Quantum Operations & CPTP Maps

Nielsen and Chuang organize quantum mechanics into four postulates that serve as the foundation for all of quantum information science. These postulates specify the state space of a quantum system, how that state evolves in time, how measurements extract classical information, and how composite systems are described.

Postulate 1 (State space): An isolated physical system is associated with a complex Hilbert space known as the state space. The system is completely described by a state vector $\lvert\psi\rangle$, which is a unit vector in the state space. Equivalently, the state can be described by a density matrix $\rho$.

Postulate 2 (Evolution): The evolution of a closed quantum system is described by a unitary transformation $\lvert\psi\rangle \mapsto U\lvert\psi\rangle$, or in the density matrix picture $\rho \mapsto U\rho U^\dagger$. The operator $U$ is unitary: $U^\dagger U = U U^\dagger = I$.

Postulate 3 (Measurement): Quantum measurements are described by a collection $\{M_m\}$ of measurement operators satisfying $\sum_m M_m^\dagger M_m = I$. The probability of outcome $m$ is $p(m) = \langle\psi\vert M_m^\dagger M_m \vert\psi\rangle$, and the post-measurement state is $M_m \lvert\psi\rangle / \sqrt{p(m)}$.

Postulate 4 (Composite systems): The state space of a composite physical system is the tensor product of the state spaces of the component systems. If system $i$ is in state $\lvert\psi_i\rangle$, the joint state is $\lvert\psi_1\rangle \otimes \lvert\psi_2\rangle \otimes \cdots \otimes \lvert\psi_n\rangle$.

• • State space: Hilbert space, unit vectors or density matrices.
• • Evolution: unitary transformations preserve inner products.
• • Measurement: operators $\{M_m\}$ with completeness relation.
• • Composite systems: tensor product structure.

In realistic settings, quantum systems are rarely perfectly isolated; they interact with their environment. The most general physically allowed transformation of a quantum state is a quantum operation $\mathcal{E}$ that maps an input state $\rho$ to an output state $\mathcal{E}(\rho)$. This framework encompasses unitary evolution, measurements, and noise processes.

A quantum operation must satisfy three physical conditions. First, it must be linear: $\mathcal{E}(a\rho_1 + b\rho_2) = a\mathcal{E}(\rho_1) + b\mathcal{E}(\rho_2)$. Linearity is required because quantum mechanics is a linear theory at the level of density matrices. Second, it must be completely positive: not only must $\mathcal{E}$ map positive operators to positive operators, but $\mathcal{E} \otimes I$ must do so on any extended system. This ensures that the map remains valid even when the system is entangled with an ancilla. Third, it must be trace-preserving: $\text{tr}(\mathcal{E}(\rho)) = \text{tr}(\rho) = 1$, which guarantees that probabilities sum to unity.

• • Quantum operations describe open-system dynamics.
• • Linearity is inherited from the linearity of quantum mechanics.
• • Complete positivity ensures validity on entangled states.
• • Trace preservation guarantees normalized probabilities.

A map $\mathcal{E}$ is positive if it sends positive semidefinite operators to positive semidefinite operators. However, positivity alone is insufficient. Consider a map that acts correctly on a single qubit but fails when that qubit is entangled with another. To rule out such pathological behavior, we require complete positivity.

A map $\mathcal{E}$ is completely positive (CP) if $\mathcal{E} \otimes I_k$ is positive for every integer $k \geq 1$, where $I_k$ is the identity map on a $k$-dimensional ancilla. This condition is physically motivated: if $\mathcal{E}$ describes a legitimate physical process, it must remain valid even when the system is part of a larger entangled state.

A map is trace-preserving (TP) if $\text{tr}(\mathcal{E}(\rho)) = \text{tr}(\rho)$ for all $\rho$. Together, completely positive and trace-preserving maps—CPTP maps—are the mathematical description of open quantum systems. They capture the reduced dynamics of a system that interacts unitarily with an environment and is then traced over.

• • Positivity: $\mathcal{E}(\rho) \geq 0$ whenever $\rho \geq 0$.
• • Complete positivity: $\mathcal{E} \otimes I$ is positive for all ancilla dimensions.
• • Trace preservation: $\text{tr}(\mathcal{E}(\rho)) = \text{tr}(\rho)$.
• • CPTP maps are the gold standard for physically valid quantum dynamics.

A fundamental theorem in quantum information theory states that any CPTP map $\mathcal{E}$ can be written in the Kraus operator representation:

The Kraus operators $\{E_k\}$ are not unique: different sets of Kraus operators can represent the same map. This freedom corresponds to unitary transformations on the environment. Specifically, if $\{E_k\}$ and $\{F_j\}$ are two Kraus representations of the same map, then there exists a unitary matrix $u_{kj}$ such that $F_j = \sum_k u_{jk} E_k$. This unitary freedom is sometimes called the 'non-uniqueness of the operator-sum representation'.

As an example, consider the depolarizing channel on a single qubit, which with probability $p$ replaces the state by the maximally mixed state $I/2$ and with probability $1-p$ leaves it untouched. Its Kraus operators can be chosen as proportional to the Pauli operators.

• • Every CPTP map has a Kraus representation.
• • The completeness relation $\sum_k E_k^\dagger E_k = I$ ensures trace preservation.
• • Kraus operators are not unique: unitary freedom on the environment.

Three canonical quantum channels illustrate the Kraus formalism in action. The depolarizing channel models a process that with probability $p$ completely randomizes the qubit, leaving it in the maximally mixed state $I/2$. Its action is $\mathcal{E}(\rho) = (1-p)\rho + p \frac{I}{2}$. Kraus operators are $\sqrt{1-p}\, I$ and $\sqrt{p/3}\, X$, $\sqrt{p/3}\, Y$, $\sqrt{p/3}\, Z$.

The dephasing (or phase damping) channel destroys coherence in the computational basis without changing the populations. Its action is $\mathcal{E}(\rho) = (1-p)\rho + p Z\rho Z$. The Kraus operators are $\sqrt{1-p}\, I$ and $\sqrt{p}\, Z$. After dephasing, the off-diagonal elements of $\rho$ decay, which models the loss of phase information due to interaction with the environment.

The amplitude damping channel models spontaneous emission: a two-level atom in the excited state $\lvert 1 \rangle$ can decay to the ground state $\lvert 0 \rangle$ by emitting a photon. The Kraus operators are $E_0 = \lvert 0 \rangle\langle 0 \rvert + \sqrt{1-\gamma}\lvert 1 \rangle\langle 1 \rvert$ and $E_1 = \sqrt{\gamma}\lvert 0 \rangle\langle 1 \rvert$, where $\gamma \in [0,1]$ is the decay probability. This channel irreversibly transfers population from $\lvert 1 \rangle$ to $\lvert 0 \rangle$.

• • Depolarizing: isotropic noise, Kraus operators proportional to Paulis.
• • Dephasing: loss of phase coherence, Kraus operators $I$ and $Z$.
• • Amplitude damping: spontaneous emission, Kraus operators $E_0$ and $E_1$.