Quantum channel

A quantum erasure channel transmits an input state unchanged with probability

1 - p

and replaces it by an orthogonal, recognizable erasure flag with probability

p

. Its output space has one more dimension than the input. This is a quantum channel for

0 \leq p \leq 1

and preserves all input coherences on the nonerased branch.

Entanglement-assisted capacity of a quantum erasure channel ( $C_{E} (E_{p}) = 2 (1 - p) lo g_{2} D$ )

For input dimension

D

, an input of entropy

S (ρ)

gives output entropy

h_{2} (p) + (1 - p) S (ρ)

and reference-output entropy

h_{2} (p) + pS (ρ)

. These follow from entropy of an orthogonal quantum mixture and purity of the input purification. Thus the quantum mutual information is

2 (1 - p) S (ρ)

, maximized at

ρ = I_{D} / D

. The entanglement-assisted classical capacity is consequently

2 (1 - p) lo g_{2} D

bits per use; for a binary input it is

2 (1 - p)

Entanglement-assisted classical capacity ( $C_{E} (N) = max_{ρ} I (R; B)$ )

This is the supremum of reliable classical bits per use of a memoryless quantum channel when sender and receiver share unlimited prior quantum entanglement. If

∣ ψ_{ρ} ⟩_{R A}

purifies the input, let

ω_{RB} = (id_{R} \otimes N) (∣ ψ_{ρ} ⟩ ⟨ ψ_{ρ} ∣)

. The single-letter capacity formula is

max_{ρ} [S (ρ) + S (N (ρ)) - S (ω_{RB})]

, the maximum quantum mutual information between reference and output. The preshared resource is independent of the message.

Amplitude damping channel ( $A_{0} = ∣0 ⟩ ⟨ 0∣ + 1 - p ∣1 ⟩ ⟨ 1∣, A_{1} = p ∣0 ⟩ ⟨ 1∣$ )

A zero-temperature energy-relaxation channel transfers the excited population of a qubit to its ground state. Its Kraus operators are

A_{0} = diag (1, 1 - p)

and

A_{1} = p ∣0 ⟩ ⟨ 1∣

for

0 \leq p \leq 1

. They satisfy

\sum_{j} A_{j}^{†} A_{j} = I

. The excited population is multiplied by

1 - p

and coherences by

1 - p

. For nonzero

p

, it is not a unital quantum channel.

Unitary dilation of amplitude damping ( $U ∣10 ⟩ = 1 - p ∣10 ⟩ + p ∣01 ⟩$ )

With the environment initially in

∣0 ⟩

, take

U

to fix

∣00 ⟩

and

∣11 ⟩

, and rotate the span of

∣10 ⟩, ∣01 ⟩

by the displayed action and

U ∣01 ⟩ = 1 - p ∣01 ⟩ - p ∣10 ⟩

. This is a unitary operator. Its environment matrix elements

⟨ 0 ∣_{E} U ∣0 ⟩_{E}

and

⟨ 1 ∣_{E} U ∣0 ⟩_{E}

are the Kraus operators of the amplitude damping channel. Tracing out the environment multiplies the excited population by

1 - p

, transfers the remainder to the ground state, and multiplies off-diagonal entries by

1 - p

Entanglement fidelity of amplitude damping ( $F_{e} = \frac{1}{4} [1 + s_{z} + 1 - p (1 - s_{z})]^{2} + \frac{p}{4} (s_{x}^{2} + s_{y}^{2})$ )

The Kraus formula for entanglement fidelity applies to any input, including mixed qubit states. For the amplitude damping channel, the diagonal Kraus trace is

[1 + s_{z} + 1 - p (1 - s_{z})] /2

and the jump Kraus trace has squared modulus

p (s_{x}^{2} + s_{y}^{2}) /4

. Summing them gives the formula. Ground-state inputs have unit entanglement fidelity; excited-state inputs have fidelity

1 - p

Bloch-vector map of amplitude damping ( $s \mapsto (1 - p s_{x}, 1 - p s_{y}, p + (1 - p) s_{z})$ )

Applying the amplitude damping channel to the Bloch vector form of a qubit density operator gives an affine map. Transverse components contract by

1 - p

; the longitudinal component contracts by

1 - p

and shifts toward the ground-state north pole. Reading the entries of the output density operator proves the map. The displacement distinguishes this channel from centered depolarization.

Repeated amplitude damping limit ( $s^{(n)} = ((1 - p)^{n /2} s_{x}, (1 - p)^{n /2} s_{y}, 1 - (1 - p)^{n} (1 - s_{z}))$ )

Under repeated identical amplitude damping channels, all excited population eventually relaxes to the ground state for any fixed

0 < p \leq 1

. The Bloch vector approaches

(0, 0, 1)

. For

p = 1

the limit is reached in one action, while

p = 0

is the identity channel and preserves every input. Keeping the zero-damping endpoint separate prevents an incorrect universal relaxation claim.

Holevo capacity ( $χ^{*} (Λ)$ )

The Holevo capacity of a quantum channel is the maximum one-use output Holevo quantity,

χ^{*} (Λ) = max_{{p_{x}, ρ_{x}}} [S (\sum_{x} p_{x} Λ (ρ_{x})) - \sum_{x} p_{x} S (Λ (ρ_{x}))] .

It is the unassisted classical communication limit for product state encodings with collective output measurements. Arbitrary entangled inputs require the regularized expression in the Holevo-Schumacher-Westmoreland theorem.

Superadditivity of Holevo capacity ( $χ^{} (Φ_{1} \otimes Φ_{2}) \geq χ^{} (Φ_{1}) + χ^{*} (Φ_{2})$ )

Take independent ensembles for two quantum channels and their Cartesian product ensemble. The Von Neumann entropy of a tensor product is the sum of the two entropies, both for its average output and for each labelled output. Thus its Holevo quantity is the sum of the two individual quantities. Taking ensembles arbitrarily close to each supremum proves the displayed inequality without requiring maximizing ensembles to exist. It does not assert additivity for arbitrary channels.

Additivity of Holevo capacity ( $χ^{} (Λ \otimes Γ) = χ^{} (Λ) + χ^{*} (Γ)$ )

A channel's Holevo capacity is additive with another channel when their joint capacity equals the sum of the individual capacities. If this holds with every other channel, the channel's tensor powers have capacity

n χ^{*} (Λ)

and regularization does not increase its unassisted classical capacity of a quantum channel. Additivity is a special property, not a general rule for quantum channels.

Holevo-capacity additivity for entanglement-breaking channels ( $χ^{} (E \otimes N) = χ^{} (E) + χ^{*} (N)$ )

Every entanglement-breaking channel

E

has additive Holevo capacity with every finite-dimensional quantum channel

N

. A measure-and-prepare channel factorization exposes a classical measurement outcome. The quantum mutual information balance identity and data processing for quantum mutual information then bound the joint output Holevo quantity by the sum of the individual capacities, using the conditional input ensemble after a local measurement for the other channel. Independent product ensembles attain the reverse inequality.

Product-input classical-capacity converse

For a code with product state inputs to

n

uses of a memoryless quantum channel, Subadditivity of Von Neumann entropy bounds its output Holevo quantity by

n χ^{*} (Λ)

. The Holevo bound and Fano's inequality then give, for asymptotic rate

R

, average and hence maximum error at least

1 - χ^{*} (Λ) / R

in the limit inferior. In particular, no code above that rate can have vanishing maximum error, even with a collective decoder measurement.

Classical capacity of a quantum channel ( $C (Λ)$ )

The unassisted classical capacity is the supremum of classical bit rates achievable with asymptotically vanishing message error. It allows entanglement among different channel inputs, but no preshared sender-receiver entanglement. The Holevo-Schumacher-Westmoreland theorem gives its regularized entropy expression.

Holevo-Schumacher-Westmoreland theorem

For a finite-dimensional memoryless quantum channel,

C (Λ) = lim_{n \to \infty} \frac{1}{n} χ^{*} (Λ^{\otimes n}) .

The Holevo bound controls information obtainable from an output ensemble, while collective decoding makes rates below the corresponding coding expression achievable. The Holevo capacity needs regularization because input states can be entangled across channel uses. Watrous's channel-coding chapter states and develops the classical-capacity theorem.

Random unitary channel

A random unitary channel has

T (ρ) = \sum_{j} p_{j} U_{j} ρ U_{j}^{†}

. Its Choi matrix is a convex combination of maximally entangled pure states.

Heisenberg-Weyl operator ( $W_{k, m} = X^{k} Z^{m}$ )

On a

d

-dimensional Hilbert space, let

X ∣ j ⟩ = ∣ j + 1 mod d ⟩

and

Z ∣ j ⟩ = e^{2 πij / d} ∣ j ⟩

. The

d^{2}

unitaries

W_{k, m} = X^{k} Z^{m}

form an orthogonal operator basis under the Hilbert-Schmidt inner product.

Heisenberg-Weyl twirling channel

Averaging conjugation by all Heisenberg-Weyl operators completely depolarizes a system:

\frac{1}{d ^{2}} \sum_{k, m} W_{k, m} A W_{k, m}^{†} = (Tr A) \frac{I}{d} .

Summing over phases

Z^{m}

kills off-diagonal matrix elements, and averaging shifts

X^{k}

makes all diagonal elements equal. Acting on subsystem

B

of a bipartite operator gives

(Tr_{B} A) \otimes I_{B} / d

. This identity turns joint convexity of quantum relative entropy into monotonicity under partial trace.

Pauli channel

A one-qubit Pauli channel applies

I, σ_{x}, σ_{y}, σ_{z}

with probabilities

p_{0}, p_{x}, p_{y}, p_{z}

. It maps a Bloch vector diagonally by independently contracting or reversing its three components.

Quantum bit-flip channel ( $E_{p} (ρ) = (1 - p) ρ + pXρX$ )

A bit-flip channel applies the Pauli X gate with probability

p

and does nothing otherwise. If the classical error outcome is not recorded, its output is the convex mixture

E_{p} (ρ) = (1 - p) ρ + pXρX

, not a coherent superposition of the two alternatives.

// Target: quantum-information-theory.bigb

Average pure-qubit fidelity of a bit-flip channel ( $⟨ F^{2} ⟩ = 1 - 2 p /3$ )

For pure input

∣ ψ ⟩

, squared quantum fidelity after the bit-flip channel is

(1 - p) + p ∣ ⟨ ψ ∣ X ∣ ψ ⟩ ∣^{2}

. The last factor is the square of the input's Bloch

x

component. Rotational symmetry of the uniform pure-qubit average gives

⟨ r_{x}^{2} ⟩ = 1/3

, because

r_{x}^{2} + r_{y}^{2} + r_{z}^{2} = 1

. Hence the average squared fidelity is

1 - 2 p /3

. Averaging an unsquared fidelity or over a different ensemble gives a different quantity.

// Target: quantum-mechanics.bigb

Quantum depolarizing channel

A qubit depolarizing channel shrinks every Bloch vector by the same factor:

Λ_{p} (ρ) = pρ + (1 - p) I /2

. In this retention-parameter convention it is completely positive for

- 1/3 \leq p \leq 1

; the usual mixture of unchanged and fully randomized states uses

0 \leq p \leq 1

Entanglement-assisted capacity of a qubit depolarizing channel ( $C_{E} = 2 - H_{4} (1 - p, p /3, p /3, p /3)$ )

For the quantum depolarizing channel applying the identity with probability

1 - p

and each nonidentity Pauli operator with probability

p /3

, the entanglement-assisted classical capacity is the displayed number of bits per use. Here

H_{4}

is Shannon entropy of the four listed probabilities. Its unassisted product-state capacity is

1 - h_{2} (2 p /3)

. Both vanish at

p = 3/4

. If

q

instead denotes replacement by

I /2

, then

p = 3 q /4

and complete depolarization is at

q = 1

. The original entanglement-assisted capacity paper gives the depolarizing-channel formula.

High-noise capacity ratio for qubit depolarization ( $lim C_{E} / C_{prod} = 3$ )

With Bloch vector retention

η \to 0

, the unassisted product-state capacity of a quantum depolarizing channel is

η^{2} / (2 ln 2) + O (η^{4})

. Its entanglement-assisted capacity is

3 η^{2} / (2 ln 2) + O (η^{3})

, obtained by expanding Shannon entropy around the uniform four-outcome distribution. Consequently their limiting ratio is three, independent of the linear noise-parameter convention.

Depolarizing noise weight above one ( $Φ_{q} (ρ) = (1 - q) ρ + q I /2, 0 \leq q \leq 4/3$ )

The displayed qubit quantum channel is completely positive for

0 \leq q \leq 4/3

. Indeed its Pauli channel probabilities are

1 - 3 q /4

for the identity and

q /4

for each nonidentity Pauli operator. In the range

q > 1

the coefficient

1 - q

is negative, so the displayed expression is not a convex mixture of the unchanged and maximally mixed states, even though the Pauli representation is a valid random unitary channel. Its Bloch vector is contracted and reversed by the factor

1 - q

. The Holevo capacity of a qubit depolarizing channel remains

1 - h_{2} (q /2)

throughout this interval.

Pauli-mixture parametrization of qubit depolarization ( $η = (4 p - 1) /3$ )

If the identity is applied with probability

p

and the three nonidentity Pauli operators each with probability

(1 - p) /3

, the Bloch vector retention factor is

(4 p - 1) /3

. Thus complete depolarization occurs at

p = 1/4

, not at zero. The Holevo capacity of a qubit depolarizing channel becomes

1 - h_{2} ((2 p + 1) /3)

in this identity-probability parametrization.

Holevo capacity of a qubit depolarizing channel

For the retention parameter

p

, the Holevo capacity is

χ^{*} (Λ_{p}) = 1 - h_{2} (\frac{1 + p}{2}) .

Every output has Von Neumann entropy at least the value on a pure input,

h_{2} ((1 + p) /2)

. Two equiprobable orthogonal pure inputs attain this minimum entropy individually and have maximally mixed average output, attaining the bound. Additivity of Holevo capacity makes the unassisted classical capacity of a quantum channel equal to this value. King's depolarizing-channel capacity paper proves additivity with an arbitrary other channel.

Product-input block bound for a qubit depolarizing channel ( $χ ({p_{x}, Φ^{\otimes n} (ρ_{x})}) \leq n [1 - h ((1 - p) /2)]$ )

For

Φ (ρ) = pρ + (1 - p) I /2

, every output qubit has Von Neumann entropy at least

h ((1 - p) /2)

. A product input therefore has output entropy at least

nh ((1 - p) /2)

. The average output has entropy at most

n

, so its Holevo quantity is bounded by the displayed expression. This allows correlated classical message labels and joint output measurements, but requires product quantum inputs. The Holevo bound and Fano's inequality turn it into the corresponding asymptotic rate bound.

Phase-flip channel

The phase-flip channel is

Λ_{p} (ρ) = (1 - p) ρ + p σ_{z} ρ σ_{z}

. It multiplies the

x

and

y

components of the Bloch vector by

1 - 2 p

and leaves the

z

component fixed.

Iterated phase-flip channel ( $ϵ_{n} = [1 - (1 - 2 ϵ)^{n}] /2$ )

Independent successive applications of the phase-flip channel

D_{ϵ} (ρ) = (1 - ϵ) ρ + ϵ ZρZ

preserve diagonal entries and multiply off-diagonal entries by

(1 - 2 ϵ)^{n}

. Hence

D_{ϵ}^{n} = D_{ϵ_{n}}

with the displayed parameter, for nonnegative integers

n

. If

0 < ϵ < 1/2

, the limit is the completely dephasing channel, equivalently a nonselective projective measurement in the computational basis. A time-interval law can be iterated this way only under the memoryless composition assumption.

Dephasing channel

A dephasing channel suppresses off-diagonal entries of a density matrix in a preferred basis while preserving quantum state populations. Complete dephasing is identical to a projective measurement in that basis with its outcome discarded.

For the Bell correlations versus a classical mixture example, completely dephasing Alice's qubit in the computational basis maps

ρ_{Bell}

ρ_{mix}

. It preserves their perfect

Z

agreement and each reduced density matrix, yet destroys the

X

correlation. This is why testing only populations can miss damage to a quantum memory: the stored quantum coherence may have disappeared while every preferred-basis measurement still looks correct. The joint Von Neumann entropy rises from zero to one bit, although each local entropy stays fixed.

Complex environment overlap in qubit dephasing ( $c = ⟨ R_{0} ∣ R_{1} ⟩$ )

An environment interaction preserving the computational populations multiplies a qubit's upper off-diagonal density entry by

\overline{c}

. A real phase-flip channel representation exists for every input precisely when

c

is real, with probability

(1 - c) /2

. For

c = r e^{i φ}

, the channel is a phase-flip channel of probability

(1 - r) /2

followed by the diagonal unitary operator

diag (1, e^{i φ})

. The coherent phase cannot be discarded just because the populations are unchanged.

Unital quantum channel

A quantum channel is unital when

T (I) = I

. Every random unitary channel is unital, while the converse fails in dimension at least three.

Eigenvalue mixing matrix of a unital quantum channel ( $A_{ji} = ⟨ v_{j} ∣Λ (∣ u_{i} ⟩ ⟨ u_{i} ∣) ∣ v_{j} ⟩$ )

Choose input eigenvectors

u_{i}

and output eigenvectors

v_{j}

for a unital quantum channel. Positivity makes the displayed entries nonnegative. Trace preservation makes every column sum one; unitality makes every row sum one. Thus

A

is a doubly stochastic matrix and the output spectrum satisfies

s = A r

. The mixing matrix need not be a matrix of squared overlaps of a single unitary, even though its row and column constraints are the same.

Spectral majorization under a unital quantum channel ( $λ (Λ (ρ)) ≺ λ (ρ)$ )

A unital quantum channel mixes its input spectrum with a doubly stochastic matrix, so its output spectrum satisfies the majorization relation with its input spectrum. The reverse inequality is generally false: a pure state can be sent to a maximally mixed state. The direction expresses the fact that unital noise can make eigenvalues more uniform. Equality of the two spectra is possible, for example under a unitary channel.

Werner–Holevo channel

The Werner–Holevo channel is

T (ρ) = (Tr (ρ) I - ρ^{T}) / (d - 1)

. Its normalized Choi matrix is the maximally mixed state on the antisymmetric subspace.

V = \sum_{i} K_{i} \otimes ∣ i ⟩

followed by tracing out the environment.

Entanglement fidelity ( $F_{e} (ρ, Λ)$ )

For a purification

∣ Ψ_{ρ} ⟩_{R A}

, the entanglement fidelity of a channel

Λ

F_{e} (ρ, Λ) = ⟨ Ψ_{ρ} ∣ (id_{R} \otimes Λ) (∣ Ψ_{ρ} ⟩ ⟨ Ψ_{ρ} ∣) ∣ Ψ_{ρ} ⟩ .

It is independent of the chosen purification.

Entanglement fidelity bound by state fidelity ( $F_{e} (ρ, Λ) \leq F (ρ, Λ (ρ))^{2}$ )

In the unsquared quantum fidelity convention, the fidelity between a purification and its channel output is the square root of entanglement fidelity. Tracing out the purifying reference leaves the input and output marginals. Monotonicity of quantum fidelity under partial trace therefore gives the bound. Preserving an input marginal is a weaker requirement than preserving its correlations with a reference.

Quantum Fano inequality ( $S (σ) \leq h (F_{e}) + (1 - F_{e}) lo g_{2} (d^{2} - 1)$ )

For a quantum channel on a

d \geq 2

dimensional system, purify the input to

∣Ψ ⟩_{QR}

and set

σ = (N \otimes id) (∣Ψ ⟩ ⟨ Ψ∣)

. Its entanglement fidelity

F_{e} = ⟨ Ψ∣ σ ∣Ψ ⟩

satisfies

S (σ) \leq h (F_{e}) + (1 - F_{e}) lo g_{2} (d^{2} - 1)

. This is the entropy bound from overlap with a pure state in dimension

d^{2}

. For a trivial one-dimensional system the output entropy is zero.

Kraus formula for entanglement fidelity ( $F_{e} (ρ, Λ) = \sum_{a} ∣ Tr (ρ K_{a}) ∣^{2}$ )

{K_{a}}

is any Kraus representation of a quantum channel

Λ

, then

F_{e} (ρ, Λ) = \sum_{a} ∣ Tr (ρ K_{a}) ∣^{2} .

For a purification of a density operator, each branch overlap is

⟨ Ψ_{ρ} ∣ (I \otimes K_{a}) ∣ Ψ_{ρ} ⟩ = Tr (ρ K_{a})

. Summing the squared branch overlaps gives entanglement fidelity.

Operation fidelity ( $F_{op} (Λ, ρ) = F_{e} (ρ, Λ)$ )

In the unsquared quantum fidelity convention, operation fidelity compares a purification of a density operator with its image under a quantum channel acting on the source subsystem:

F_{op} (Λ, ρ) = F (∣ Ψ_{ρ} ⟩ ⟨ Ψ_{ρ} ∣, (id \otimes Λ) (∣ Ψ_{ρ} ⟩ ⟨ Ψ_{ρ} ∣)) .

Its square is entanglement fidelity. Thus this convention is distinct from the squared quantity

F_{e}

; the two should not be installed as synonyms.

Lindblad equation

A Markovian continuous-time quantum channel obeys

\overset{ρ}{˙} = - i [H, ρ] + \sum_{α} (L_{α} ρ L_{α}^{†} - \frac{1}{2} {L_{α}^{†} L_{α}, ρ}) .

Lindblad dissipator ( $D [V] ρ = V ρ V^{†} - \frac{1}{2} {V^{†} V, ρ}$ )

A Lindblad dissipator is the non-Hamiltonian generator associated with a Lindblad operator

V

. It preserves the trace because cyclicity gives

Tr (V ρ V^{†}) = Tr (V^{†} V ρ)

, cancelling the two anticommutator terms. It preserves Hermiticity and satisfies

D [V] (I) = [V, V^{†}]

; it need not preserve the identity or the purity of a density operator.

Lindblad operator

The Lindblad operators

L_{α}

describe dissipative channels in a Markovian open-system evolution. Their normalization fixes the corresponding rates, and their representation is not unique.

Lindbladian

The Lindbladian

L

is the linear superoperator on density operators defined by the right-hand side of a Lindblad equation, so that

\overset{ρ}{˙} = L (ρ)

Finite-dimensional Lindbladians have stationary states ( $\exists ρ_{} : L (ρ_{}) = 0$ )

For any initial density operator, average its time-independent Lindblad equation trajectory as

\overset{ρ}{ˉ}_{T} = T^{- 1} \int_{0}^{T} ρ (t) d t

. These averages stay in the compact convex set of density operators. Along a convergent subsequence,

L (\overset{ρ}{ˉ}_{T}) = (ρ (T) - ρ (0)) / T \to 0

, so the limit is stationary. Thus a finite-dimensional physical Affine Bloch equation has at least one physical equilibrium, even though an arbitrary affine real equation can have none.

Lindbladian gap

The Lindbladian gap is the smallest positive decay rate

- Re λ

among nonzero eigenvalues

λ

of a relaxing Lindbladian. It controls the slowest asymptotic exponential approach to the stationary state.

Unique stationary state of a Lindbladian

A finite-dimensional Lindbladian converges to one stationary state for every initial state exactly when zero is a simple eigenvalue and every other eigenvalue has strictly negative real part. A trivial commutant of the Hamiltonian and all jump operators and their adjoints is a standard irreducibility criterion leading to uniqueness under the usual finite-dimensional hypotheses.

Quantum Trajectory Theory

A quantum-trajectory unravelling represents a Lindblad equation as an ensemble of stochastic pure-state histories. In one jump unravelling, a state evolves between jumps under the non-Hermitian effective Hamiltonian

H_{eff} = H - \frac{i}{2} \sum_{α} L_{α}^{†} L_{α}

, while a jump of type

α

sends it to a normalized multiple of

L_{α} ∣ ψ ⟩

Dark state of a Lindblad equation