Communication, Cryptography, and Networks

Quantum communication studies what separated parties can do with quantum channels, shared entanglement and classical messages. Sending an unknown state, establishing a secret key and performing a remote computation are different tasks. This chapter connects their ideal protocols to the timing, noise and security assumptions that determine whether a network can deliver the intended result.

Helpful background. Density operators for QI supplies the state language, entanglement measures describes shared-state resources, and quantum channels for QI explains noisy transmission. Individual protocol pages introduce their additional requirements.

Enter this chapter

Begin with communication with quantum systems: identify the input, desired output and available resources. Then read no-cloning and no-signaling. An unknown state cannot generally be copied like an ordinary packet, and shared entanglement alone cannot transmit a chosen message. These restrictions explain why a quantum network needs different methods from a classical network that amplifies or copies signals.

Choose the route that matches the question you want to answer.

1. How are communication resources converted? Read teleportation and superdense coding, then entanglement swapping and entanglement distillation. Track which states are consumed, which messages must arrive, and whether success is guaranteed or announced by a classical signal called a herald.
2. What makes a communication protocol secure? Read the QKD overview and BB84, then compare the trust assumptions below. Security concerns the adversary’s information as well as agreement between the intended users.
3. How do links become a network? Read quantum repeaters, quantum memories and quantum transduction, followed by network architectures. Finish with distributed quantum computing and network verification: what must a delivered resource accomplish for its user?

Compare the information delivered and the resources consumed

Two ideal qubit protocols make the distinction especially clear:

Protocol	Information delivered	Resources consumed
Teleportation	One unknown qubit	One Bell pair; two forward classical bits
Superdense coding	Two classical bits	One Bell pair; one transmitted qubit

The Bell pair must already be shared. Its preparation and distribution have costs in addition to the transmission listed here. Ideal local operations and measurements are also assumed. Teleportation transfers a state without leaving the sender a second copy; the receiver needs the classical measurement outcome to interpret or correct the output (Bennett et al., 1993). Superdense coding uses the receiver’s half of the pair together with the transmitted qubit to distinguish four messages (Bennett and Wiesner, 1992). Neither protocol gives entanglement an independent signaling capacity.

The timing of classical information matters in other protocols too. In entanglement swapping, a middle measurement can establish a conditional state between distant endpoints. To know which state was established and use the appropriate correction, the endpoints need its outcome. A selected high-quality ensemble and the probability of obtaining that ensemble answer different questions.

A herald rate is not a key rate

For a simple link, suppose independent attempts each last $\tau$ and succeed with probability $p$. Include reset and required classical communication in $\tau$, and assume attempts do not overlap. The long-run accepted rate and the mean time to the first success are

$$R=\frac{p}{\tau},\qquad \mathbb E[T]=\frac{\tau}{p}.$$

With $p=0.02$ and $\tau=1\,\mathrm{ms}$, these are $20\,\mathrm{s}^{-1}$ and $50\,\mathrm{ms}$. The waiting time fluctuates; its mean is not a delivery deadline. A high conditional Bell fidelity does not increase this rate. Conversely, frequent heralds alone do not establish entanglement or secrecy. Memories, parallel attempts, correlations and scheduling require the fuller models developed in the network pages.

Quantum key distribution also needs an authenticated classical channel: users must detect replacement of their classical messages by an adversary. The messages need not all be secret, but QKD does not establish the users’ identities from nothing. Its security statement specifies the trusted devices, randomness, isolation and finite-data guarantees, as well as what information the adversary may hold.

Choose the security assumptions before interpreting the result

E91 and entanglement-based QKD connects shared states, correlations and key generation. An entangled source does not by itself make an implementation device independent. Decoy-state QKD uses varied source intensities to constrain photon-number contributions under a specified source model.

Measurement-device-independent QKD permits an untrusted measurement relay while retaining assumptions about the endpoint sources (Lo, Curty and Qi, 2012). Device-independent QKD instead uses Bell correlations to constrain the adversary without a detailed model of the measurement devices. It still needs appropriate private settings, laboratory isolation, treatment of detection events and a security proof for the available data; a Bell violation by itself is not a complete key-generation claim (Vazirani and Vidick, 2014).

Two related tasks have different outputs. Quantum randomness asks how much unpredictability remains against allowed side information and how it can be extracted into usable bits. Blind and delegated quantum computation distinguishes hiding a computation from verifying its answer. A private computation need not be correctly performed, just as an apparently random string need not be secret from its generator.

Connect a component to the task it must support

A memory must preserve the relevant state for the actual waiting time. A transducer must preserve coherence between different physical carriers despite loss and added noise. A repeater must coordinate elementary links and their classical outcomes. The staged network picture of Wehner, Elkouss and Hanson (2018) helps explain why demonstrating one of these capabilities does not establish every network application.

For implementation details, continue to hardware platforms. For protected operations, use error correction and fault tolerance. For parameter estimation using shared resources, see distributed sensing. Network case studies puts link quality, timing and application requirements together in concrete examples.

Check your understanding

1. Can teleportation send a message before the classical bits arrive?

Alice chooses to teleport either $|0\rangle$ or $|1\rangle$ using the ideal protocol. Can Bob distinguish her choice before receiving the Bell-measurement outcome?

Solution

No. The four equally likely outcomes leave Bob with four possible Pauli corrections. Without the outcome, his state is their average:

$$\begin{aligned} \rho_B&=\frac14\sum_{P\in\{I,X,Y,Z\}}P\rho P^\dagger\\ &=\frac I2. \end{aligned}$$

For either chosen input, two possibilities give $|0\rangle$ and two give $|1\rangle$, up to irrelevant phases. The same averaged state occurs for any input qubit. Bob’s local statistics therefore cannot reveal Alice’s choice. The arriving classical outcome tells him which correction to use.

2. Does a mean waiting time guarantee delivery?

For the link with $p=0.02$ and $\tau=1\,\mathrm{ms}$, what is the probability of no success in the first $100$ attempts?

Solution

Independence gives

$$\begin{aligned} \Pr(T>100\tau)&=(1-p)^{100}\\ &\approx0.133. \end{aligned}$$

About $13\%$ of runs still have no success after $100\,\mathrm{ms}$, twice the mean waiting time. A network application with a deadline needs a waiting-time distribution, not just the mean rate.

3. Do high-fidelity heralds establish a secure bit rate?

A link produces $10^4$ heralds per second with reported Bell fidelity $0.95$. Has it established a secure key rate of $10^4$ bits per second?

Solution

No. The herald rate counts accepted events, while the fidelity describes a specified conditional state ensemble. Key generation additionally needs an applicable security proof, measurements, sifting, parameter estimation, error correction, privacy amplification and authentication. These can reduce the key rate or make it zero under the chosen assumptions. The fidelity estimate must also account for which events were measured or excluded. The two quoted numbers alone do not determine a secure key rate.

References

• Bennett, Charles H., Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and William K. Wootters. “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels.” Physical Review Letters 70, 1895–1899 (1993). doi:10.1103/PhysRevLett.70.1895.
• Bennett, Charles H., and Stephen J. Wiesner. “Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states.” Physical Review Letters 69, 2881–2884 (1992). doi:10.1103/PhysRevLett.69.2881.
• Lo, Hoi-Kwong, Marcos Curty, and Bing Qi. “Measurement-device-independent quantum key distribution.” Physical Review Letters 108, 130503 (2012). arXiv:1109.1473; doi:10.1103/PhysRevLett.108.130503.
• Vazirani, Umesh, and Thomas Vidick. “Fully device-independent quantum key distribution.” Physical Review Letters 113, 140501 (2014). arXiv:1210.1810; doi:10.1103/PhysRevLett.113.140501.
• Wehner, Stephanie, David Elkouss, and Ronald Hanson. “Quantum internet: A vision for the road ahead.” Science 362, eaam9288 (2018). doi:10.1126/science.aam9288; author manuscript.