Erasure and Loss Channels

Erasure and loss channels describe situations in which the intended system may fail to arrive, leave the retained Hilbert space, or be replaced by a known loss flag. The central modeling distinction is whether the loss event is flagged.

In a flagged erasure channel, the receiver knows that the system was lost because the output contains an orthogonal erasure state. In unflagged loss, the system may be attenuated, leaked, or missing without a reliable classical record. Those are different physical and information-theoretic situations.

The useful first split is:

Model	What happens	Is loss known?	Typical use
flagged erasure	output includes an orthogonal loss flag	yes	communication erasures, located errors, heralded loss
survival branch	keep only successful events	yes, after postselection	conditional analysis, heralded experiments
leakage	population leaves a chosen subspace	maybe	qubits with noncomputational states, atom loss
bosonic loss	a mode is attenuated into an environment	usually no	photon loss, fiber attenuation, cavity damping
detector inefficiency	loss before ideal detection	partially	photon counting, finite collection efficiency

The page owns the channel-level distinctions. Full continuous-variable Gaussian formalism, coding theory for erasure correction, and hardware-specific loss budgets belong to their own canonical pages. Photonic Qubits applies these distinctions to source failure, propagation loss, detector inefficiency, heralded absence, and architecture-level erasure accounting.

Flagged Erasure Channel

Let $\mathcal H_{\mathrm{in}}$ be the input Hilbert space. The output Hilbert space for an erasure channel is enlarged:

$$\mathcal H_{\mathrm{out}} = \mathcal H_{\mathrm{surv}} \oplus \mathbb C\lvert e\rangle,$$

where $\lvert e\rangle$ is orthogonal to every valid surviving output state. Let

$$V: \mathcal H_{\mathrm{in}} \to \mathcal H_{\mathrm{surv}}$$

be an isometric embedding of the input into the non-erased output subspace. With erasure probability $\epsilon$, the channel is

$$\mathcal E_\epsilon(\rho) = (1-\epsilon)V\rho V^\dagger + \epsilon\,\operatorname{Tr}(\rho) \lvert e\rangle\langle e\rvert, \qquad 0\le\epsilon\le1.$$

For normalized inputs, this says:

$$\text{survive with probability }1-\epsilon, \qquad \text{erased with probability }\epsilon.$$

The output state itself contains the flag. Measuring the projector

$$\Pi_e=\lvert e\rangle\langle e\rvert$$

reveals whether erasure occurred without confusing the flag with an ordinary input state.

Kraus Representation

Choose an orthonormal input basis

$$\{\lvert j\rangle\}_{j=1}^d.$$

A Kraus representation of the flagged erasure channel is

$$K_s = \sqrt{1-\epsilon}\,V, \qquad K_j = \sqrt{\epsilon}\, \lvert e\rangle\langle j\rvert, \quad j=1,\ldots,d.$$

The completeness relation is

$$\begin{aligned} K_s^\dagger K_s + \sum_{j=1}^dK_j^\dagger K_j &= (1-\epsilon)I + \epsilon\sum_{j=1}^d \lvert j\rangle\langle j\rvert\\ &= I. \end{aligned}$$

Therefore the flagged erasure map is trace preserving. The different Kraus labels should not be overread as unique microscopic histories; what is physical here is the orthogonal output flag.

This example is a clean application of the trace conditions discussed in Trace-Preserving and Trace-Nonincreasing Maps.

Survival Branch and Postselection

If one keeps only the successful branch, the operation is

$$\mathcal S_\epsilon(\rho) = (1-\epsilon)V\rho V^\dagger.$$

This is completely positive but trace decreasing:

$$\operatorname{Tr}\mathcal S_\epsilon(\rho) = (1-\epsilon)\operatorname{Tr}\rho.$$

For a normalized input, the survival probability is $1-\epsilon$. If survival is observed and the output is conditioned on survival, then

$$\rho_{\mathrm{surv}} = \frac{\mathcal S_\epsilon(\rho)} {\operatorname{Tr}\mathcal S_\epsilon(\rho)} = V\rho V^\dagger.$$

This conditional normalized map is not the same object as the physical operation before conditioning. It has discarded the probability that the system was lost.

Postselection is often useful in experiments, but it can bias statistics. A calculation that keeps only successful detections must state which events were removed and whether the removed events carry information about the state or setting.

Erasure Versus Depolarizing Noise

Erasure is not depolarizing noise. In a depolarizing channel, the output remains in the system Hilbert space and the receiver does not receive a reliable marker telling where the damage occurred. In an erasure channel, the damaged system is replaced by a known orthogonal flag.

The contrast is visible in the outputs:

$$\text{erasure:}\quad \rho \longmapsto (1-\epsilon)V\rho V^\dagger + \epsilon\lvert e\rangle\langle e\rvert,$$

while for a $d$-dimensional depolarizing replacement model,

$$\text{depolarizing:}\quad \rho \longmapsto (1-q)\rho+q\frac{I}{d}.$$

The flag makes erasures located errors. A known missing qubit, photon, or atom is often easier to handle than an unknown corrupted one. The detailed coding theory belongs to quantum information; the channel-level lesson is that the output alphabet has changed.

See Depolarizing Channel for the unflagged isotropic benchmark model.

Leakage and Loss from a Subspace

In many platforms the computational Hilbert space is only a subspace of a larger physical Hilbert space:

$$\mathcal H_{\mathrm{phys}} = \mathcal H_{\mathrm{comp}} \oplus \mathcal H_{\mathrm{leak}}.$$

Leakage means population leaves $\mathcal H_{\mathrm{comp}}$. If leakage is detected reliably, it can behave like an erasure flag. If not, it is an unflagged loss process on the computational subspace.

Let $\Phi_{\mathrm{phys}}$ be a channel on the full physical space and let $P_{\mathrm{comp}}$ project onto the computational subspace. The retained-subspace operation is

$$\mathcal L_{\mathrm{ret}}(\rho) = P_{\mathrm{comp}} \Phi_{\mathrm{phys}}(\rho) P_{\mathrm{comp}}.$$

This map is generally trace nonincreasing:

$$\operatorname{Tr}\mathcal L_{\mathrm{ret}}(\rho) \le \operatorname{Tr}\rho.$$

The missing trace is the probability that population leaked out of the retained subspace. If the leaked sector is included as part of the output, the full physical map can remain trace preserving.

This is why atom loss, molecular loss, ion shelving outside a detection manifold, and leakage to noncomputational states should not be modeled as pure dephasing inside a fixed qubit unless that approximation has been justified. Effective Hamiltonians in Quantum Information separates coherent logical evolution from the contraction of a projected physical propagator. The neutral-atom platform page gives a concrete example in Neutral Atoms.

Leakage and Crosstalk owns the QI-facing state, average, worst-case, coherent-return, flag, population-dynamics, and context-composition diagnostics for computational leakage; this page retains formal loss and erasure channels, vacuum and orthogonal flags, trace-decreasing survival operations, bosonic attenuation, detector inefficiency, capacities, and platform examples.

Bosonic Pure-Loss Channel

For an optical or oscillator mode, loss is usually modeled by coupling the signal mode $a$ to an environment mode $e$ through a beam splitter of transmissivity $\eta$. In the Heisenberg picture,

$$a_{\mathrm{out}} = \sqrt{\eta}\,a_{\mathrm{in}} + \sqrt{1-\eta}\,e_{\mathrm{in}}, \qquad 0\le\eta\le1.$$

If the environment mode is initially vacuum, this defines the quantum-limited pure-loss channel. A coherent state transforms as

$$\lvert\alpha\rangle \longmapsto \lvert\sqrt{\eta}\,\alpha\rangle$$

after the environment is traced out. Photon number is attenuated:

$$\langle a^\dagger a\rangle \longmapsto \eta\langle a^\dagger a\rangle.$$

A useful Kraus representation is

$$A_\ell = \sqrt{\frac{(1-\eta)^\ell}{\ell!}}\, \eta^{\hat n/2}a^\ell, \qquad \ell=0,1,2,\ldots,$$

where

$$\hat n=a^\dagger a.$$

Acting on a Fock state,

$$A_\ell\lvert n\rangle = \begin{cases} \sqrt{\binom n\ell} (1-\eta)^{\ell/2} \eta^{(n-\ell)/2} \lvert n-\ell\rangle, & \ell\le n,\\ 0, & \ell>n. \end{cases}$$

The label $\ell$ counts photons lost to the environment in this representation. If the environment is not monitored, the receiver does not usually know $\ell$.

Bosonic loss is therefore not automatically the same as a flagged erasure channel. A lost photon may produce an identifiable vacuum in a fixed time bin, or it may be confused with source failure, detector inefficiency, mode mismatch, or another unobserved event. The modeling depends on what the receiver can actually distinguish.

Detector Inefficiency

Detector inefficiency is often modeled as loss before an ideal detector. For a photon-counting detector with quantum efficiency $\eta$, place a beam splitter of transmissivity $\eta$ before a perfect detector and trace out the lost port.

For an ideal on/off detector after this loss, the no-click POVM element is

$$\Pi_0 = \sum_{n=0}^\infty (1-\eta)^n \lvert n\rangle\langle n\rvert,$$

and

$$\Pi_{\mathrm{click}} = I-\Pi_0.$$

This formula says that an $n$-photon input fails to produce a click with probability $(1-\eta)^n$ in the idealized independent-loss model.

Detector inefficiency belongs to the measurement model, not only to the state channel. The same physical loss can be represented as a channel before measurement, as a POVM with inefficient effects, or as part of a quantum instrument. The right representation depends on whether one is predicting transmitted states, click probabilities, or conditional post-measurement states.

Relation to Amplitude Damping

Amplitude damping and loss are related but not identical.

For a two-level atom, amplitude damping describes relaxation

$$\lvert1\rangle \to \lvert0\rangle$$

inside the retained two-dimensional Hilbert space. If the emitted photon is not monitored, the final atom is still present in the ground state. That is not an erasure of the atom.

For a bosonic mode, photon loss changes excitation number and can remove a photon from an encoded subspace. In a single-excitation code, the vacuum may function as a loss state; whether it is a flagged erasure depends on the protocol and detector model.

The finite-time qubit amplitude-damping channel is treated at Amplitude-Damping Channel. The continuous-time photon-loss master equation appears in Quantum Optical Master Equation.

Thermal Loss and Gaussian Boundary

If the environment mode in the beam-splitter model is thermal instead of vacuum, attenuation is accompanied by added thermal noise. At the covariance-matrix level for a single mode, one often writes schematically

$$V \longmapsto \eta V+(1-\eta)V_E,$$

where $V_E$ is the covariance matrix of the environment mode in the same quadrature convention.

This is the entry point to Gaussian channels: attenuation, amplification, additive noise, and thermal-loss models. This page stops at the loss distinction; Gaussian Channels owns the continuous-variable covariance formalism.

Common Mistakes

• Treating unheralded photon loss as flagged erasure without an actual flag.
• Modeling atom or qubit leakage as pure dephasing inside the computational subspace.
• Normalizing a survival branch and then forgetting the survival probability.
• Calling every missing detector click a lost photon; source failure, detector inefficiency, dark counts, and mode mismatch may be operationally different.
• Treating bosonic attenuation as a qubit amplitude-damping channel without specifying the encoded subspace.
• Ignoring thermal environment photons when microwave or low-frequency modes are not in a vacuum bath.
• Comparing loss rates without checking whether the parameter is transmissivity $\eta$, loss probability $1-\eta$, or continuous-time decay $e^{-\kappa t}$.

References

• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press (2010).
• A. S. Holevo, Quantum Systems, Channels, Information, De Gruyter (2012).
• M. M. Wilde, Quantum Information Theory, Cambridge University Press, 2nd ed. (2017).
• C. W. Gardiner and P. Zoller, Quantum Noise, Springer, 3rd ed. (2004).
• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press (2002).
• C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Reviews of Modern Physics 84, 621-669 (2012).

Exercises

1. Trace preservation of erasure. Verify that the Kraus operators $K_s=\sqrt{1-\epsilon}\,V$ and $K_j=\sqrt{\epsilon}\lvert e\rangle\langle j\rvert$ define a trace-preserving channel.

Solution

Since $V^\dagger V=I$,

$$K_s^\dagger K_s = (1-\epsilon)I.$$

Also,

$$\sum_{j=1}^dK_j^\dagger K_j = \epsilon \sum_{j=1}^d \lvert j\rangle\langle j\rvert = \epsilon I.$$

Therefore

$$K_s^\dagger K_s+\sum_{j=1}^dK_j^\dagger K_j=I.$$

2. Survival conditioning. For the survival operation $\mathcal S_\epsilon(\rho)=(1-\epsilon)V\rho V^\dagger$, compute the success probability and the normalized surviving state for a normalized input.

Solution

The success probability is

$$p_{\mathrm{surv}} = \operatorname{Tr}\mathcal S_\epsilon(\rho) = (1-\epsilon)\operatorname{Tr}\rho = 1-\epsilon.$$

The normalized surviving state is

$$\rho_{\mathrm{surv}} = \frac{(1-\epsilon)V\rho V^\dagger}{1-\epsilon} = V\rho V^\dagger.$$

The probability $1-\epsilon$ is lost if one records only the normalized conditional state.

3. Fock-state loss. Use the pure-loss Kraus operator $A_\ell$ to find the probability of losing exactly $\ell$ photons from an input number state $\lvert n\rangle$.

Solution

For $\ell\le n$,

$$A_\ell\lvert n\rangle = \sqrt{\binom n\ell} (1-\eta)^{\ell/2} \eta^{(n-\ell)/2} \lvert n-\ell\rangle.$$

The probability is the squared norm:

$$p(\ell|n) = \binom n\ell (1-\eta)^\ell \eta^{n-\ell}.$$

For $\ell>n$, the probability is zero. This is the binomial distribution for independently transmitting each of the $n$ photons with probability $\eta$ in the ideal loss model.

4. Flagged or unflagged? A receiver expects one photon in a fixed time bin. The detector does not click. Name two different physical models that could give this observation and explain why they are not automatically the same channel.

Solution

One model is a flagged erasure: the source and timing are trusted, and the absence of a photon is a reliable orthogonal flag that transmission failed. Another model is detector inefficiency: the photon may have arrived but failed to trigger the detector. A third possibility is source failure.

These models can give the same observed no-click event but imply different state transformations, different conditioning, and different error correlations. A channel model must specify which degrees of freedom are part of the output and which records are trusted.