B}$

Stage 1 Distribution and measurement

1. For i=1,2,...,n
   1. Sender chooses random bits Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_i\epsilon\{0,1\}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_i\epsilon_R\{0,1\}} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(X_i=1)=\gamma}
   2. Sender prepares Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H^{X_i}|A_i\rangle} and sends it to Bob
   3. Receiver announces receiving a state
   4. Receiver chooses bit Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_i\in_R\{0,1\}} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(Y_i=1)=\gamma}
   5. Receiver measures $H^{X_{i}}|A_{i}\rangle$ in basis Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{H^{Y_i}|0\rangle, H^{Y_i}|1\rangle\}} with outcome Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_i}

• At this stage Sender holds strings Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_1^n, A_1^n} and Receiver Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_1^n, B_1^n} , all of length Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n}

Stage 2 Sifting

1. Alice and Bob publicly announce Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_1^n, Y_1^n}
2. For i=1,2,....,n
   1. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_i=Y_i}
      1. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_1^{n'} = A_1^{n'}.} append</math>(A_i)</math>
      2. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_1^{n'} = B_1^{n'}.} appendFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (B_i)}
      3. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_1^{n'} = X_1^{n'}.} appendFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X_i)}
      4. $Y_{1}^{n'}=Y_{1}^{n'}.$appendFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (Y_i)}

• Now Sender holds strings Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_1^{n'}, A_1^{n'}} and Receiver Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_1^{n'}, B_1^{n'}} , all of length Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n'\leq n}

Stage 3 Parameter estimation

1. For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=1,...,n}
   1. sizeFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q} = 0
   2. If{Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_i = Y_i = 1}
      1. Sender and Receiver publicly announce Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_i, B_i}
      2. Sender and Receiver compute Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_i = 1 - \delta_{A_iB_i}} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta_{A_iB_i}} is the Kronecker delta
   3. sizeFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q} += 1\;

• Both Sender and Receiver, each, compute Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_X = \frac{1}{\text{size}Q} \sum_{i=1}^{n'}Q_i}
  

Stage 4 Error correction

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C(\cdot,\cdot)} is an error correction subroutine determined by the previously estimated value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_Z} and with error parameters Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon'_{\rm EC}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon_{\rm EC}}

1. Both Sender and Receiver run Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C(A_1^{n'},B_1^{n'})} .
2. Receiver obtains Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde{B}_1^{n'}}

Stage 5 Privacy amplification

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle PA(\cdot,\cdot)} is a privacy amplification subroutine determined by the size Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ell} , computed from equation for key length Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ell} (see Properties), and with secrecy parameter Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon_{\rm PA}}

1. Sender and Receiver run $PA(A_1^{n'},\tilde{B}_1^{n'})$ and obtain secret keys $K_A, K_B$\;

Further Information

1. BB(1984) introduces the BB84 protocol, as the name says, by Charles Bennett and Gilles Brassard.
2. TL(2017) The derivation of the key length in Properties, combines the techniques developed in this article and minimum leakage error correcting codes.
3. GL03 gives an extended analysis of the BB84 in the finite regime.
4. Sifting: the BB84 protocol can also be described in a symmetric way. This means that the inputs Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} are chosen with the same probability. In that case only Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/2} of the generated bits are discarded during the sifting process. Indeed, in the symmetric protocol, Alice and Bob measure in the same basis in about half of the rounds.
5. LCA05 the asymmetric protocol was introduced to make this more efficient protocol presented in this article.
6. A post-processing of the key using 2-way classical communication, denoted Advantage distillation, can increase the QBER tolarance up to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 18.9\%} (3).
7. We remark that in Pseudo Code, the QBER in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z} basis is not estimated during the protocol. Instead Alice and Bob make use of a previous estimate for the value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_Z} and the error correction step, Step 4 in the pseudo-code, will make sure that this estimation is correct. Indeed, if the real QBER is higher than the estimated value Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_Z} , Pseudo Code will abort in the Step 4 with very high probability.
8. The BB84 can be equivalently implemented by distributing EPR pairs and Alice and Bob making measurements in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} basis, however this required a entanglement distribution network stage.