Partial Trace

Formula

Let

$$\mathcal H_{AB} = \mathcal H_A \otimes \mathcal H_B$$

and let $X_{AB}$ be an operator on the composite space. For an orthonormal basis $\{\lvert b\rangle\}$ of $\mathcal H_B$,

$$\operatorname{Tr}_B X_{AB} = \sum_b \left( I_A\otimes\langle b\rvert \right) X_{AB} \left( I_A\otimes\lvert b\rangle \right).$$

The result is an operator on $\mathcal H_A$. For a bipartite density operator,

$$\rho_A = \operatorname{Tr}_B\rho_{AB}$$

is the reduced state of subsystem $A$.

Similarly,

$$\rho_B = \operatorname{Tr}_A\rho_{AB}.$$

At a Glance

Setting	Formula
Reduced state on $A$	$\rho_A=\operatorname{Tr}_B\rho_{AB}$
Basis definition	$\operatorname{Tr}_BX=\sum_b(I_A\otimes\langle b\rvert)X(I_A\otimes\lvert b\rangle)$
Matrix indices	$(\operatorname{Tr}_BX)_{ij}=\sum_\alpha X_{i\alpha,j\alpha}$
Product operator	$\operatorname{Tr}_B(A\otimes B)=A\,\operatorname{Tr}B$
Local expectation	$\operatorname{Tr}_{AB}[X(M_A\otimes I_B)]=\operatorname{Tr}_A[(\operatorname{Tr}_BX)M_A]$
Pure-state coefficients	$(\rho_A)_{ij}=\sum_\alpha C_{i\alpha}C_{j\alpha}^*$
Continuous kernel	$\rho_A(x,x')=\int dy\,\rho_{AB}(x,y;x',y)$
Full trace	$\operatorname{Tr}_A(\operatorname{Tr}_BX)=\operatorname{Tr}_{AB}X$

The subscript on $\operatorname{Tr}_B$ names the subsystem discarded. The output acts on the complementary subsystem.

Meaning

The partial trace discards selected tensor factors while retaining exactly the information required for observables on the factors that remain.

For every operator $M_A$ on subsystem $A$,

$$\operatorname{Tr}_{AB} \left[ \rho_{AB} (M_A\otimes I_B) \right] = \operatorname{Tr}_A \left( \rho_AM_A \right).$$

This characterizing identity is the safest way to interpret $\rho_A$: it reproduces all local probabilities and expectation values available from measurements on $A$ alone.

The reduced state does not retain all correlations with $B$. Many globally different states can have the same $\rho_A$.

Matrix-Index Formula

Choose product bases

$$\left\lbrace \lvert i\rangle_A \right\rbrace, \qquad \left\lbrace \lvert\alpha\rangle_B \right\rbrace.$$

Write

$$X_{i\alpha,j\beta} = \langle i,\alpha\rvert X_{AB} \lvert j,\beta\rangle.$$

Then

$$\left( \operatorname{Tr}_B X \right)_{ij} = \sum_\alpha X_{i\alpha,j\alpha}.$$

To trace out $B$, set the $B$ bra and ket indices equal and sum them. The $A$ indices remain as the output row and column.

Likewise,

$$\left( \operatorname{Tr}_A X \right)_{\alpha\beta} = \sum_i X_{i\alpha,i\beta}.$$

This rule is reliable only after the tensor-factor order and basis order are fixed.

Rank-One and Product Rules

For rank-one product operators,

$$\operatorname{Tr}_B \left[ \left( \lvert a\rangle \langle c\rvert \right)_A \otimes \left( \lvert b\rangle \langle d\rvert \right)_B \right] = \langle d\rvert b\rangle \lvert a\rangle \langle c\rvert.$$

For general product operators,

$$\operatorname{Tr}_B (A\otimes B) = A\,\operatorname{Tr}B.$$

Therefore, for a product density operator,

$$\operatorname{Tr}_B (\rho_A\otimes\rho_B) = \rho_A \operatorname{Tr}\rho_B = \rho_A.$$

These identities extend by linearity to sums of product operators.

Pure-State Coefficient Formula

Write a normalized bipartite pure state as

$$\lvert\Psi\rangle = \sum_{i,\alpha} C_{i\alpha} \lvert i\rangle_A \otimes \lvert\alpha\rangle_B.$$

Then

$$(\rho_A)_{ij} = \sum_\alpha C_{i\alpha} C_{j\alpha}^*.$$

If $C$ is the coefficient matrix with $A$ labels as rows and $B$ labels as columns, this is

$$\rho_A = CC^\dagger.$$

For subsystem $B$, the safest convention-independent statement is the indexed formula

$$(\rho_B)_{\alpha\beta} = \sum_i C_{i\alpha} C_{i\beta}^*.$$

Compact matrix formulas for $\rho_B$ depend on whether coefficient arrays are stored with transposed or conjugated index conventions. The indexed expression fixes the meaning.

Schmidt Form

For a pure state in Schmidt form,

$$\lvert\Psi\rangle = \sum_r \sqrt{\lambda_r} \lvert u_r\rangle_A \otimes \lvert v_r\rangle_B,$$

where

$$\lambda_r\geq0, \qquad \sum_r\lambda_r=1,$$

the reduced states are

$$\rho_A = \sum_r \lambda_r \lvert u_r\rangle \langle u_r\rvert,$$ $$\rho_B = \sum_r \lambda_r \lvert v_r\rangle \langle v_r\rvert.$$

The nonzero eigenvalues of $\rho_A$ and $\rho_B$ are therefore identical for a pure bipartite state.

The state is a product state exactly when one Schmidt coefficient equals one. It is entangled exactly when more than one Schmidt coefficient is nonzero.

Two-Qubit Block Formula

Use the basis order

$$\lvert00\rangle, \quad \lvert01\rangle, \quad \lvert10\rangle, \quad \lvert11\rangle.$$

Group a two-qubit operator into blocks labeled by subsystem $A$:

$$X_{AB} = \begin{pmatrix} B_{00}&B_{01}\\ B_{10}&B_{11} \end{pmatrix},$$

where each $B_{ij}$ is a $2\times2$ matrix acting on $B$. Then

$$\operatorname{Tr}_B X_{AB} = \begin{pmatrix} \operatorname{Tr}B_{00} & \operatorname{Tr}B_{01} \\ \operatorname{Tr}B_{10} & \operatorname{Tr}B_{11} \end{pmatrix}.$$

This rule changes appearance under a different basis ordering. Declaring the register and basis order is part of the calculation.

Continuous-Variable Form

For a two-particle density kernel

$$\rho_{AB} (x,y;x',y') = \langle x,y\rvert \rho_{AB} \lvert x',y'\rangle,$$

tracing over coordinate $y$ gives

$$\rho_A(x,x') = \int dy\, \rho_{AB} (x,y;x',y).$$

For a pure wavefunction $\Psi(x,y)$,

$$\rho_A(x,x') = \int dy\, \Psi(x,y) \Psi^*(x',y).$$

The integration measure must match the coordinate basis normalization. In curvilinear coordinates or field variables, the appropriate measure replaces $dy$.

Multiple Subsystems

For

$$\mathcal H_{ABC} = \mathcal H_A \otimes \mathcal H_B \otimes \mathcal H_C,$$

the reduced state on $A$ is

$$\rho_A = \operatorname{Tr}_{BC} \rho_{ABC}.$$

Partial traces over distinct factors commute:

$$\operatorname{Tr}_B \operatorname{Tr}_C \rho_{ABC} = \operatorname{Tr}_C \operatorname{Tr}_B \rho_{ABC}.$$

The notation

$$\operatorname{Tr}_{BC}$$

means that both $B$ and $C$ are discarded. It does not mean that the output acts on $B\otimes C$.

Algebraic Properties

The partial trace is linear:

$$\operatorname{Tr}_B (\alpha X+\beta Y) = \alpha \operatorname{Tr}_B X +\beta \operatorname{Tr}_B Y.$$

It preserves the full trace:

$$\operatorname{Tr}_A \left( \operatorname{Tr}_B X \right) = \operatorname{Tr}_{AB}X.$$

It preserves adjoints:

$$\left( \operatorname{Tr}_B X \right)^\dagger = \operatorname{Tr}_B (X^\dagger).$$

It preserves positivity:

$$X\geq0 \quad\Longrightarrow\quad \operatorname{Tr}_B X\geq0.$$

For operators $A_1$ and $A_2$ on the retained subsystem,

$$\operatorname{Tr}_B \left[ (A_1\otimes I_B) X (A_2\otimes I_B) \right] = A_1 \left( \operatorname{Tr}_B X \right) A_2.$$

A unitary acting only on the discarded subsystem leaves the reduced operator unchanged:

$$\operatorname{Tr}_B \left[ (I_A\otimes U_B) X (I_A\otimes U_B^\dagger) \right] = \operatorname{Tr}_B X.$$

As a map from operators on $AB$ to operators on $A$, the partial trace is completely positive and trace preserving. It is not unital in the dimension-independent sense:

$$\operatorname{Tr}_B I_{AB} = d_BI_A$$

when $B$ has finite dimension $d_B$.

Reduced-State Checks

If $\rho_{AB}$ is a density operator, then

$$\rho_A^\dagger=\rho_A,$$ $$\rho_A\geq0,$$ $$\operatorname{Tr}\rho_A=1.$$

For a pure joint state, local purity is

$$\gamma_A = \operatorname{Tr}(\rho_A^2).$$

Then

$$\gamma_A=1$$

exactly for a product pure state, while

$$\gamma_A<1$$

signals entanglement across the $A$–$B$ split. For a mixed joint state, local mixedness alone does not diagnose entanglement.

Bell-State Example

For

$$\lvert\Phi^+\rangle = \frac{1}{\sqrt2} \left( \lvert00\rangle +\lvert11\rangle \right),$$

the joint density operator is

$$\rho_{AB} = \frac12 \left( \lvert00\rangle\langle00\rvert +\lvert00\rangle\langle11\rvert +\lvert11\rangle\langle00\rvert +\lvert11\rangle\langle11\rvert \right).$$

Tracing out $B$, the cross terms vanish because their $B$ bra and ket are orthogonal. The remaining terms give

$$\rho_A = \frac12 \left( \lvert0\rangle\langle0\rvert +\lvert1\rangle\langle1\rvert \right) = \frac12I_A.$$

The same holds for $\rho_B$. The global state is pure, but each one-qubit reduced state is maximally mixed.

Partial Trace Is Not Projection

Projecting subsystem $B$ onto $\lvert b\rangle$ gives the unnormalized conditional operator

$$\widetilde\rho_{A\mid b} = \left( I_A\otimes\langle b\rvert \right) \rho_{AB} \left( I_A\otimes\lvert b\rangle \right).$$

The partial trace sums these terms over a complete orthonormal basis:

$$\rho_A = \sum_b \widetilde\rho_{A\mid b}.$$

One projection selects a measurement-conditioned branch. The partial trace computes the unconditional marginal and does not choose an outcome or a measurement basis.

Measuring $B$ and forgetting the outcome is a physical channel on the joint state. Computing $\operatorname{Tr}_B\rho_{AB}$ is a mathematical reduction. A trace-preserving local operation on $B$ cannot change $\rho_A$, but it can change the joint state and its correlations.

Numerical Implementation

For finite dimensions $d_A$ and $d_B$, reshape the operator into a rank-four array with logical indices

$$X_{a b,a'b'}.$$

Then contract the two $B$ indices:

$$(\rho_A)_{aa'} = \sum_{b=0}^{d_B-1} X_{a b,a'b}.$$

A reliable implementation records:

• subsystem dimensions;
• tensor-factor order;
• basis order within each factor;
• which row and column axes correspond to the subsystem traced out.

Inferring these from the total matrix dimension is unsafe because the same total dimension can have several factorizations.

Symbols

Symbol	Mathematical type	Meaning
$\mathcal H_A$, $\mathcal H_B$	Hilbert spaces	Subsystem state spaces
$X_{AB}$	Composite operator	General input to the partial trace
$\rho_{AB}$	Positive trace-one operator	Joint state
$\rho_A$	Positive trace-one operator	Reduced state on $A$
$\operatorname{Tr}_B$	Linear map	Trace over subsystem $B$
$I_A$, $I_B$	Identity operators	Identities on specified factors
$C_{i\alpha}$	Complex coefficient	Pure-state amplitude in a product basis
$\lambda_r$	Nonnegative number	Schmidt probability
$d_B$	Positive integer	Finite dimension of subsystem $B$

The tensor-product factorization is part of the meaning. An abstract Hilbert space without a declared subsystem split does not determine a partial trace.

Units and Dimensions

Abstract density operators and the partial-trace map are dimensionless. The trace of a normalized density operator is dimensionless and equals one.

Coordinate kernels can carry units inherited from their normalization measure. In

$$\rho_A(x,x') = \int dy\, \rho_{AB}(x,y;x',y),$$

the units of $dy$ cancel the $B$-coordinate density units so the reduced kernel has the correct $A$-space normalization.

For a finite-dimensional identity,

$$\operatorname{Tr}_B I_{AB} = d_BI_A,$$

so tracing an unnormalized operator can introduce a dimension factor. Trace preservation means preservation of the scalar full trace, not preservation of the identity operator.

Assumptions

• A tensor-product factorization into retained and discarded subsystems is specified.
• The basis used in the sum is complete and orthonormal.
• The operator and basis use the same tensor-factor and basis ordering.
• Density operators are positive and trace one.
• Infinite-dimensional inputs are trace class when a trace-class output is required.
• Continuous-variable formulas use the correct measure and kernel normalization.
• Local-observable identities use operators supported on the retained subsystem.

Validity and Limitations

The partial trace applies to any trace-class operator on a declared tensor-product Hilbert space. In finite dimensions it is always defined for matrices of the correct composite size.

The reduced state:

• determines all local statistics on the retained subsystem;
• does not determine joint correlations;
• does not identify a unique global state;
• does not by itself distinguish classical from quantum correlations for a mixed joint state;
• depends on the chosen subsystem factorization;
• is not a conditional state unless a measurement outcome and normalization are supplied.

For quantum field theory and gauge theories, Hilbert-space factorization across spatial regions can fail or require extra structure. The elementary tensor-product formula should not be transferred to those settings without checking the operator-algebraic framework.

Calculation Checks

• The output dimension after tracing $B$ must be $d_A\times d_A$.
• The output trace must equal the input trace.
• Hermiticity and positivity must be preserved.
• A product state must reduce to its retained factor.
• Results computed in two different orthonormal bases of $B$ must agree.
• Local expectations from $\rho_A$ must match those from $\rho_{AB}$ with $M_A\otimes I_B$.
• For a pure bipartite state, $\rho_A$ and $\rho_B$ must have the same nonzero eigenvalues.
• Tensor reshaping and contractions must respect register ordering.
• Small negative eigenvalues at numerical roundoff scale should shrink with increased precision; large negative eigenvalues indicate an error.

Derivation and Canonical Home

Partial Trace owns the basis-independence proof, index rules, block formulas, computational implementation, and worked reductions.

Reduced Density Operators owns the subsystem interpretation. Local Measurement Statistics owns the operational characterization, and Schmidt Decomposition owns the pure-state spectral structure.

Worked Examples

• Density-Matrix Examples
• Partial-Trace Exercises
• Bell States
• Product States
• Position-Space Two-Particle States

Common Mistakes

• Tracing over the subsystem that should have been retained.
• Forgetting that the trace subscript names the discarded factor.
• Treating the operation as projection onto one state of the discarded subsystem.
• Deleting rows and columns instead of contracting matching subsystem indices.
• Reshaping a matrix with the wrong tensor-factor or basis order.
• Assuming a pure joint state must have pure reduced states.
• Assuming a mixed reduced state proves entanglement when the joint state is mixed.
• Expecting the reduced state to preserve joint correlations.
• Using a nonorthonormal basis in the simple basis-sum formula without a dual basis or Gram-matrix correction.
• Forgetting measure factors in continuous variables.
• Assuming the map is unital because it is trace preserving.

Related Formulas

• Density-Matrix Expectation
• Kraus Map
• Von Neumann Entropy
• Born Rule
• Expectation Value
• Tensor-Product Ordering
• Trace-Class and Hilbert–Schmidt Operators

References

• J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955.
• A. Peres, Quantum Theory: Concepts and Methods, Kluwer, 1995, ch. 5.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th anniversary ed., Cambridge University Press, 2010, sec. 2.4.
• J. Watrous, The Theory of Quantum Information, Cambridge University Press, 2018, sec. 1.1.
• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press, 2002, ch. 2.