Stinespring Dilation

Stinespring dilation is the channel-side counterpart of the indirect-measurement idea. It says that every completely positive map can be represented by embedding the system into a larger Hilbert space, evolving there, and then ignoring an environment.

For the quantum-channel working form, examples, and model-selection cautions, see Stinespring Representation. This page keeps the theorem and measurement-side dilation perspective as the canonical home.

For a quantum channel, the operational slogan is:

$$\text{channel} \quad=\quad \text{unitary system-environment dynamics} \quad+\quad \text{discarded environment}.$$

This theorem is one of the main bridges between generalized measurements, quantum channels, decoherence, and open-system dynamics.

Finite-Dimensional Statement

Let

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

be a completely positive map. In finite dimensions, there exists an auxiliary Hilbert space $\mathcal H_E$ and a linear operator

$$V:\mathcal H_{\mathrm{in}} \to \mathcal H_{\mathrm{out}}\otimes\mathcal H_E$$

such that

$$\Phi(\rho) = \operatorname{Tr}_E \!\left[ V\rho V^\dagger \right].$$

If $\Phi$ is trace preserving, then $V$ is an isometry:

$$V^\dagger V=I_{\mathrm{in}}.$$

If $\Phi$ is trace nonincreasing, then

$$V^\dagger V\le I_{\mathrm{in}}.$$

The trace-preserving case is the usual quantum-channel case. The trace-nonincreasing case describes a selected measurement outcome or a branch with postselection.

Construction from Kraus Operators

Suppose

$$\Phi(\rho) = \sum_{\alpha=1}^r K_\alpha\rho K_\alpha^\dagger,$$

where

$$K_\alpha: \mathcal H_{\mathrm{in}} \to \mathcal H_{\mathrm{out}}.$$

Let the environment have orthonormal basis

$$\{\lvert\alpha\rangle_E\}_{\alpha=1}^r.$$

Define

$$V\lvert\psi\rangle = \sum_{\alpha=1}^r K_\alpha\lvert\psi\rangle \otimes \lvert\alpha\rangle_E.$$

Then

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

Therefore, if $\Phi$ is trace preserving,

$$\sum_\alpha K_\alpha^\dagger K_\alpha=I_{\mathrm{in}},$$

and $V$ is an isometry.

Tracing out the environment recovers the Kraus representation:

$$\begin{aligned} \operatorname{Tr}_E \!\left[ V\rho V^\dagger \right] &= \operatorname{Tr}_E \!\left[ \sum_{\alpha,\beta} K_\alpha\rho K_\beta^\dagger \otimes \lvert\alpha\rangle\langle\beta\rvert \right]\\ &= \sum_\alpha K_\alpha\rho K_\alpha^\dagger\\ &= \Phi(\rho). \end{aligned}$$

This construction is the simplest finite-dimensional proof of the theorem.

Unitary Form

When the map is trace preserving, the isometry can be realized by a unitary on a larger space. In the common case $\mathcal H_{\mathrm{in}}=\mathcal H_{\mathrm{out}}=\mathcal H_S$, prepare an environment in a fixed state $\lvert0\rangle_E$ and choose a unitary $U$ such that

$$U(\lvert\psi\rangle_S\otimes\lvert0\rangle_E) = V\lvert\psi\rangle_S$$

for every $\lvert\psi\rangle_S$. Then

$$\Phi(\rho) = \operatorname{Tr}_E \!\left[ U(\rho\otimes\lvert0\rangle\langle0\rvert_E) U^\dagger \right].$$

This is the standard reduced-dynamics formula for a system initially uncorrelated with its environment. The theorem says that every channel has such a representation, although the environment in a representation need not be a literal microscopic bath.

Physical Interpretation

The environment basis label $\alpha$ can be read as a noise alternative:

$$K_\alpha \quad\leftrightarrow\quad \text{environment record }\alpha.$$

If the environment is ignored, the alternatives add incoherently:

$$\Phi(\rho) = \sum_\alpha K_\alpha\rho K_\alpha^\dagger.$$

If the environment is measured, one obtains an outcome-resolved instrument:

$$\mathcal I_\alpha(\rho) = K_\alpha\rho K_\alpha^\dagger.$$

Changing the environment measurement basis mixes the Kraus operators and gives a different unraveling of the same channel. This is the physical meaning of Kraus nonuniqueness.

Minimal Environment Dimension

The construction above uses one environment basis state per Kraus operator. A nonminimal Kraus representation therefore gives a nonminimal dilation.

In finite dimensions, the smallest environment dimension needed for a Stinespring dilation of a channel equals the Kraus rank:

$$\dim\mathcal H_E^{\min} = \operatorname{rank}J_\Phi,$$

where $J_\Phi$ is the Choi matrix of the channel. This number is independent of the chosen Kraus representation.

For examples:

• a unitary channel needs a one-dimensional environment;
• a qubit amplitude-damping channel with $0\lt p\lt1$ has Kraus rank $2$;
• a fully depolarizing qudit channel has Kraus rank $d^2$.

Example: Dephasing from an Environment Record

Let a qubit interact with an environment so that

$$\lvert0\rangle\lvert e_0\rangle \mapsto \lvert0\rangle\lvert e_0'\rangle, \qquad \lvert1\rangle\lvert e_0\rangle \mapsto \lvert1\rangle\lvert e_1'\rangle.$$

For an input density matrix

$$\rho = \begin{pmatrix} \rho_{00} & \rho_{01}\\ \rho_{10} & \rho_{11} \end{pmatrix},$$

tracing out the environment gives

$$\rho' = \begin{pmatrix} \rho_{00} & \gamma\rho_{01}\\ \gamma^*\rho_{10} & \rho_{11} \end{pmatrix},$$

where

$$\gamma = \langle e_1'|e_0'\rangle.$$

When the environment states are orthogonal, $\gamma=0$, and the environment has a perfect record of the qubit basis state. When they are nearly identical, $|\gamma|\approx1$, little information has leaked and little dephasing occurs.

This is the same structural calculation behind decoherence.

Example: Amplitude Damping

For a zero-temperature decay channel with decay probability $p$, use an environment with states $\lvert0\rangle_E$ and $\lvert1\rangle_E$ and define

$$\begin{aligned} U\lvert0\rangle_S\lvert0\rangle_E &= \lvert0\rangle_S\lvert0\rangle_E,\\ U\lvert1\rangle_S\lvert0\rangle_E &= \sqrt{1-p}\, \lvert1\rangle_S\lvert0\rangle_E + \sqrt p\, \lvert0\rangle_S\lvert1\rangle_E. \end{aligned}$$

Tracing out $E$ gives the channel with Kraus operators

$$K_0 = \begin{pmatrix} 1 & 0\\ 0 & \sqrt{1-p} \end{pmatrix}, \qquad K_1 = \begin{pmatrix} 0 & \sqrt p\\ 0 & 0 \end{pmatrix}.$$

The environment state $\lvert1\rangle_E$ records that a quantum of energy was emitted. If that record is ignored, the system undergoes amplitude damping. If it is monitored, the same dilation supports a quantum-jump description.

Example: Random Unitary Noise

A random unitary channel has the form

$$\Phi(\rho) = \sum_j p_j U_j\rho U_j^\dagger.$$

A Stinespring isometry is

$$V\lvert\psi\rangle = \sum_j \sqrt{p_j}\, U_j\lvert\psi\rangle \otimes \lvert j\rangle_E.$$

The environment can be interpreted as a classical register storing which unitary occurred. Tracing out that register gives the random unitary mixture.

Not every channel is random unitary. Stinespring dilation is more general because the environment can become genuinely entangled with the system rather than only store a classical random label.

Relation to Naimark Dilation

Naimark dilation starts with a POVM and represents outcome probabilities as a projective measurement on a larger Hilbert space:

$$F_m=V^\dagger\Pi_mV.$$

Stinespring dilation starts with a completely positive map and represents the state transformation as an isometry plus a partial trace:

$$\Phi(\rho) = \operatorname{Tr}_E[V\rho V^\dagger].$$

In a full measurement model, both appear. A system couples unitarily to an apparatus and environment; a pointer may be projectively measured; ignoring or reading different parts produces channels, instruments, and POVMs.

Relation to Reduced Dynamics

The reduced-dynamics formula

$$\rho_S(t) = \operatorname{Tr}_E \!\left[ U_{SE}(t) (\rho_S(0)\otimes\rho_E) U_{SE}^\dagger(t) \right]$$

is a Stinespring-type representation when the initial state factorizes. If $\rho_E$ is mixed,

$$\rho_E=\sum_\beta q_\beta \lvert e_\beta\rangle\langle e_\beta\rvert,$$

one obtains Kraus operators

$$K_{\alpha\beta}(t) = \sqrt{q_\beta}\, \langle e_\alpha|U_{SE}(t)|e_\beta\rangle_E.$$

If the initial system-environment state is correlated, the simple channel representation for arbitrary system inputs may fail. That caveat belongs to the physics of reduced dynamics, not to a failure of Stinespring’s theorem.

See Reduced Dynamics for the system-environment setting.

Common Mistakes

Treating the dilation as unique

Stinespring dilations are not unique. Different environment dimensions, unitary extensions, and environment bases can represent the same channel.

Reifying every environment

A dilation is an exact representation of a channel, but the auxiliary environment need not be the actual microscopic bath. Additional physical evidence is needed to identify it with a real environment.

Forgetting the product initial state

The unitary formula with $\rho\otimes\lvert0\rangle\langle0\rvert$ assumes a fixed environment state independent of the input. Initial correlations require separate analysis.

Confusing trace preservation with complete positivity

The isometry condition $V^\dagger V=I$ encodes trace preservation. Complete positivity comes from the structure $V\rho V^\dagger$ followed by a partial trace.

Exercises

Verify the isometry condition

Given $V\lvert\psi\rangle=\sum_\alpha K_\alpha\lvert\psi\rangle\otimes\lvert\alpha\rangle$, show that $V^\dagger V=\sum_\alpha K_\alpha^\dagger K_\alpha$.

Solution

For arbitrary $\lvert\phi\rangle$ and $\lvert\psi\rangle$,

$$\langle\phi\rvert V^\dagger V\lvert\psi\rangle = \sum_{\alpha,\beta} \langle\phi\rvert K_\alpha^\dagger K_\beta\lvert\psi\rangle \langle\alpha|\beta\rangle.$$

Since $\langle\alpha|\beta\rangle=\delta_{\alpha\beta}$,

$$\langle\phi\rvert V^\dagger V\lvert\psi\rangle = \sum_\alpha \langle\phi\rvert K_\alpha^\dagger K_\alpha\lvert\psi\rangle.$$

Thus

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

Recover the Kraus map

Show that tracing out the environment in the same construction gives $\Phi(\rho)=\sum_\alpha K_\alpha\rho K_\alpha^\dagger$.

Solution

First write

$$V\rho V^\dagger = \sum_{\alpha,\beta} K_\alpha\rho K_\beta^\dagger \otimes \lvert\alpha\rangle\langle\beta\rvert.$$

Using

$$\operatorname{Tr}_E (\lvert\alpha\rangle\langle\beta\rvert) = \delta_{\alpha\beta},$$

we obtain

$$\operatorname{Tr}_E[V\rho V^\dagger] = \sum_\alpha K_\alpha\rho K_\alpha^\dagger.$$

Amplitude-damping probabilities

For the amplitude-damping unitary above, what is the probability that the environment is found in $\lvert1\rangle_E$ for input state $\rho$?

Solution

The environment outcome $\lvert1\rangle_E$ corresponds to the Kraus operator

$$K_1 = \begin{pmatrix} 0 & \sqrt p\\ 0 & 0 \end{pmatrix}.$$

Thus

$$p(1) = \operatorname{Tr}(K_1\rho K_1^\dagger) = \operatorname{Tr}(K_1^\dagger K_1\rho).$$

Since

$$K_1^\dagger K_1 = p\lvert1\rangle\langle1\rvert,$$

the probability is

$$p(1)=p\,\rho_{11}.$$

The environment clicks only if the system had excited-state population.

References

• W. F. Stinespring, “Positive functions on $C^*$-algebras,” Proceedings of the American Mathematical Society 6, 211–216 (1955).
• 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 (2010).
• J. Watrous, The Theory of Quantum Information, Cambridge University Press (2018).
• V. Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge University Press (2002).
• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press (2002).