Trace-Preserving and Trace-Nonincreasing Maps

Trace preservation is the probability-conservation condition for deterministic quantum operations. Trace nonincrease is the corresponding condition for selected outcomes, filters, loss branches, and postselected operations.

For a normalized state $\rho$,

$$\operatorname{Tr}\rho=1.$$

If a map is deterministic, the output should also have trace $1$. If a map describes only one possible branch, the trace of the unnormalized output is the probability of that branch.

The guiding distinction is:

$$\text{trace preserving} \quad\Longleftrightarrow\quad \text{all outcomes included},$$

while

$$\text{trace nonincreasing} \quad\Longleftrightarrow\quad \text{one selected branch or filter}.$$

Trace Preservation

Let

$$\Phi: \mathcal B(\mathcal H_{\mathrm{in}}) \to \mathcal B(\mathcal H_{\mathrm{out}})$$

be a linear map on operators. The map is trace preserving if

$$\operatorname{Tr}\Phi(X) = \operatorname{Tr}X$$

for all operators $X$ in its domain.

For normalized input states, this gives

$$\operatorname{Tr}\Phi(\rho)=1.$$

A completely positive trace-preserving map is a quantum channel. It may represent unitary evolution, noise, state preparation, discard-and-reprepare dynamics, or reduced dynamics after tracing out an environment.

Trace Nonincrease

A map $\mathcal E$ is trace nonincreasing on positive inputs if

$$X\ge0 \quad\Longrightarrow\quad \operatorname{Tr}\mathcal E(X) \le \operatorname{Tr}X.$$

For an input state $\rho$, the branch probability is

$$p_{\mathcal E}(\rho) = \operatorname{Tr}\mathcal E(\rho).$$

The conditional output state, when the branch occurs, is

$$\rho_{\mathcal E} = \frac{\mathcal E(\rho)} {\operatorname{Tr}\mathcal E(\rho)}.$$

This normalization is nonlinear in $\rho$ because it conditions on a selected event. The underlying operation $\mathcal E$ is the linear trace-nonincreasing map.

Kraus Trace Conditions

For a completely positive map with Kraus representation

$$\mathcal E(\rho) = \sum_\alpha K_\alpha\rho K_\alpha^\dagger,$$

the trace is

$$\operatorname{Tr}\mathcal E(\rho) = \operatorname{Tr} \!\left( \rho\sum_\alpha K_\alpha^\dagger K_\alpha \right).$$

Therefore $\mathcal E$ is trace preserving exactly when

$$\sum_\alpha K_\alpha^\dagger K_\alpha=I.$$

It is trace nonincreasing exactly when

$$\sum_\alpha K_\alpha^\dagger K_\alpha\le I.$$

The positive operator

$$G_{\mathcal E} = \sum_\alpha K_\alpha^\dagger K_\alpha$$

is the effect associated with the selected operation:

$$p_{\mathcal E}(\rho) = \operatorname{Tr}(\rho G_{\mathcal E}).$$

The details of Kraus representations are developed in Kraus Representation.

Adjoint-Map Condition

The adjoint map $\Phi^\dagger$ is defined by the trace pairing:

$$\operatorname{Tr} \!\left[ A\,\Phi(B) \right] = \operatorname{Tr} \!\left[ \Phi^\dagger(A)\,B \right].$$

With this convention, $\Phi$ is trace preserving exactly when

$$\Phi^\dagger(I_{\mathrm{out}}) = I_{\mathrm{in}}.$$

It is trace nonincreasing exactly when

$$\Phi^\dagger(I_{\mathrm{out}}) \le I_{\mathrm{in}}.$$

This form is often the cleanest way to remember the relation between Schrödinger-picture states and Heisenberg-picture observables:

$$\text{trace preservation of states} \quad\Longleftrightarrow\quad \text{identity preservation by the adjoint}.$$

Choi Trace Condition

With the Choi convention used in Choi Matrix, a map

$$\Phi: \mathcal B(\mathcal H_{\mathrm{in}}) \to \mathcal B(\mathcal H_{\mathrm{out}})$$

is trace preserving exactly when

$$\operatorname{Tr}_{\mathrm{out}}J_\Phi = I_{\mathrm{in}}.$$

It is trace nonincreasing exactly when

$$\operatorname{Tr}_{\mathrm{out}}J_\Phi \le I_{\mathrm{in}}.$$

Together with $J_\Phi\ge0$, these are semidefinite constraints. They are useful in channel tomography, optimization over operations, and physicality checks for fitted models.

Instruments and Missing Branches

A quantum instrument is a family of operations $\{\mathcal I_m\}$ whose sum is trace preserving:

$$\Phi = \sum_m\mathcal I_m.$$

Each individual operation is trace nonincreasing:

$$\operatorname{Tr}\mathcal I_m(\rho) = p(m|\rho).$$

The total map is trace preserving because the probabilities of all outcomes sum to one:

$$\sum_m p(m|\rho) = \operatorname{Tr} \sum_m\mathcal I_m(\rho) = 1.$$

If an analysis keeps only one outcome and renormalizes, it is doing postselection. The missing trace is not a mathematical defect; it is the probability assigned to the outcomes not retained.

Trace Preserving Is Not Unital

A map is unital if

$$\Phi(I_{\mathrm{in}}) = I_{\mathrm{out}}.$$

Trace preserving and unital are different conditions. Trace preservation says normalized states keep total probability. Unitality says the identity operator is fixed by the Schrödinger-picture map.

For a channel $\Phi$, trace preservation is equivalent to unitality of the adjoint map:

$$\Phi^\dagger(I_{\mathrm{out}}) = I_{\mathrm{in}}.$$

But $\Phi(I_{\mathrm{in}})=I_{\mathrm{out}}$ need not hold. Amplitude damping is trace preserving but not unital: it preserves total probability while relaxing excited-state population toward the ground state.

Loss: Two Equivalent Descriptions

Loss can be represented in two common ways.

First, if one keeps only the surviving subspace, the survival operation may be trace decreasing:

$$\mathcal E_{\mathrm{surv}}(\rho) = \eta\rho, \qquad 0\le\eta\le1.$$

The trace is

$$\operatorname{Tr}\mathcal E_{\mathrm{surv}}(\rho) = \eta.$$

Second, if one includes an explicit erasure flag $\lvert e\rangle$, the full map can be trace preserving:

$$\Phi(\rho) = \eta\rho \oplus (1-\eta) \operatorname{Tr}(\rho) \lvert e\rangle\langle e\rvert.$$

The two descriptions answer different modeling questions. The trace-decreasing map describes the selected survival branch. The flagged channel includes both survival and loss as outcomes. See Erasure and Loss Channels for the canonical channel-level treatment.

Common Mistakes

Normalizing selected branches too early

The map $\rho\mapsto\mathcal E(\rho)/\operatorname{Tr}\mathcal E(\rho)$ is conditional and nonlinear. The physical operation before conditioning is the trace-nonincreasing linear map $\mathcal E$.

Calling a selected outcome a channel

An individual measurement outcome is generally not trace preserving. The sum over all outcomes is the channel.

Confusing trace preserving with unital

Trace preserving keeps $\operatorname{Tr}\rho$ fixed. Unital keeps $I$ fixed. The two coincide only in special cases.

Forgetting hidden loss outcomes

A detector model that only describes successful clicks may be trace decreasing. To make a channel, include no-click, loss, or erasure outcomes.

Checking only normalized inputs

Trace conditions are linear statements. They should hold on the operator space or, at minimum, on all positive inputs, not just on one favorite normalized state.

Exercises

Kraus proof

Prove that a Kraus map

$$\mathcal E(\rho) = \sum_\alpha K_\alpha\rho K_\alpha^\dagger$$

is trace preserving if and only if $\sum_\alpha K_\alpha^\dagger K_\alpha=I$.

Solution

Using cyclicity of trace,

$$\operatorname{Tr}\mathcal E(\rho) = \sum_\alpha \operatorname{Tr}(K_\alpha\rho K_\alpha^\dagger) = \operatorname{Tr} \!\left[ \rho \sum_\alpha K_\alpha^\dagger K_\alpha \right].$$

This equals $\operatorname{Tr}\rho=\operatorname{Tr}(\rho I)$ for every $\rho$ exactly when

$$\sum_\alpha K_\alpha^\dagger K_\alpha=I.$$

Projective measurement branches

Let $\{P_m\}$ be a projective measurement. Show that each branch $\mathcal I_m(\rho)=P_m\rho P_m$ is trace nonincreasing, while $\sum_m\mathcal I_m$ is trace preserving.

Solution

For one branch,

$$\operatorname{Tr}\mathcal I_m(\rho) = \operatorname{Tr}(P_m\rho P_m) = \operatorname{Tr}(\rho P_m) \le \operatorname{Tr}\rho,$$

because $0\le P_m\le I$.

For the sum,

$$\operatorname{Tr} \sum_mP_m\rho P_m = \sum_m\operatorname{Tr}(\rho P_m) = \operatorname{Tr} \!\left( \rho\sum_mP_m \right) = \operatorname{Tr}\rho.$$

Amplitude damping is not unital

For the qubit amplitude-damping Kraus operators

$$K_0 = \lvert0\rangle\langle0\rvert + \sqrt{1-\gamma}\, \lvert1\rangle\langle1\rvert, \qquad K_1 = \sqrt{\gamma}\, \lvert0\rangle\langle1\rvert,$$

with $0\le\gamma\le1$, show that the channel is trace preserving but not unital when $\gamma\ne0$.

Solution

First,

$$K_0^\dagger K_0 = \lvert0\rangle\langle0\rvert + (1-\gamma) \lvert1\rangle\langle1\rvert,$$

and

$$K_1^\dagger K_1 = \gamma \lvert1\rangle\langle1\rvert.$$

Therefore

$$K_0^\dagger K_0+K_1^\dagger K_1=I,$$

so the channel is trace preserving.

For unitality, compute

$$\Phi(I) = K_0K_0^\dagger+K_1K_1^\dagger = (1+\gamma)\lvert0\rangle\langle0\rvert + (1-\gamma)\lvert1\rangle\langle1\rvert.$$

This equals $I$ only when $\gamma=0$.

Cross-Links

• Quantum Operations
• Completely Positive Maps
• Kraus Representation
• Choi Matrix
• Quantum Instruments
• Selective and Nonselective Measurements
• Amplitude-Damping Channel
• Common Noise Channels
• Formula Sheet
• Glossary

References

• K. Kraus, States, Effects, and Operations: Fundamental Notions of Quantum Theory, Springer (1983).
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 10th anniversary edition (2010).
• J. Watrous, The Theory of Quantum Information, Cambridge University Press (2018).
• A. S. Holevo, Quantum Systems, Channels, Information: A Mathematical Introduction, De Gruyter (2012).
• M. M. Wolf, Quantum Channels and Operations: Guided Tour, lecture notes (2012).
• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press (2002).