Partial trace

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In linear algebra and functional analysis, the partial trace is a generalization of the trace. Whereas the trace is a scalar-valued function on operators, the partial trace is an operator-valued function. The partial trace has applications in quantum information and decoherence which is relevant for quantum measurement and thereby to the decoherent approaches to interpretations of quantum mechanics, including consistent histories and the relative state interpretation.

Suppose V {\displaystyle V} , W {\displaystyle W} are finite-dimensional vector spaces over a field, with dimensions m {\displaystyle m} and n {\displaystyle n} , respectively. For any space ⁠ A {\displaystyle A} ⁠, let L ( A ) {\displaystyle L(A)} denote the space of linear operators on A {\displaystyle A} . The partial trace over W {\displaystyle W} is then written as ⁠ Tr W : L ⁡ ( V ⊗ W ) → L ⁡ ( V ) {\displaystyle \operatorname {Tr} _{W}:\operatorname {L} (V\otimes W)\to \operatorname {L} (V)} ⁠, where ⊗ {\displaystyle \otimes } denotes the Tensor Product.

It is defined as follows: For ⁠ T ∈ L ⁡ ( V ⊗ W ) {\displaystyle T\in \operatorname {L} (V\otimes W)} ⁠, let ⁠ e 1 , … , e m {\displaystyle e_{1},\ldots ,e_{m}} ⁠, and ⁠ f 1 , … , f n {\displaystyle f_{1},\ldots ,f_{n}} ⁠, be bases for V and W respectively; then T has a matrix representation

{ a k ℓ , i j } 1 ≤ k , i ≤ m , 1 ≤ ℓ , j ≤ n {\displaystyle \{a_{k\ell ,ij}\}\quad 1\leq k,i\leq m,\quad 1\leq \ell ,j\leq n}

relative to the basis e k ⊗ f ℓ {\displaystyle e_{k}\otimes f_{\ell }} of V ⊗ W {\displaystyle V\otimes W} .

Now for indices k, i in the range 1, ..., m, consider the sum

b k , i = ∑ j = 1 n a k j , i j {\displaystyle b_{k,i}=\sum _{j=1}^{n}a_{kj,ij}}

This gives a matrix b_k,i. The associated linear operator on V is independent of the choice of bases and is by definition the partial trace.

Among physicists, this is often called "tracing out" or "tracing over" W to leave only an operator on V in the context where W and V are Hilbert spaces associated with quantum systems (see below).

The partial trace operator can be defined invariantly (that is, without reference to a basis) as follows: it is the unique linear map

Tr W : L ⁡ ( V ⊗ W ) → L ⁡ ( V ) {\displaystyle \operatorname {Tr} _{W}:\operatorname {L} (V\otimes W)\rightarrow \operatorname {L} (V)}

such that

Tr W ⁡ ( R ⊗ S ) = Tr ⁡ ( S ) R ∀ R ∈ L ⁡ ( V ) ∀ S ∈ L ⁡ ( W ) . {\displaystyle \operatorname {Tr} _{W}(R\otimes S)=\operatorname {Tr} (S)\,R\quad \forall R\in \operatorname {L} (V)\quad \forall S\in \operatorname {L} (W).}

To see that the conditions above determine the partial trace uniquely, let v 1 , … , v m {\displaystyle v_{1},\ldots ,v_{m}} form a basis for V {\displaystyle V} , let w 1 , … , w n {\displaystyle w_{1},\ldots ,w_{n}} form a basis for W {\displaystyle W} , let E i j : V → V {\displaystyle E_{ij}:V\to V} be the map that sends v i {\displaystyle v_{i}} to v j {\displaystyle v_{j}} (and all other basis elements to zero), and let F k l : W → W {\displaystyle F_{kl}\colon W\to W} be the map that sends w k {\displaystyle w_{k}} to w l {\displaystyle w_{l}} . Since the vectors v i ⊗ w k {\displaystyle v_{i}\otimes w_{k}} form a basis for V ⊗ W {\displaystyle V\otimes W} , the maps E i j ⊗ F k l {\displaystyle E_{ij}\otimes F_{kl}} form a basis for L ⁡ ( V ⊗ W ) {\displaystyle \operatorname {L} (V\otimes W)} .

From this abstract definition, the following properties follow:

Tr W ⁡ ( I V ⊗ W ) = dim ⁡ W   I V {\displaystyle \operatorname {Tr} _{W}(I_{V\otimes W})=\dim W\ I_{V}}

Tr W ⁡ ( T ( I V ⊗ S ) ) = Tr W ⁡ ( ( I V ⊗ S ) T ) ∀ S ∈ L ⁡ ( W ) ∀ T ∈ L ⁡ ( V ⊗ W ) . {\displaystyle \operatorname {Tr} _{W}(T(I_{V}\otimes S))=\operatorname {Tr} _{W}((I_{V}\otimes S)T)\quad \forall S\in \operatorname {L} (W)\quad \forall T\in \operatorname {L} (V\otimes W).}

It is the partial trace of linear transformations that is the subject of Joyal, Street, and Verity's notion of Traced monoidal category. A traced monoidal category is a monoidal category ( C , ⊗ , I ) {\displaystyle (C,\otimes ,I)} together with, for objects X, Y, U in the category, a function of Hom-sets,

Tr X , Y U : Hom C ⁡ ( X ⊗ U , Y ⊗ U ) → Hom C ⁡ ( X , Y ) {\displaystyle \operatorname {Tr} _{X,Y}^{U}:\operatorname {Hom} _{C}(X\otimes U,Y\otimes U)\to \operatorname {Hom} _{C}(X,Y)}

satisfying certain axioms.

Another case of this abstract notion of partial trace takes place in the category of finite sets and bijections between them, in which the monoidal product is disjoint union. One can show that for any finite sets, X,Y,U and bijection X + U ≅ Y + U {\displaystyle X+U\cong Y+U} there exists a corresponding "partially traced" bijection X ≅ Y {\displaystyle X\cong Y} .

The partial trace generalizes to operators on infinite dimensional Hilbert spaces. Suppose V, W are Hilbert spaces, and let

{ f i } i ∈ I {\displaystyle \{f_{i}\}_{i\in I}}

be an orthonormal basis for W. Now there is an isometric isomorphism

⨁ ℓ ∈ I ( V ⊗ C f ℓ ) → V ⊗ W {\displaystyle \bigoplus _{\ell \in I}(V\otimes \mathbb {C} f_{\ell })\rightarrow V\otimes W}

Under this decomposition, any operator T ∈ L ⁡ ( V ⊗ W ) {\displaystyle T\in \operatorname {L} (V\otimes W)} can be regarded as an infinite matrix of operators on V

[ T 11 T 12 … T 1 j … T 21 T 22 … T 2 j … ⋮ ⋮ ⋮ T k 1 T k 2 … T k j … ⋮ ⋮ ⋮ ] , {\displaystyle {\begin{bmatrix}T_{11}&T_{12}&\ldots &T_{1j}&\ldots \\T_{21}&T_{22}&\ldots &T_{2j}&\ldots \\\vdots &\vdots &&\vdots \\T_{k1}&T_{k2}&\ldots &T_{kj}&\ldots \\\vdots &\vdots &&\vdots \end{bmatrix}},}

where T k ℓ ∈ L ⁡ ( V ) {\displaystyle T_{k\ell }\in \operatorname {L} (V)} .

First suppose T is a non-negative operator. In this case, all the diagonal entries of the above matrix are non-negative operators on V. If the sum

∑ ℓ T ℓ ℓ {\displaystyle \sum _{\ell }T_{\ell \ell }}

converges in the strong operator topology of L(V), it is independent of the chosen basis of W. The partial trace Tr_W(T) is defined to be this operator. The partial trace of a self-adjoint operator is defined if and only if the partial traces of the positive and negative parts are defined.

Suppose W has an orthonormal basis, which we denote by ket vector notation as ⁠ { | ℓ ⟩ } ℓ {\displaystyle \{\vert \ell \rangle \}_{\ell }} ⁠. Then

Tr W ⁡ ( ∑ k , ℓ T ( k ℓ ) ⊗ | k ⟩ ⟨ ℓ | ) = ∑ j T ( j j ) . {\displaystyle \operatorname {Tr} _{W}\left(\sum _{k,\ell }T^{(k\ell )}\,\otimes \,|k\rangle \langle \ell |\right)=\sum _{j}T^{(jj)}.}

The superscripts in parentheses do not represent matrix components, but instead label the matrix itself.

In the case of finite dimensional Hilbert spaces, there is a useful way of looking at partial trace involving integration with respect to a suitably normalized Haar measure μ over the unitary group U(W) of W. Suitably normalized means that μ is taken to be a measure with total mass dim(W).

Theorem. Suppose V, W are finite dimensional Hilbert spaces. Then

∫ U ⁡ ( W ) ( I V ⊗ U ∗ ) T ( I V ⊗ U )   d μ ( U ) {\displaystyle \int _{\operatorname {U} (W)}(I_{V}\otimes U^{*})T(I_{V}\otimes U)\ d\mu (U)}

commutes with all operators of the form I V ⊗ S {\displaystyle I_{V}\otimes S} and hence is uniquely of the form R ⊗ I W {\displaystyle R\otimes I_{W}} . The operator R is the partial trace of T.

The partial trace can be viewed as a quantum operation. Consider a quantum mechanical system whose state space is the tensor product H A ⊗ H B {\displaystyle H_{A}\otimes H_{B}} of Hilbert spaces. A mixed state is described by a density matrix ρ, that is, a non-negative trace-class operator of trace 1 on the tensor product H A ⊗ H B . {\displaystyle H_{A}\otimes H_{B}.} The partial trace of ρ with respect to the system B, denoted by ρ A {\displaystyle \rho ^{A}} , is called the reduced state of ρ on system A. In symbols,^[1] ρ A = Tr B ⁡ ρ . {\displaystyle \rho ^{A}=\operatorname {Tr} _{B}\rho .}

To show that this is indeed a sensible way to assign a state on the A subsystem to ρ, we offer the following justification. Let M be an observable on the subsystem A, then the corresponding observable on the composite system is M ⊗ I {\displaystyle M\otimes I} . However one chooses to define a reduced state ρ A {\displaystyle \rho ^{A}} , there should be consistency of measurement statistics. The expectation value of M after the subsystem A is prepared in ρ A {\displaystyle \rho ^{A}} and that of M ⊗ I {\displaystyle M\otimes I} when the composite system is prepared in ρ should be the same, i.e. the following equality should hold:

Tr A ⁡ ( M ⋅ ρ A ) = Tr ⁡ ( M ⊗ I ⋅ ρ ) . {\displaystyle \operatorname {Tr} _{A}(M\cdot \rho ^{A})=\operatorname {Tr} (M\otimes I\cdot \rho ).}

We see that this is satisfied if ρ A {\displaystyle \rho ^{A}} is as defined above via the partial trace. Furthermore, such operation is unique.

Let T(H) be the Banach space of trace-class operators on the Hilbert space H. It can be easily checked that the partial trace, viewed as a map

Tr B : T ( H A ⊗ H B ) → T ( H A ) {\displaystyle \operatorname {Tr} _{B}:T(H_{A}\otimes H_{B})\rightarrow T(H_{A})}

is completely positive and trace-preserving.

The density matrix ρ is Hermitian, positive semi-definite, and has a trace of 1. It has a spectral decomposition:

ρ = ∑ m p m | Ψ m ⟩ ⟨ Ψ m | ;   0 ≤ p m ≤ 1 ,   ∑ m p m = 1 {\displaystyle \rho =\sum _{m}p_{m}|\Psi _{m}\rangle \langle \Psi _{m}|;\ 0\leq p_{m}\leq 1,\ \sum _{m}p_{m}=1}

Its easy to see that the partial trace ρ A {\displaystyle \rho ^{A}} also satisfies these conditions. For example, for any pure state | ψ A ⟩ {\displaystyle |\psi _{A}\rangle } in H A {\displaystyle H_{A}} , we have

⟨ ψ A | ρ A | ψ A ⟩ = ∑ m p m Tr B ⁡ [ ⟨ ψ A | Ψ m ⟩ ⟨ Ψ m | ψ A ⟩ ] ≥ 0 {\displaystyle \langle \psi _{A}|\rho ^{A}|\psi _{A}\rangle =\sum _{m}p_{m}\operatorname {Tr} _{B}[\langle \psi _{A}|\Psi _{m}\rangle \langle \Psi _{m}|\psi _{A}\rangle ]\geq 0}

Note that the term Tr B ⁡ [ ⟨ ψ A | Ψ m ⟩ ⟨ Ψ m | ψ A ⟩ ] {\displaystyle \operatorname {Tr} _{B}[\langle \psi _{A}|\Psi _{m}\rangle \langle \Psi _{m}|\psi _{A}\rangle ]} represents the probability of finding the state | ψ A ⟩ {\displaystyle |\psi _{A}\rangle } when the composite system is in the state | Ψ m ⟩ {\displaystyle |\Psi _{m}\rangle } . This proves the positive semi-definiteness of ρ A {\displaystyle \rho ^{A}} .

The partial trace map as given above induces a dual map Tr B ∗ {\displaystyle \operatorname {Tr} _{B}^{*}} between the C*-algebras of bounded operators on H A {\displaystyle \;H_{A}} and H A ⊗ H B {\displaystyle H_{A}\otimes H_{B}} given by

Tr B ∗ ⁡ ( A ) = A ⊗ I . {\displaystyle \operatorname {Tr} _{B}^{*}(A)=A\otimes I.}

Tr B ∗ {\displaystyle \operatorname {Tr} _{B}^{*}} maps observables to observables and is the Heisenberg picture representation of ⁠ Tr B {\displaystyle \operatorname {Tr} _{B}} ⁠.

Suppose instead of quantum mechanical systems, the two systems A and B are classical. The space of observables for each system are then abelian C*-algebras. These are of the form C(X) and C(Y) respectively for compact spaces X, Y. The state space of the composite system is simply

C ( X ) ⊗ C ( Y ) = C ( X × Y ) . {\displaystyle C(X)\otimes C(Y)=C(X\times Y).}

A state on the composite system is a positive element ρ of the dual of C(X × Y), which by the Riesz–Markov theorem corresponds to a regular Borel measure on X × Y. The corresponding reduced state is obtained by projecting the measure ρ to X. Thus the partial trace is the quantum mechanical equivalent of this operation.

• Filipiak, Katarzyna; Klein, Daniel; Vojtková, Erika (2018). "The properties of partial trace and block trace operators of partitioned matrices". Electronic Journal of Linear Algebra. 33. doi:10.13001/1081-3810.3688.
• Johnston, Nathaniel (2021). Advanced Linear and Matrix Algebra. Springer. p. 367. doi:10.1007/978-3-030-52815-7. ISBN 978-3-030-52815-7.