GHZ-based Quantum Anonymous Transmission

The GHZ-based quantum anonymous transmission protocol implements the task of Anonymous Transmission in a $n$-node quantum network. The protocol uses $n$-partite GHZ state to enable two nodes, sender $S$ and receiver $R$, to establish a link which they use to transmit a quantum message. Importantly, the quantum message is transmitted in a way that the identity of $S$ is unknown to every other node, and the identity of $R$ is known only 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 S} .

Tags: Quantum Enhanced Classical Functionality, Multi Party Protocols, Specific Task, GHZ state, anonymous transmission

Assumptions

• Pairwise authenticated private classical channels.
• Broadcast channel.
• Trusted multipartite source.
• Adversarial model: active adversary who does not control the source.

Outline

The presented GHZ-based quantum anounymous transmission protocol is based on the work of [6] . The goal of the protocol is to transmit a quantum state $|\psi \rangle$ from the sender $S$ to the receiver $R$, while keeping the identities of $S$ and $R$ anonymous. We assume that there is exactly one receiver which is determined before the start of the protocol. The protocol consists of the following steps.

• Collision detection: Nodes run a collision detection protocol to determine a single sender 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 S} .
• State distribution: A trusted source distributes the $n$-partite GHZ state.
• Anonymous entanglement: $n-2$ nodes (all except for $S$ 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 R} ) measure in the $X$ basis and broadcast their measurement 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 S} and $R$ broadcast random dummy bits. The parity of measurement outcomes allows to establish an entangled link between $S$ 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 R} which is called anonymous entanglement (AE).
• Teleportation: Sender 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 S} teleports the message state $|\psi \rangle$ to the receiver $R$ using the established anonymous entanglement. Classical message 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 m} associated with teleportation is also sent anonymously.

Notation

• $n$ number of network nodes taking part in the anonymous transmission.
• 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 |\psi\rangle} quantum message which the sender wants to send anonymously
• $S$ the sender of the quantum message
• $R$ the receiver of the quantum message

Hardware Requirements

• Network stage: (Fault-tolerant) Quantum computing network stage
• Relevant parameters to establish one anonymous link: 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=1} round of quantum communication per node, circuit depth 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 m=1} , $q=1$ physical qubits per node.
• Quantum memories, single-qubit Pauli gates and single-qubit measurements at the end nodes.

Properties

See Quantum Anonymous Transmission for the precise security definition. Pseudocode given below implements secure anonymous transmission, i.e. it hides the identities of the sender and the receiver from other nodes in the network. That is, the maximum probability that adversaries guess the identity of $S$ or $R$ given all the classical and quantum information they have available at the end of the protocol is no larger than the uncertainty the adversaries have about the identities of $S$ and $R$ before the protocol begins. More formally, the anonymous transmission protocol with the GHZ state, $P_{\text{guess}}[S|C,S\notin {\mathcal {A}}]\leq \max _{i\in [n]}P[S=i|S\notin {\mathcal {A}}]={\frac {1}{n-t}},$
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_{\text{guess}}[R|C,S\notin \mathcal{A}] \leq \max_{i\in[n]} P[R=i|S\notin \mathcal{A}] = \frac{1}{n-t},}
where ${\mathcal {A}}$ is the subset of $t$ adversaries among $n$ nodes and $C$ is the register that contains all classical and quantum side information accessible to the adversaries. Note that this implies that the protocol is also traceless, since even if the adversary hijacks any $t\leq n-2$ players and gains access to all of their classical and quantum information after the end of the protocol, she cannot learn the identities of $S$ and $R$. For a formal argument see [6] .

Pseudocode

Receiver $R$ is determined before the start of the protocol. $S$ holds a message qubit $|\psi \rangle$.

1. Nodes run a collision detection protocol and determine a single sender $S$.
2. A trusted source distributes $n$-partite GHZ state to every player, $|GHZ\rangle ={\frac {1}{\sqrt {2}}}(|0^{n}\rangle +|1^{n}\rangle )$.

• Anonymous entanglement:

1. 1. Sender $S$ and receiver $R$ do not do anything to their part of the state.
   2. Every player $j\in [n]\setminus \{S,R\}$:
      1. Applies a Hadamard transform to her qubit,
      2. Measures this qubit in the computational basis with outcome $m_{j}$,
      3. Broadcasts $m_{j}$.
   3. $S$ picks a random bit $b\in _{R}\{0,1\}$ and broadcasts $b$.
   4. $S$ applies a phase flip $Z$ to her qubit if $b=1$.
   5. $R$ picks a random bit $b'\in _{R}\{0,1\}$ and broadcasts $b'$.
   6. $R$ applies a phase flip $Z$ to her qubit, if $b\oplus \bigoplus _{j\in [n]\setminus \{S,R\}}m_{j}=1$.
      $S$ and $R$ share anonymous entanglement $|\Gamma \rangle _{SR}={\frac {1}{\sqrt {2}}}(|00\rangle +|11\rangle )$.
2. $S$ uses the quantum teleportation circuit with input $|\psi \rangle$ and anonymous entanglement $|\Gamma \rangle _{SR}$, and obtains measurement outcomes $m_{0},m_{1}$.
3. The players run a protocol to anonymously send bits $m_{0},m_{1}$ from $S$ to $R$ (see Discussion for details).
4. $R$ applies the transformation described by $m_{0},m_{1}$ on his part of $|\Gamma \rangle _{SR}$ and obtains $|\psi \rangle _{SR}$.

Further Information

• To determine the sender $S$ (Step 1) one can run either a classical collision detection protocol of [4] or a quantum collision detection protocol of [6] . The quantum version of the protocol requires additional $(\left\lceil \log n\right\rceil +1)$ GHZ states.
• To determine the receiver $R$ during the protocol one can incorporate an additional step using a classical receiver notification protocol of [4] .
• To send classical teleportation bits $m_{0},m_{1}$ (Step 5) the players can run a classical logical OR protocol of [4] or anonymous transmission protocol for classical bits with quantum resources of [6] . The quantum protocol requires one additional GHZ state for transmitting one classical bit.
• The anonymous transmission of quantum states was introduced in [6] .
• The problem was subsequently developed to consider the preparation and certification of the GHZ state [3], [5] .
• In [5] , it was first shown that the proposed protocol is information-theoretically secure against an active adversary.
• In [1] a protocol using another multipartite state, the W state, was introduced. The reference discusses noise robustness of both GHZ-based and W-based protocols and compares the performance of both protocols.
• Other protocols were proposed, which do not make use of multipartite entanglement, but utilize solely Bell pairs to create anonymous entanglement [2] .

References

1. Lipinska et al (2018)
2. Yang et al (2016)
3. Bouda et al (2007)
4. Broadbent et al (2007)
5. Brassard et al (2007)
6. Christandl et al (2005)

contributed by Victoria Lipinska