Communication with Quantum Systems

Quantum communication uses quantum systems and channels to transmit a classical message, preserve an unknown quantum state, establish entanglement, generate a secret key, or coordinate a distributed quantum task. These goals are not interchangeable. The same physical optical fiber can have a high rate for classical data, zero unassisted quantum capacity over a chosen operating regime, and a nonzero secret-key rate under a specific protocol and adversary model.

The central question is what must reach the receiver, with which help and at what error. A shared Bell pair, a reliable classical message and an unknown quantum state are different resources. We first define those tasks, then compare their capacities using a flagged erasure channel that can be solved exactly.

Required background. Bits, qubits, qudits and modes distinguishes carrier dimensions and coherence. Density operators for QI supplies mixed states and purification. Quantum channels introduces completely positive trace-preserving maps and their environments.

Classical messages and quantum states require different tests

Classical message transmission

For the capacity formulas below, assume a finite-dimensional memoryless channel: each use applies the same map $\mathcal N$ independently. Correlated channel noise instead requires a joint multi-use model. Logarithms and entropies use base two.

Let Alice receive a message $M$ drawn from a distribution $p(m)$. An encoder maps $m$ to an input state $\rho_m^{A^n}$ for $n$ uses of a channel $\mathcal N_{A\to B}$. Bob measures the output and reports $\widehat M$:

$$m \xrightarrow{\mathcal E} \rho_m^{A^n} \xrightarrow{\mathcal N^{\otimes n}} \sigma_m^{B^n} \xrightarrow{\mathcal D} \widehat m.$$

The average error probability is

$$P_e=\Pr(\widehat M\ne M)=1-\sum_m p(m)\operatorname{Tr}\!\left[D_m\sigma_m^{B^n}\right],$$

where $\{D_m\}$ is Bob’s decoding measurement. The encoder may use product or entangled codewords across channel uses, and Bob may need a collective measurement. A receiver restricted to measuring every signal separately is a different communication model.

Quantum-state transmission

To test whether a channel preserves unknown quantum information, include a reference system $R$ that never enters the channel. If $A_0$ is maximally entangled with $R$, successful encoding, transmission, and decoding should produce a state close to $|\Phi\rangle_{R\widehat A}$:

$$\begin{aligned} \mathcal L_n&=\mathcal D\circ\mathcal N^{\otimes n}\circ\mathcal E,\\ \rho_{R\widehat A}&=(\operatorname{id}_R\otimes\mathcal L_n)(\Phi_{RA_0}). \end{aligned}$$

The entanglement fidelity is

$$F_e = \langle\Phi|\rho_{R\widehat A}|\Phi\rangle.$$

This test detects loss of coherence that a test of a few orthogonal inputs would miss (Schumacher, 1996). Exactly $F_e=1$ for the maximally entangled state means the effective channel is the identity on the code space, including its correlations with any reference. For approximate transmission, however, a high $F_e$ is an average criterion: by itself it is not a dimension-independent worst-case bound. Exercise 7 gives a channel whose $F_e$ approaches one while one superposition is completely changed. State the intended error criterion when using a quantum-capacity theorem.

The reference $R$ never enters the encoder, channel or decoder. It remains available to test coherence with the final output $\widehat A$. The complementary output $E^n$ describes information lost from the receiver and may be accessible to an adversary. Classical-message transmission uses the same physical channel but scores a decoded label instead of reference entanglement.

The environment is part of the model

Every channel has a Stinespring isometry

$$U_{A\to BE}$$

whose receiver output is $B$ and complementary output is $E$. For ordinary reliability, $E$ explains lost information. For private communication, $E$ represents what an adversary may obtain. A channel can transmit a classical label reliably to Bob while leaking the same label to $E$; it then has useful classical communication but no privacy for that code.

Four communication tasks

Classical information in quantum states

Alice can encode a classical label $x$ in a quantum state $\rho_x$. Orthogonal states can be distinguished perfectly in one shot. Nonorthogonal states cannot, but long block codes and collective measurements can transmit classical information reliably below a channel-dependent rate.

The signal ensemble is the classical–quantum state

$$\rho_{XB} = \sum_x p_x |x\rangle\langle x|_X \otimes \rho_x^B.$$

Before measurement, Bob has a quantum system correlated with the classical label. After choosing a measurement, he obtains an ordinary random variable $Y$. The accessible classical information depends on the whole ensemble and the allowed measurement, not on a classical description of each ket.

Quantum information

Quantum transmission preserves superpositions and entanglement with references. It is not equivalent to estimating the state and sending a classical description. No finite classical message can specify an arbitrary unknown state exactly, and measurement generally disturbs the correlations that a quantum channel must preserve.

A noiseless qubit channel can also distribute one ebit: Alice creates a Bell pair locally and sends one half. The converse conversion needs classical communication as well as shared entanglement, as teleportation makes explicit.

Shared entanglement

Entanglement distribution aims to create a state close to a target shared state, often a Bell pair. Its outputs are correlations rather than a message chosen at transmission time. Shared entanglement can later support teleportation, dense coding, device-independent tests, clock synchronization, or distributed computation.

An entanglement-generation rate must state fidelity, success probability, heralding rule, memory time, and whether failed attempts consume the channel. “Pairs per second” without a quality threshold is not a complete service metric.

Secret classical information

Private communication asks Bob to learn a message while the environment learns asymptotically nothing about it. Key agreement is related but distinct: the parties generate a fresh random shared string rather than transmit a preexisting message. Interactive public discussion can change key-agreement rates even when it does not change a one-way message capacity.

Security statements require an adversary model, authentication, finite-size statistics, device assumptions, and a composable error criterion. A low bit error rate or a visible disturbance alone does not establish secrecy; quantum key distribution explains the additional requirements.

Ideal resource conversions

It is useful to keep ideal resources symbolically distinct:

Symbol	Resource
$[c\to c]$	one noiseless classical bit from Alice to Bob
$[q\to q]$	one noiseless qubit channel use
$[qq]$	one shared maximally entangled qubit pair, or ebit
$[cc]_{\mathrm{secret}}$	one shared secret classical bit

A resource inequality means that the resources on the left can simulate those on the right under the declared allowed operations.

Entanglement distribution

Sending half of a locally prepared Bell pair gives

$$[q\to q] \succeq [qq].$$

This use establishes entanglement; it does not also leave an independent qubit channel use available.

Teleportation

One ebit and two forward classical bits simulate one noiseless qubit channel:

$$[qq] +2[c\to c] \succeq [q\to q].$$

The ebit is consumed. The classical message is indispensable, so the protocol does not transmit usable information faster than light.

Superdense coding

Superdense Coding shows that one shared ebit and one transmitted qubit can carry two classical bits:

$$[qq] +[q\to q] \succeq 2[c\to c].$$

The gain is relative to a model with preshared entanglement. Preparing and distributing that entanglement has a cost unless it is explicitly supplied as an auxiliary resource. The protocol article derives the four orthogonal codewords and explains how noise changes the result.

These identities are not claims that a qubit “contains two bits” or that an ebit alone carries a message. They describe conversions among resources under specific protocols.

Entanglement alone does not let Alice change Bob’s local statistics. A local trace-preserving operation can change correlations, and a selected measurement outcome can change Bob’s conditional state. When the outcome is unavailable to him, he must average over it. No-cloning and no-signaling gives the general proof; the distinction between conditioning and averaging is essential in both teleportation and heralded links.

Classical capacity and the Holevo quantity

For an ensemble $\{p_x,\rho_x\}$ with average $\overline\rho=\sum_xp_x\rho_x$, define

$$\chi = S(\overline\rho) - \sum_xp_xS(\rho_x),$$

where $S(\rho)=-\operatorname{Tr}(\rho\log_2\rho)$. Equivalently, $\chi=I(X:B)_\rho$ for the classical–quantum state $\rho_{XB}$. For every measurement outcome $Y$,

$$I(X:Y) \le \chi.$$

This is the Holevo bound (Holevo, 1973). It limits the classical information accessible from one ensemble, but it does not say that a $d$-dimensional system stores only one fixed classical variable. The bound depends on the ensemble, and coding over many channel uses can make collective measurements essential.

For a channel $\mathcal N$, the one-use Holevo information optimized over input ensembles is

$$\chi(\mathcal N) =\max_{\{p_x,\rho_x\}} \chi\bigl(\{p_x,\mathcal N(\rho_x)\}\bigr).$$

The Holevo–Schumacher–Westmoreland coding theorem gives the product-state coding rate from the optimized Holevo quantity (Holevo, 1998; Schumacher and Westmoreland, 1997). Allowing entangled codewords across many uses gives the regularized unassisted classical capacity:

$$C(\mathcal N) = \lim_{n\to\infty} \frac{1}{n} \chi(\mathcal N^{\otimes n}).$$

Regularization allows an ensemble of entangled inputs to a larger block to outperform the best independent one-use ensembles; Hastings (2009) established that such additivity failures exist. Collective output measurements can already be needed for product-state coding, so they are not themselves the reason for input-ensemble nonadditivity. For channels with proved additivity, the formula simplifies to one use.

Quantum capacity and coherent information

Let $|\psi\rangle_{RA}$ purify an input $\rho_A$, and send $A$ through a channel isometry to outputs $B$ and $E$. The coherent information is

$$I_c(\rho,\mathcal N) = S(B)-S(E) = -S(R|B).$$

It measures a balance between correlation recoverable by Bob and information lost to the environment. Unlike ordinary mutual information, coherent information can be negative and is not generally additive.

The unassisted quantum capacity is the regularized optimized coherent information, as in the quantum coding theorem of Lloyd, Shor and Devetak (see Devetak, 2005):

$$Q(\mathcal N) = \lim_{n\to\infty} \frac{1}{n} \max_{\rho_{A^n}} I_c \left( \rho_{A^n}, \mathcal N^{\otimes n} \right).$$

This is an asymptotic coding theorem: rates below $Q$ can preserve quantum information with error tending to zero as block length grows. It is not a one-shot fidelity formula, and its regularization can make exact evaluation difficult.

Channels whose environment can simulate Bob’s output are antidegradable. If both Bob and the environment could receive high-fidelity quantum copies, one could violate no-cloning. Antidegradable channels therefore have zero unassisted quantum capacity. Zero $Q$ does not imply zero classical capacity, and some pairs of zero-capacity channels have positive joint capacity (Smith and Yard, 2008). This does not mean that every pair does, or that a single antidegradable channel becomes useful merely by repeating it.

A family of capacities

There is no single capacity of a quantum channel because the communication task and free assistance matter.

Quantity	Output per channel use	Free assistance
$C$	Reliable classical bits	None
$Q$	Reliable qubits or ebits	None
$P$	Private classical bits	None
$C_E$	Reliable classical bits	Unlimited shared entanglement
$K_{\leftrightarrow}$	Shared secret bits	Authenticated two-way public discussion

Privacy from the complementary output is a success criterion for $P$, not free assistance. Here $K_{\leftrightarrow}$ explicitly allows interactive public discussion; other allowed resources define other key capacities.

The entanglement-assisted classical capacity has the single-letter formula (Bennett, Shor, Smolin and Thapliyal, 2002)

$$C_E(\mathcal N) = \max_{\rho_A} I(R:B)_\sigma,$$

where $|\psi\rangle_{RA}$ purifies $\rho_A$ and $\sigma_{RB}=(\operatorname{id}_R\otimes\mathcal N)(\psi_{RA})$. The shared entanglement is an explicit resource; comparing $C_E$ with $C$ without noting that assistance is not a like-for-like hardware comparison.

Private capacity and key-agreement capacity involve decoupling an adversarial environment and may differ when public interaction or additional resources are permitted. A protocol-specific secret-key rate is not automatically the private capacity of the underlying physical channel.

Feedback and side information

Classical feedback can improve finite-block performance and coordinate heralded attempts even when it does not change a particular asymptotic capacity. Quantum feedback, two-way classical communication, entanglement assistance, degradability flags, and trusted repeaters define still other capacity regions. Every quoted rate should list the allowed side resources.

Finite blocks, time and energy constraints

A capacity is the boundary of a limiting statement. For classical messages, a rate $R$ is achievable if there exists a sequence of block codes with

$$\begin{aligned} \lim_{n\to\infty}P_e^{(n)}&=0,\\ \liminf_{n\to\infty}\frac{\log_2M_n}{n}&\ge R. \end{aligned}$$

Real links operate at finite $n$, finite latency, and nonzero target error. For a finite block, give the number $n$ of channel uses, the message-set size or code dimension $M$, and the permitted error $\varepsilon$. The rate $\log_2M/n$ counts information per use. Converting it to information per second additionally requires the elapsed time, including coding, synchronization, waiting and control operations. A short code meeting a stringent error target can operate well below capacity.

For bosonic channels an unconstrained input Hilbert space can make an unqualified classical capacity infinite. Mean energy, peak photon number, bandwidth, mode count, and time must be constrained. “Bits per photon,” “bits per mode,” and “bits per second” describe different operating limits.

Strong converse and error conventions

Some coding theorems have a strong converse: attempting a rate above capacity forces error toward one. Others require more careful error and assistance conditions. Average message error, maximal message error, entanglement infidelity, trace-distance error, and purified-distance error are different criteria. State which convergence defines the capacity being used.

A flagged erasure channel separates the capacities

The $d$-dimensional erasure channel transmits its input perfectly with probability $1-\epsilon$ and replaces it with an orthogonal erasure flag with probability $\epsilon$:

$$\mathcal N_\epsilon(\rho) = (1-\epsilon)\rho +\epsilon|e\rangle\langle e|.$$

The flag tells Bob which uses were erased, a major difference from an unflagged replacement error. For this channel, the exact classical and quantum capacities are known (Bennett, DiVincenzo and Smolin, 1997):

$$C(\mathcal N_\epsilon) = (1-\epsilon)\log_2d,$$ $$Q(\mathcal N_\epsilon) = \max\{0,1-2\epsilon\}\log_2d,$$ $$C_E(\mathcal N_\epsilon) = 2(1-\epsilon)\log_2d.$$

For a qubit and $\epsilon=0.25$, these become

$$C=0.75, \qquad Q=0.50, \qquad C_E=1.50$$

bits, qubits, and entanglement-assisted bits per channel use, respectively. The three values differ because they answer different operational questions.

At $\epsilon\ge1/2$, $Q=0$: the environment’s complementary channel transmits the input on a fraction $\epsilon$ of uses, while Bob receives it on a fraction $1-\epsilon$. The environment can simulate Bob by discarding additional successful outputs, so the channel is antidegradable. Yet $C=(1-\epsilon)\log_2d$ remains positive for every $\epsilon<1$. The channel can still carry some classical information even though it cannot reliably preserve arbitrary quantum information without additional assistance.

With two-way classical assistance, successful transmissions of halves of Bell pairs can be retained using Bob’s erasure flags. This gives an entanglement-distribution rate $(1-\epsilon)\log_2d$, even above the unassisted threshold. Teleporting an unknown input then also requires classical messages. It is the changed assistance that permits this conversion; one cannot simply resend a lost unknown state.

Erasure and loss channels develops the map and its complementary output in detail.

From channel uses to delivered pairs

A link that makes independent, nonoverlapping attempts every $\tau$ seconds and succeeds with probability $p$ has long-run raw rate $p/\tau$. Include reset and required heralding communication in that cycle time. If a subsequent protocol produces $y$ usable output pairs per accepted input pair on average, the supply-limited output rate is $yp/\tau$, provided processing and storage do not become bottlenecks. The yield $y$ must include both rejected events and the consumption of multiple input pairs.

A repeater waits for successful neighboring links, stores the earlier success and performs an entanglement swap. Memory decoherence, swap failures and contention change the final distribution of quality and delivery time. Quantum repeaters develops that stochastic problem. A mean elementary rate alone does not answer whether an application receives an adequate state before its deadline.

Network architectures explains how links and memories support shared requests; network verification explains what measurements establish that the intended resource was delivered. These network questions add physical constraints to the capacity problem rather than replacing its definition.

Loss, noise and conditional efficiencies

The abstract map $\mathcal N$ must describe the relevant physical processes. For optical links this can include attenuation, coupling efficiency, detector efficiency, dark counts, background photons, mode mismatch, dispersion, phase noise, and source multiphoton probability. For microwave or matter links it can include thermal occupancy, conversion efficiency, added noise, bandwidth, and impedance matching.

If each stage efficiency is conditional on the signal having reached that stage, the probability chain rule gives a product without assuming independent losses. Multiplying independently measured marginal efficiencies instead needs additional assumptions about correlations and matching operating conditions. For correctly defined conditional efficiencies,

$$\begin{aligned} \eta_{\mathrm{tot}}&=\prod_j\eta_j,\\ L_{\mathrm{tot}}^{(\mathrm{dB})}&=-10\log_{10}\eta_{\mathrm{tot}}\\ &=\sum_jL_j^{(\mathrm{dB})}. \end{aligned}$$

Loss and noise are not interchangeable. A heralded erasure can be easier for a code or protocol than an unflagged Pauli error at the same raw frequency. Likewise, a transducer with high efficiency but enough added thermal noise may destroy the entanglement needed by the application.

Interconnects and transduction relates coupling efficiency to added noise and bandwidth. Quantum-memory hardware treats storage and retrieval. A measured loss probability and an unflagged logical error probability are not interchangeable inputs to these communication models.

Exercises

1. Distinguish four tasks

Classify each output as classical communication, quantum communication, entanglement distribution, or key agreement: (a) Bob learns a chosen index; (b) Bob receives one half of a state still entangled with Alice’s reference; (c) Alice and Bob end with a Bell pair but no chosen message; (d) Alice and Bob end with matching random strings secret from Eve.

Solution

(a) is classical message transmission. (b) is quantum transmission because correlation with a reference is preserved. (c) is entanglement distribution. (d) is secret-key agreement. A single physical channel may support all four, but the success criterion and capacity differ.

2. Compute a Holevo quantity

Alice chooses $|0\rangle$ with probability $1/2$ and $|+\rangle$ with probability $1/2$. Compute the Holevo quantity of the pure-state ensemble.

Solution

The individual states are pure, so their entropies vanish. The average state is

$$\overline\rho = \frac12|0\rangle\langle0| +\frac12|+\rangle\langle+|.$$

Its eigenvalues are

$$\lambda_\pm = \frac12 \left( 1\pm\frac{1}{\sqrt2} \right).$$

Therefore

$$\chi = h_2 \left[ \frac12 \left( 1+\frac{1}{\sqrt2} \right) \right] \approx0.601 \text{ bits},$$

where $h_2$ is binary entropy. This upper-bounds the accessible information from one signal; the nonorthogonal states do not carry one perfectly recoverable bit in a one-shot measurement.

3. Explain the role of the classical message

A report says that one shared Bell pair lets Alice send a qubit to Bob instantaneously. Identify the missing resource and the causal error.

Solution

Teleportation also consumes two classical bits sent from Alice to Bob. Conditioned on Alice’s outcome, Bob’s state generally differs by a known Pauli correction. Before the outcome arrives, he sees the average over all four outcomes, which is independent of the input state. He cannot interpret it as the transmitted state without the correction information. The shared ebit changes the conversion of communication resources but does not remove the causal classical channel.

4. Erasure-channel capacities

For a qubit erasure channel with $\epsilon=0.6$, compute $C$, $Q$, and $C_E$.

Solution

Since $\log_2d=1$,

$$C=1-0.6=0.4 \text{ bits/use},$$ $$Q=\max\{0,1-1.2\}=0,$$

and

$$C_E=2(1-0.6)=0.8 \text{ bits/use}.$$

The channel still transmits classical labels on the unerased uses, but it is antidegradable and has no unassisted quantum capacity.

5. Convert loss units

A source-to-memory path has efficiencies $0.8$, $0.25$, and $0.6$ for collection, transmission, and storage/retrieval. Find the end-to-end efficiency and loss in decibels.

Solution

The efficiency is

$$\eta_{\mathrm{tot}} = 0.8\times0.25\times0.6 =0.12.$$

Thus

$$L^{(\mathrm{dB})} = -10\log_{10}(0.12) \approx9.21\,\mathrm{dB}.$$

This multiplication assumes the efficiencies are conditional stages with no double counting.

6. Include heralding in delivered rate

A link attempts entanglement every $20\,\mu\mathrm{s}$, succeeds with probability $0.01$, and feeds accepted pairs to a protocol with net yield $0.70$ usable output pairs per input pair. The yield includes rejected blocks and all pair consumption. Assuming processing is fast enough and ignoring memory contention, find the raw and delivered mean pair rates.

Solution

The attempt rate is $50{,}000\,\mathrm{s}^{-1}$. The raw heralded rate is

$$R_{\mathrm{raw}} = 50{,}000\times0.01 =500\,\mathrm{s}^{-1}.$$

The delivered retained rate is

$$R_{\mathrm{del}} = 0.7\times500 =350\,\mathrm{s}^{-1}.$$

A real repeater would also include pair consumption, waiting-time decoherence, swap success, and buffer limits.

7. Distinguish average fidelity from a worst-case guarantee

First explain why perfect transmission of $|0\rangle$ and $|1\rangle$ is insufficient. Then consider a unitary channel on a $d$-dimensional code with $U=\operatorname{diag}(-1,1,\ldots,1)$. Find its entanglement fidelity for a maximally entangled input, and its action on $(|0\rangle+|1\rangle)/\sqrt2$.

Solution

Complete dephasing preserves both computational-basis states but destroys their coherent superpositions. For a Bell input it gives the equal mixture of $|00\rangle$ and $|11\rangle$, whose entanglement fidelity is only $1/2$.

For the unitary example, the maximally entangled overlap is $\operatorname{Tr}U/d$, so

$$F_e=\frac{|\operatorname{Tr}U|^2}{d^2} =\left(1-\frac2d\right)^2.$$

This tends to one as $d$ grows. Yet the stated superposition becomes $(-|0\rangle+|1\rangle)/\sqrt2$, orthogonal to its input. Thus a high average entanglement fidelity alone is not a dimension-independent uniform guarantee. A worst-case claim requires its own norm or a suitable dimension-dependent bound.

8. Compare like communication tasks

System A reports $1.2$ entanglement-assisted bits per channel use. System B reports $0.9$ unassisted bits per channel use at a finite block length. Can the numbers rank the systems? State the missing information.

Solution

No. The assistance models differ, and one quantity may be asymptotic while the other is finite-block. A fair comparison must align the channel definition, task, shared-entanglement supply and cost, block length, target error, input energy, receiver restrictions, use duration, and delivered throughput. The larger numerical rate does not by itself identify the better implementation.

Exact capacities remain difficult to evaluate for many channels because the block optimizations can be nonadditive. A finite device also has energy, time and receiver constraints that an unconstrained asymptotic theorem may omit. The distinction between a coding limit and a delivered rate is therefore part of interpreting the physics, not merely a choice of units.

References

• Bennett, Charles H., David P. DiVincenzo, and John A. Smolin. “Capacities of quantum erasure channels.” Physical Review Letters 78, 3217–3220 (1997). arXiv:quant-ph/9701015; doi:10.1103/PhysRevLett.78.3217.
• Bennett, Charles H., Peter W. Shor, John A. Smolin, and Ashish V. Thapliyal. “Entanglement-assisted capacity of a quantum channel and the reverse Shannon theorem.” IEEE Transactions on Information Theory 48, 2637–2655 (2002). arXiv:quant-ph/0106052; doi:10.1109/TIT.2002.802612.
• Devetak, Igor. “The private classical capacity and quantum capacity of a quantum channel.” IEEE Transactions on Information Theory 51, 44–55 (2005). arXiv:quant-ph/0304127; doi:10.1109/TIT.2004.839515.
• Hastings, Matthew B. “Superadditivity of communication capacity using entangled inputs.” Nature Physics 5, 255–257 (2009). doi:10.1038/nphys1224.
• Holevo, Alexander S. “Bounds for the quantity of information transmitted by a quantum communication channel.” Problems of Information Transmission 9, 177–183 (1973).
• Holevo, Alexander S. “The capacity of the quantum channel with general signal states.” IEEE Transactions on Information Theory 44, 269–273 (1998). arXiv:quant-ph/9611023; doi:10.1109/18.651037.
• Schumacher, Benjamin. “Sending entanglement through noisy quantum channels.” Physical Review A 54, 2614–2628 (1996). arXiv:quant-ph/9604023; doi:10.1103/PhysRevA.54.2614.
• Schumacher, Benjamin, and Michael D. Westmoreland. “Sending classical information via noisy quantum channels.” Physical Review A 56, 131–138 (1997). doi:10.1103/PhysRevA.56.131.
• Smith, Graeme, and Jon Yard. “Quantum communication with zero-capacity channels.” Science 321, 1812–1815 (2008). doi:10.1126/science.1162242.