Partial trace

In quantum mechanics and quantum information theory, the partial trace is a fundamental linear operation that reduces the density operator of a composite quantum system, defined on a tensor product of Hilbert spaces H_A \otimes H_B, to the reduced density operator of one subsystem (say, H_A) by tracing out the degrees of freedom of the other subsystem (H_B).^[1] Mathematically, for a density operator \hat{\rho}_{AB} on H_A \otimes H_B, the partial trace over B is given by \hat{\rho}_A = \operatorname{Tr}_B(\hat{\rho}_{AB}) = \sum_j \langle b_j | \hat{\rho}_{AB} | b_j \rangle, where \{ |b_j \rangle \} forms an orthonormal basis for H_B, yielding an operator acting solely on H_A that describes the local state of subsystem A.^[1] This operation preserves the trace, ensuring \operatorname{Tr}(\hat{\rho}_A) = 1 if \hat{\rho}_{AB} is normalized, and is unique in determining the marginal state of the subsystem.^[2] The partial trace plays a central role in describing open quantum systems, where interactions with an environment are modeled by tracing out the environmental degrees of freedom to obtain the effective state of the system of interest.^[2] For instance, in a bipartite system, if \hat{\rho}_{AB} = \hat{\rho}_A \otimes \hat{\rho}_B, the subsystems are uncorrelated, and the partial trace recovers the individual states; however, for entangled states, \hat{\rho}_A is generally mixed even if the global state is pure, quantifying quantum correlations.^[1] This property is crucial for entanglement theory, where the reduced density operator \hat{\rho}_A from a pure bipartite state enables computation of entanglement measures like the von Neumann entropy S(\hat{\rho}_A) = -\operatorname{Tr}(\hat{\rho}_A \log \hat{\rho}_A), which serves as an entanglement monotone for pure states.^[2] Beyond basic state reduction, the partial trace underpins key protocols in quantum information processing, such as quantum teleportation, where tracing over the sender's qubits yields the receiver's state after applying corrections, and quantum channels, modeled as \mathcal{T}^*(\rho) = \operatorname{Tr}_E [ U (\rho \otimes \rho_E) U^\dagger ] with U a unitary on system plus environment E.^[2] It also facilitates separability criteria, like the reduction criterion for detecting entanglement, and is invariant under local unitary transformations on the traced-out subsystem, ensuring consistency in local descriptions.^[2] Computationally, efficient algorithms for partial traces are vital in simulating large quantum systems, with recent advances focusing on randomized estimation to handle high-dimensional cases.^[3] Overall, the partial trace bridges global quantum dynamics to local observables, making it indispensable for theoretical and practical advancements in quantum technologies.^[1]

General definitions

Invariant definition

The partial trace over subsystem B is defined as the unique linear map \mathrm{Tr}_B: \mathrm{End}(H_A \otimes H_B) \to \mathrm{End}(H_A) that is invariant under the action of unitary operators on H_B, satisfying \mathrm{Tr}_B \bigl( (I_A \otimes U_B) \rho (I_A \otimes U_B)^\dagger \bigr) = \mathrm{Tr}_B(\rho) for all \rho \in \mathrm{End}(H_A \otimes H_B) and all unitary U_B on H_B.^[4] This formulation emphasizes the map's independence from choices of basis or inner product, relying solely on the algebraic structure of the tensor product and the representation of the unitary group.^[5] This invariant map also derives from and satisfies the full trace property, ensuring \mathrm{Tr}_A \bigl( \mathrm{Tr}_B(\rho) \bigr) = \mathrm{Tr}(\rho) for the trace \mathrm{Tr} on H_A \otimes H_B and \mathrm{Tr}_A on H_A. For an operator expressed as \rho = \sum_{i,j} |i\rangle\langle j| \otimes \sigma_{ij}, the partial trace aligns with this relation by the linearity of the trace and its cyclic invariance, confirming that the reduced description on H_A preserves the total trace when further traced.^[6] The invariant formulation of the partial trace draws from representation theory and invariant theory in Lie groups, with roots in early applications of group theory to quantum mechanics. The concept of the partial trace itself originates from John von Neumann's development of density operators.^[7] In finite-dimensional settings, the invariance condition explicitly determines the map as the average over the unitary group action with respect to the Haar measure, projecting onto the trivial representation component in the decomposition of the space of linear maps under the induced group representation. This averaging procedure uniquely identifies the partial trace among all possible linear maps, coinciding with the standard reduction in quantum subsystems.^[8]

Category-theoretic formulation

In category theory, the partial trace arises as a structure in symmetric monoidal categories, generalizing the notion of tracing out a subsystem while preserving the monoidal tensor product. Specifically, for objects A and B, the partial trace is defined as a family of maps \mathrm{Tr}_B: \mathrm{Hom}(A \otimes B, A \otimes B) \to \mathrm{Hom}(A, A), forming a natural transformation in A that satisfies compatibility with the monoidal structure and dagger involution in the case of dagger categories. This formulation captures the partial trace as the unique such transformation ensuring the category supports feedback-like operations akin to quantum decoherence, axiomatized within the framework of traced monoidal categories where the trace acts on endomorphisms of tensor products.^[9]^[10] The key axioms include dinaturality with respect to the traced subsystem B. For a morphism f: X \otimes B' \to A' \otimes B and g: B \to B', the partial trace satisfies \mathrm{Tr}_{B'}(f \circ (\mathrm{id}_X \otimes g)) = \mathrm{Tr}_B( (\mathrm{id}_{A'} \otimes g) \circ f ), ensuring consistency under substitutions in the traced factor while maintaining naturality in the retained factor A. Additional axioms enforce vanishing conditions (e.g., \mathrm{Tr}_B( u \otimes v ) = 0 if u or v factors through the zero object) and supervenience (compatibility with direct sums), making the partial trace the maximal such structure in additive categories. These properties generalize the linear algebraic partial trace to arbitrary symmetric monoidal settings, with the dagger structure in dagger-compact categories ensuring hermiticity preservation, \mathrm{Tr}_B(f^\dagger) = (\mathrm{Tr}_B(f))^\dagger.^[9]^[11] In compact closed categories, the partial trace acquires a concrete realization via the dual pairings of coevaluation \eta_B: I \to B \otimes B^* and evaluation \mathrm{ev}_B: B^* \otimes B \to I. For an endomorphism f: A \otimes B \to A \otimes B, it is given by composing f with these maps to "cap off" the B-leg, yielding

\mathrm{Tr}_B(f) = ( \mathrm{id}_A \otimes \mathrm{ev}_B ) \circ ( f \otimes \mathrm{id}_{B^*} ) \circ ( \mathrm{id}_A \otimes \eta_B ),

up to braiding if non-symmetric. A pivotal property follows: \mathrm{Tr}_B( \mathrm{id}_A \otimes \mathrm{ev}_B ) = \dim(B) \, \mathrm{id}_A, where \dim(B) is the categorical dimension, reflecting the trace of the identity on B. This construction is unique in rigid symmetric monoidal categories like finite-dimensional Hilbert spaces.^[10]^[12] As an example, consider the category \mathrm{FinVect}_\mathbb{C} of finite-dimensional complex vector spaces and linear maps, which is dagger-compact. Here, the partial trace \mathrm{Tr}_B recovers the standard linear algebraic operation: choosing bases \{e_i\} for A and \{f_j\} for B, the matrix elements of \mathrm{Tr}_B(f) are

\sum_k _{ik,kj}

, effectively summing over the B-basis indices, thus aligning the categorical abstraction with concrete computations in quantum mechanics.^[12]

Partial trace in Hilbert spaces

Definition for operators

In the context of Hilbert spaces, the partial trace builds upon the full trace operation, which for a bounded operator \rho on a Hilbert space H with orthonormal basis \{\psi_i\} is defined as \operatorname{Tr}(\rho) = \sum_i \langle \psi_i | \rho | \psi_i \rangle, yielding a scalar that is independent of the basis choice.^[13] For a composite system with Hilbert space H_A \otimes H_B, the partial trace over the subsystem B, denoted \operatorname{Tr}_B, is defined for a bounded operator \rho \in \mathcal{B}(H_A \otimes H_B) as

\operatorname{Tr}_B(\rho) = \sum_k (I_A \otimes \langle \phi_k |) \, \rho \, (I_A \otimes | \phi_k \rangle),

where \{ \phi_k \} is an orthonormal basis of H_B and I_A is the identity operator on H_A.^[13] This operation produces a bounded operator on H_A alone, independent of the choice of basis for H_B, and for density operators \rho (positive trace-class operators with \operatorname{Tr}(\rho) = 1), \operatorname{Tr}_B(\rho) yields the reduced density operator describing the local state on subsystem A.^[13] The partial trace preserves key structural properties of the input operator relevant to quantum mechanics. If \rho is Hermitian, then \operatorname{Tr}_B(\rho) is also Hermitian.^[13] In the special case where \dim(H_A) = 1, the partial trace reduces to the full trace \operatorname{Tr}(\rho_{AB}), yielding a scalar, and applying the trace to this scalar (as the operator on the 1D space \mathbb{C}) reproduces the same scalar. In general, the partial trace acts as a projection onto the subspace of operators supported solely on H_A, effectively discarding information about correlations with B while retaining the marginal on A.^[13]

Computation methods

In finite-dimensional Hilbert spaces, the partial trace over subsystem B of a density operator \rho on \mathcal{H}_A \otimes \mathcal{H}_B can be computed explicitly using its matrix representation in a product basis \{|i\rangle_A\} \otimes \{|m\rangle_B\}. Expanding \rho = \sum_{i,j,m,n} \rho_{ijmn} |i\rangle\langle j|_A \otimes |m\rangle\langle n|_B, the partial trace is given by

\operatorname{Tr}_B(\rho) = \sum_{i,j} \left( \sum_m \rho_{ijmm} \right) |i\rangle\langle j|_A,

which corresponds to summing the diagonal blocks of the coefficient matrix corresponding to the traced-out basis indices.\ An alternative computational approach leverages vectorization for efficient numerical implementation, though direct summation over the basis indices is often straightforward.\ For illustration, consider the density operator \rho = |\Phi^+\rangle\langle\Phi^+| of the Bell state |\Phi^+\rangle = \frac{1}{\sqrt{2}} (|00\rangle + |11\rangle), which has matrix elements \rho_{0000} = \rho_{1111} = \frac{1}{2} and \rho_{0011} = \rho_{1100} = \frac{1}{2} (with other off-diagonals zero in the reduced). Tracing over the second qubit yields \operatorname{Tr}_2(\rho) = \frac{1}{2} (|0\rangle\langle 0| + |1\rangle\langle 1|), the maximally mixed state on the first qubit, obtained by summing \sum_m \rho_{ijmm}: for i=j=0, \rho_{0000} + \rho_{0101} = \frac{1}{2} + 0 = \frac{1}{2}; similarly for i=j=1; off-diagonals vanish.^[13] In infinite-dimensional systems, such as continuous-variable quantum optics, the partial trace over a continuous-degree-of-freedom subsystem B requires integration using a resolution of the identity. For a complete orthonormal basis \{|\phi\rangle\} (or overcomplete frame) resolving the identity as \int |\phi\rangle\langle \phi| \, d\mu(\phi) = I_B, the partial trace is

\operatorname{Tr}_B(\rho) = \int \langle \phi | \rho | \phi \rangle \, d\mu(\phi),

provided \rho is trace-class and the integral converges in the trace norm to ensure well-definedness.\

Properties and connections

Relation to invariant integration

The partial trace operation admits a formulation in terms of invariant integration over a compact group acting on the traced subsystem. Consider a bipartite Hilbert space \mathcal{H} = \mathcal{H}_A \otimes \mathcal{H}_B and a density operator \rho \in \mathcal{B}(\mathcal{H}). When the group G is the unitary group on \mathcal{H}_B, the partial trace \operatorname{Tr}_B(\rho) corresponds to the conditional expectation onto the fixed-point subalgebra \{ \sigma \otimes I_B \mid \sigma \in \mathcal{B}(\mathcal{H}_A) \} under the adjoint action \alpha_g(X) = (I_A \otimes U_g) X (I_A \otimes U_g)^\dagger, where U_g is the unitary representation of g \in G. This expectation is uniquely given by the invariant average

E(\rho) = \int_G (I_A \otimes U_g) \rho (I_A \otimes U_g)^\dagger \, d\mu(g),

with \mu the normalized Haar measure on G (so \mu(G) = 1), yielding E(\rho) = [\operatorname{Tr}_B(\rho)] \otimes I_B / \dim \mathcal{H}_B. Identifying the fixed-point subalgebra with \mathcal{B}(\mathcal{H}_A) via the canonical isomorphism recovers \operatorname{Tr}_B(\rho). This construction leverages the Peter–Weyl theorem, which decomposes the regular representation of a compact group into irreducible components and characterizes the projection onto invariants (the trivial representation) via Haar averaging. For an irreducible representation \pi of G on \mathcal{H}_B, the integral \int_G \pi(g) T \pi(g)^\dagger \, d\mu(g) projects any operator T onto the invariant subspace; if \pi contains no trivial component, the projection vanishes. In the partial trace context, this ensures the map isolates the G-invariant part of \rho, projecting away non-invariant correlations in the B subsystem. An illustrative example arises in SU(2)-invariant spin systems, where subsystems transform under irreducible representations labeled by spin quantum numbers j. Tracing over a higher-spin subsystem (representation of dimension $2j_B + 1) in a total state from the coupled representation space yields a reduced density operator on the lower-spin subsystem that is SU(2)-invariant, i.e., proportional to the identity operator in the remaining representation space of dimension $2j_A + 1 < 2j_B + 1. For instance, in the maximally entangled state within the highest total spin J = j_A + j_B, the partial trace over the higher spin j_B produces the maximally mixed state on j_A, embodying the invariant subspace under joint SU(2) rotations. The uniqueness of this invariant map follows from the general theory of conditional expectations for minimal actions of compact groups on von Neumann algebras: the Haar average provides the unique trace-preserving projection onto the fixed-point algebra, ensuring the partial trace is the sole G-equivariant extension preserving the trace on invariants.

As a quantum channel

In quantum information theory, the partial trace over subsystem B, denoted \mathrm{Tr}_B: \mathcal{L}(\mathcal{H}_A \otimes \mathcal{H}_B) \to \mathcal{L}(\mathcal{H}_A), acts as a quantum channel, mapping density operators on the composite Hilbert space \mathcal{H}_A \otimes \mathcal{H}_B to reduced density operators on \mathcal{H}_A. This map is completely positive and trace-preserving (CPTP), representing the most general form of a physically realizable quantum operation that discards information about subsystem B without introducing correlations.^[14] The CPTP nature arises from its Kraus operator representation. Let \{|k\rangle\} be an orthonormal basis for \mathcal{H}_B. The Kraus operators are E_k = I_A \otimes \langle k|_B for k = 1, \dots, d_B, where d_B = \dim \mathcal{H}_B and I_A is the identity on \mathcal{H}_A. The action is then

\mathrm{Tr}_B(\rho_{AB}) = \sum_k E_k \rho_{AB} E_k^\dagger,

with the dagger denoting the Hermitian adjoint. The completeness relation \sum_k E_k^\dagger E_k = I_A \otimes \sum_k |k\rangle\langle k|_B = I_{AB} ensures trace preservation, as \mathrm{Tr}_A[\mathrm{Tr}_B(\rho_{AB})] = \mathrm{Tr}_{AB}(\rho_{AB}) for any trace-class operator \rho_{AB}. This form directly follows from the definition of the partial trace as a sum over basis projections.^[14] Complete positivity can be verified using the Choi-Jamiołkowski isomorphism, where the corresponding Choi matrix is positive semidefinite, as expected for any CP map. Trace preservation corresponds to the condition that the partial trace over the reference system of the Choi matrix yields the identity on the output space. The adjoint channel \mathrm{Tr}_B^*, defined in the Heisenberg picture via the Hilbert-Schmidt inner product \mathrm{Tr}(Y_A \cdot \mathrm{Tr}_B(\rho_{AB})) = \mathrm{Tr}((\mathrm{Tr}_B^*(Y_A)) \rho_{AB}), is given by \mathrm{Tr}_B^*(Y_A) = Y_A \otimes I_B. This unital map (\mathrm{Tr}_B^*(I_A) = I_{AB}) evolves observables backward. Equivalently, the predual channel from \mathcal{L}(\mathcal{H}_A) to \mathcal{L}(\mathcal{H}_A \otimes \mathcal{H}_B) that tensor-products with the maximally mixed state \pi_B = I_B / d_B on B has adjoint d_B \cdot \mathrm{Tr}_B, linking the two via normalization.^[14]

Comparisons and applications

Classical analog

In classical probability theory, the partial trace finds a direct analog in the process of marginalization, where one obtains the marginal probability distribution for a subset of variables from a joint distribution. For a joint probability distribution p(a, b) over discrete variables a and b, the marginal p(a) is given by summing over all possible outcomes of b:

p(a) = \sum_b p(a, b).

This operation discards information about b while preserving the normalization and positivity of the resulting distribution. Similarly, in quantum mechanics, the partial trace over subsystem B, denoted \mathrm{Tr}_B(\rho_{AB}), reduces the bipartite density operator \rho_{AB} on \mathcal{H}_A \otimes \mathcal{H}_B to the reduced density operator \rho_A on \mathcal{H}_A, effectively tracing out the degrees of freedom associated with B. This analogy highlights how both operations yield a description of one subsystem by integrating out the other, maintaining key probabilistic features like normalization (\mathrm{Tr}(\rho_A) = 1) and positivity (\rho_A \geq 0).^[15] A fundamental difference arises from the non-commutativity inherent in quantum operators, which has no counterpart in classical marginalization. In the classical case, the summation is fully commutative—the order of integrating out variables does not affect the result, as probabilities are scalars that commute under addition and multiplication. In contrast, quantum operators generally do not commute ([A, B] \neq 0 for many pairs), so the partial trace must respect the algebraic structure of the tensor product Hilbert space, leading to operations that are sensitive to basis choices and operator ordering. This non-commutativity implies that there is no straightforward classical equivalent to the completely positive (CP) property required for quantum maps like the partial trace, as classical probability operations are inherently commutative and positivity-preserving without needing the "complete" extension to tensor products.^[15] Consider an example with binary variables, akin to classical bits. For two classical bits with joint probabilities p(0,0) = 0.3, p(0,1) = 0.2, p(1,0) = 0.4, p(1,1) = 0.1, the marginal for the first bit is p(0) = 0.3 + 0.2 = 0.5 and p(1) = 0.4 + 0.1 = 0.5, preserving the probabilistic interpretation directly. In the quantum case, for a two-qubit density operator, the partial trace yields a reduced single-qubit density matrix whose eigenvalues correspond to these marginal probabilities in the computational basis (e.g., 0.5 and 0.5 for a maximally mixed state), but the eigenvectors define the specific quantum basis in which the state is diagonal, reflecting additional coherent structure absent in the classical probabilities.^[16] One limitation of the analogy is that classical marginalization always produces a valid probability distribution (non-negative and normalized), whereas in quantum mechanics, the partial trace, while positivity-preserving for density operators, requires the complete positivity of the map to ensure physical validity across all input states, a condition without a direct classical parallel due to the lack of non-commutativity.^[17]

Role in quantum information

In quantum information theory, the partial trace is essential for deriving the reduced density operator of a subsystem within a larger composite quantum system. For a bipartite density operator \rho_{AB} acting on \mathcal{H}_A \otimes \mathcal{H}_B, the reduced density operator for subsystem A is \rho_A = \Tr_B(\rho_{AB}), which fully describes the local quantum state of A as perceived by measurements confined to that subsystem alone.^[18] This operation discards information about correlations with B, yet preserves all locally observable statistics. The von Neumann entropy of the reduced state satisfies S(\rho_A) \leq S(\rho_{AB}), with equality holding if and only if \rho_{AB} is a product state, quantifying the information loss due to tracing out the second subsystem.^[18] The partial trace plays a pivotal role in entanglement detection and characterization. For entangled states, performing the partial trace over one subsystem yields a mixed reduced density operator, indicating the presence of nonlocal correlations. In contrast, separable states produce reduced operators that mirror the local marginals without added mixture from entanglement. A canonical example is the Bell state |\Phi^+\rangle = \frac{1}{\sqrt{2}} (|00\rangle + |11\rangle), whose density operator \rho = |\Phi^+\rangle\langle \Phi^+| satisfies \rho_A = \Tr_B(\rho) = \frac{I}{2}, the maximally mixed state on a qubit, with entropy S(\rho_A) = 1.^[18] In open quantum systems, the partial trace models interactions with an uncontrollable environment, enabling the study of decoherence and dissipation. The reduced state of the system S evolves as \rho_S(t) = \Tr_E \left( U(t) \rho_{SE}(0) U^\dagger(t) \right), where U(t) governs the unitary dynamics of the joint system-environment Hilbert space and \rho_{SE}(0) is the initial composite state; this traces out environmental degrees of freedom, resulting in effective non-unitary evolution for \rho_S that captures loss of coherence and entanglement with the bath. Post-2020 advancements in quantum networks have leveraged the partial trace for managing multipartite states across distributed nodes. In protocols for generating and verifying genuine multipartite entanglement, subsystem tracing isolates reduced density operators to assess local fidelities and entanglement witnesses, facilitating scalable quantum communication and sensing applications without full state tomography.^[19]