18 Channel Representations and Normal Forms

This chapter builds on the Choi–Jamiołkowski foundations established in Chapter 4 and collects the remaining representation material for finite-dimensional quantum channels from [ Wol12 , Chapter 2 ] : Kraus representation theorems, Stinespring and Naimark dilations, ordered completely positive maps, Radon–Nikodym and open-system representations, trace-pairing expansions, SVD and Lorentz normal forms, and the channel determinant. These results are important for the later channel theory, but they are not prerequisites for the matrix-product-state Fundamental Theorem.

18.1 Representations

Chapter 4 established the maximally entangled state, the Choi matrix, the Choi criterion for complete positivity, and unitary mixing of Kraus operators. This chapter develops the remaining finite-dimensional representation results from [ Wol12 , Chapter 2 ] , beginning with partial traces.

Definition 18.1.1 Partial traces

✓

Let $X \in M_{d \cdot d'}(\mathbb {C})$ be a bipartite matrix indexed by $(\{ 0,\ldots ,d-1\} \times \{ 0,\ldots ,d'-1\} )^2$. The left partial trace $\operatorname{tr}_A(X)$ and right partial trace $\operatorname{tr}_B(X)$ are the $d' \times d'$ and $d \times d$ matrices defined by

\begin{align} (\operatorname{tr}_A(X))_{ij} & = \sum _{k} X_{(k,i),(k,j)}, \notag \\ (\operatorname{tr}_B(X))_{ij} & = \sum _{k} X_{(i,k),(j,k)}. \notag \end{align}

18.2 Further Choi-matrix identities

The Choi foundations were established in Chapter 4. The remaining identities express trace preservation and Hermiticity preservation in the Choi matrix.

Theorem 18.2.1 Trace preservation implies the partial-trace condition

✓

If $T$ is trace-preserving, then $\operatorname{tr}_A(\tau ) = \frac{1}{D} \mathbb {1}_D$.

Proof ▶

By trace preservation,

\begin{align} (\operatorname{tr}_A(\tau ))_{ij} & = \operatorname{tr}(T(\Omega ^{(ij)})) = \operatorname{tr}(\Omega ^{(ij)}), \notag \end{align}

where $\Omega ^{(ij)} = \frac{1}{D} E_{ij}$ is the $(i,j)$-slice of $|\Omega \rangle \! \langle \Omega |$. Taking the trace gives $\frac{1}{D}\delta _{ij}$.

Theorem 18.2.2 Hermiticity preservation and the Choi matrix

✓

The Choi matrix $\tau $ is Hermitian if and only if $T(B^\dagger ) = T(B)^\dagger $ for all $B \in M_{D}(\mathbb {C})$.

Proof ▶

The Choi matrix entries satisfy $\tau _{(a,i),(b,j)} = \frac{1}{D}(T(E_{ij}))_{ab}$. Hermiticity of $\tau $ says $\overline{\tau _{(b,j),(a,i)}} = \tau _{(a,i),(b,j)}$, which unravels to $T(E_{ji})_{ba} = \overline{T(E_{ij})_{ab}}$, i.e., $T(E_{ij}^\dagger ) = T(E_{ij})^\dagger $. By linearity this extends to all $B$.

Theorem 18.2.3 Trace of the Choi matrix of a trace-preserving map

✓

If $D\geq 1$ and $T$ is trace-preserving, then $\operatorname{tr}(\tau ) = 1$.

Proof ▶

One has $\operatorname{tr}(\tau ) = \operatorname{tr}(\operatorname{tr}_A(\tau )) = \operatorname{tr}(\frac{1}{D}\mathbb {1}_D) = 1$.

18.3 Representation corollaries for channel decompositions

The next three corollaries support channel decompositions and correspond to [ Wol12 , Propositions 2.2–2.4 ] .

Theorem 18.3.1 Polarization of sesquilinear sandwiches

✓

For any $A, B, X \in M_{D}(\mathbb {C})$,

\begin{align} 4\, A X B^\dagger & = (A+B) X (A+B)^\dagger - (A-B) X (A-B)^\dagger \notag \\ & \quad + i\, (A + iB) X (A + iB)^\dagger - i\, (A - iB) X (A - iB)^\dagger . \label{eq:representations_polarization} \end{align}

Each of the four summands on the right is a single-Kraus CP map, so every sesquilinear sandwich decomposes into a signed complex linear combination of CP maps.

Proof ▶

Working entrywise reduces the claim to the scalar polarization identity

\begin{align} 4\, \alpha \, \overline{\delta } & = \sum _{k=0}^{3} i^k (\alpha + i^k \beta )\overline{(\gamma + i^k \delta )} \notag \end{align}

in $\mathbb {C}$, which holds after substituting $i^2 = -1$.

Theorem 18.3.2 CP decomposition of sandwich sums

✓

For any finite Kraus-like families $\{ A_i\} , \{ B_i\} $, the map $T(X) = \sum _i A_i X B_i^\dagger $ satisfies $4\, T = T_1 - T_2 + i\, T_3 - i\, T_4$ with each $T_k$ a CP map given explicitly by a single-side Kraus sum of the combined families.

Proof ▶

Apply the polarization identity summand-by-summand.

Theorem 18.3.3 No information without disturbance

✓

Let $T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})$ be linear. If $T(|v\rangle \! \langle v|) = |v\rangle \! \langle v|$ for every $v \in \mathbb {C}^D$, then $T = \operatorname{id}$. In particular, a quantum channel preserving every pure state equals the identity.

Proof ▶

Rank-one ray scalar rigidity gives $T=c\, \operatorname{id}$ for some $c\in \mathbb {C}$. If $D{\gt}0$, evaluating at the first coordinate projector gives $P_{e_0}=T(P_{e_0})=cP_{e_0}$, whose $(0,0)$ entry yields $c=1$. For $D=0$ the matrix algebra is trivial, so the conclusion also holds.

Definition 18.3.4 Pure-state ensemble density

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Given a finite family $\{ \psi _i\} _{i \in \iota }$ of (unnormalized) vectors in $\mathbb {C}^D$, its pure-ensemble density is $\rho = \sum _i |\psi _i\rangle \! \langle \psi _i|$. Weights $p_i \geq 0$ with $\sum p_i = 1$ can be absorbed by replacing $\psi _i \mapsto \sqrt{p_i} \psi _i$.

Theorem 18.3.5 Isometric mixing preserves the density

✓

If two pure-state ensembles $\{ \psi _i\} _{i \in \iota _1}$ and $\{ \phi _j\} _{j \in \iota _2}$ are related by an isometric mixing matrix $V \in \mathbb {C}^{\iota _1 \times \iota _2}$ with $V^\dagger V = \mathbb {1}$ and $\psi _i = \sum _j V_{ij} \phi _j$, then they induce the same pure-ensemble density operator. This is the sufficient direction of the Hughston–Jozsa–Wootters theorem; the converse is Theorem 18.3.6.

Proof ▶

Expanding and applying the orthogonality relation $\sum _i V_{ij}\overline{V_{ij'}} = \delta _{j j'}$ from $V^\dagger V = \mathbb {1}$ gives

\begin{align} \sum _i |\psi _i\rangle \! \langle \psi _i| & = \sum _{j, j'} \left(\sum _i V_{ij} \overline{V_{ij'}}\right) |\phi _j\rangle \! \langle \phi _{j'}| = \sum _j |\phi _j\rangle \! \langle \phi _j|. \notag \end{align}

Theorem 18.3.6 Hughston–Jozsa–Wootters converse

✓

If two pure-state ensembles $\{ \psi _i\} _{i \in \iota _1}$ and $\{ \phi _j\} _{j \in \iota _2}$ induce the same pure-ensemble density operator and $|\iota _2| \le |\iota _1|$, then there exists a tall isometric mixing matrix $V \in \mathbb {C}^{\iota _1 \times \iota _2}$ with $V^\dagger V = \mathbb {1}$ and $\psi _i = \sum _j V_{ij} \phi _j$. The cardinality hypothesis is what makes $V$ a tall isometry; the symmetric case $|\iota _1| \le |\iota _2|$ follows by swapping the roles of the ensembles.

Proof ▶

Embed each vector $\psi _i$, $\phi _j$ as the $0$-th column of a $D \times D$ matrix with zeros elsewhere; call these $K_i$, $L_j \in M_{D}(\mathbb {C})$. A direct entry-wise computation shows that for any $X \in M_{D}(\mathbb {C})$ both $\sum _i K_i X K_i^\dagger $ and $\sum _j L_j X L_j^\dagger $ collapse to $X_{00} \cdot \rho $, so the density equality $\rho _\psi = \rho _\phi $ forces the two Kraus families to define the same CP map. Theorem 18.4.8 then supplies an isometry $V$ with $V^\dagger V = \mathbb {1}$ and $K_i = \sum _j V_{ij}\, L_j$; reading the equation off at column $0$ recovers the vector relation $\psi _i = \sum _j V_{ij} \phi _j$.

Theorem 18.3.7 Hughston–Jozsa–Wootters equivalence

✓

Under the cardinality hypothesis $|\iota _2| \le |\iota _1|$, two pure-state ensembles induce the same density operator iff they are related by a tall isometric mixing matrix.

Proof ▶

Combine the two directions above.

18.4 Kraus representation theorem

A Kraus representation writes the channel as $T(\rho )=\sum _i K_i\rho K_i^\dagger $. In tensor-network form the Kraus index $i$ is summed between the $K_i$ layer and the $K_i^\dagger $ layer:

$\begin{tenkz}[ sandwich, east={cup=$\rho$}, west label=$T(\rho)$ ] \tn{K_i} \\ \tn*{K_i} \end{tenkz}$

Theorem 18.4.1 Trace preservation implies Kraus normalization

✓

If $T(X) = \sum _i K_i X K_i^\dagger $ is trace-preserving, then $\sum _i K_i^\dagger K_i = \mathbb {1}$.

Proof ▶

For any $N$,

\begin{align} \operatorname{tr}\left(\left(\sum _i K_i^\dagger K_i\right)N\right) & = \sum _i \operatorname{tr}(K_i^\dagger K_i N) = \sum _i \operatorname{tr}(K_i N K_i^\dagger ) = \operatorname{tr}(T(N)) = \operatorname{tr}(N). \notag \end{align}

Non-degeneracy of the trace pairing forces $\sum _i K_i^\dagger K_i = \mathbb {1}$.

Theorem 18.4.2 Kraus normalization implies trace preservation

✓

If $\sum _i K_i^\dagger K_i = \mathbb {1}$, then $T(X) = \sum _i K_i X K_i^\dagger $ is trace-preserving.

Proof ▶

One has

\begin{align} \operatorname{tr}(T(X)) & = \sum _i \operatorname{tr}(K_i X K_i^\dagger ) = \sum _i \operatorname{tr}(K_i^\dagger K_i X) = \operatorname{tr}\left(\left(\sum _i K_i^\dagger K_i\right)X\right) = \operatorname{tr}(X). \notag \end{align}

Theorem 18.4.3 Unitality implies Kraus unital normalization

✓

If $T(X) = \sum _i K_i X K_i^\dagger $ satisfies $T(\mathbb {1}) = \mathbb {1}$, then $\sum _i K_i K_i^\dagger = \mathbb {1}$.

Proof ▶

Evaluate the Kraus formula at the identity matrix.

The unitary mixing result was proved in Theorem 4.5.9. The same calculation extends to isometric mixing between different Kraus index spaces.

Theorem 18.4.4 Isometric mixing preserves the Kraus map

✓

Let $W$ be an isometry between two Kraus index spaces, so $W^\dagger W = \mathbb {1}$. If $K_j = \sum _\ell W_{j\ell } \widetilde K_\ell $, then the Kraus families $\{ K_j\} $ and $\{ \widetilde K_\ell \} $ define the same completely positive map.

Proof ▶

Expand the Kraus sums, interchange the finite summations, and use $W^\dagger W = \mathbb {1}$ to collapse the coefficient matrix.

Theorem 18.4.5 The transition matrix of orthonormal Kraus families is unitary

✓

Let $\{ K_j\} _{j=0}^{r-1}$ and $\{ K'_j\} _{j=0}^{r-1}$ be two Hilbert–Schmidt orthonormal Kraus families, and suppose $K_j = \sum _{\ell =0}^{r-1} U_{j\ell }\, K'_\ell $. Then the transition matrix $U$ is unitary: $U^\dagger U = \mathbb {1}_r$.

Proof ▶

Expand $\operatorname{tr}(K_j^\dagger K_i)$ using the change of basis. Hilbert–Schmidt orthonormality of both Kraus families identifies the Gram matrix with the identity, yielding the matrix identity $U U^\dagger = \mathbb {1}_r$, hence also $U^\dagger U = \mathbb {1}_r$.

Theorem 18.4.6 Dual map equality from primal map equality

✓

If two Kraus families satisfy $\sum _\alpha B_\alpha X B_\alpha ^\dagger = \sum _j A_j X A_j^\dagger $ for all $X$, then $\sum _\alpha B_\alpha ^\dagger Y B_\alpha = \sum _j A_j^\dagger Y A_j$ for all $Y$.

Proof ▶

Use the trace pairing: for all $X$,

\begin{align} \operatorname{tr}\left(X^\dagger \left(\sum _\alpha B_\alpha ^\dagger Y B_\alpha \right)\right) & = \operatorname{tr}\left(\left(\sum _\alpha B_\alpha X^\dagger B_\alpha ^\dagger \right)Y\right) \notag \\ & = \operatorname{tr}\left(\left(\sum _j A_j X^\dagger A_j^\dagger \right)Y\right) = \operatorname{tr}\left(X^\dagger \left(\sum _j A_j^\dagger Y A_j\right)\right). \notag \end{align}

Nondegeneracy of the trace pairing gives the result.

Theorem 18.4.7 Equal Stinespring Gramians

✓

If two Kraus families define the same CPM, then $\sum _\alpha B_\alpha ^\dagger B_\alpha = \sum _j A_j^\dagger A_j$.

Proof ▶

Specialise Theorem 18.4.6 at $Y = \mathbb {1}$.

Theorem 18.4.8 Rectangular Kraus freedom with general finite index types

✓

Variant of Theorem 4.5.10 with general finite index sets $\iota _1, \iota _2$ in place of standard index sets of cardinalities $r_1$ and $r_2$.

Proof ▶

Reindex by choosing enumerations of those finite index sets and apply Theorem 4.5.10.

Theorem 18.4.9 Isometric freedom of Kraus representations

✓

Let $\{ B_\alpha \} _{\alpha \in \iota _1}$ and $\{ A_j\} _{j \in \iota _2}$ be finite Kraus families with $|\iota _2| \le |\iota _1|$. Then the following are equivalent:

$\sum _\alpha B_\alpha X B_\alpha ^\dagger = \sum _j A_j X A_j^\dagger $ for all $X \in M_{D}(\mathbb {C})$.
There exists a matrix $V = (V_{\alpha j})$ with $V^\dagger V = \mathbb {1}$ such that $B_\alpha = \sum _j V_{\alpha j} A_j$ for every $\alpha $.

Proof ▶

The implication $(1) \Rightarrow (2)$ is Theorem 18.4.8. Conversely, expand $\sum _\alpha B_\alpha X B_\alpha ^\dagger $ using $B_\alpha = \sum _j V_{\alpha j} A_j$, interchange the finite sums, and use $V^\dagger V = \mathbb {1}$ to collapse the coefficient matrix to the diagonal.

Theorem 18.4.10 Unitary freedom of Kraus representations

✓

Let $\{ B_\alpha \} _{\alpha \in \iota }$ and $\{ A_j\} _{j \in \iota }$ be two Kraus families with the same finite index set. Then the following are equivalent:

$\sum _\alpha B_\alpha X B_\alpha ^\dagger = \sum _j A_j X A_j^\dagger $ for all $X \in M_{D}(\mathbb {C})$.
There exists a unitary matrix $U = (U_{\alpha j})$ such that $B_\alpha = \sum _j U_{\alpha j} A_j$ for every $\alpha $.

Proof ▶

Apply Theorem 18.4.9 with $|\iota | = |\iota |$. In the square case an isometry matrix is unitary, and the converse is immediate.

Remark 18.4.11 Choi rank and minimal Kraus cardinality

✓

If a completely positive map admits a Kraus representation with exactly $r$ operators, then its Choi matrix is a sum of $r$ rank-one outer products. Consequently, $\operatorname{rank}(\tau _E) \le r$. Conversely, diagonalizing the positive semidefinite Choi matrix and keeping only the nonzero eigenvalue summands produces a Kraus family with exactly $\operatorname{rank}(\tau _E)$ operators. Hence the Choi rank is precisely the minimal Kraus cardinality.

Proof ▶

For the forward implication, expand the Choi matrix using the Kraus formula and write $\tau _E = \sum _j v_j v_j^\dagger $ with one vector $v_j$ for each Kraus operator. The column space of this sum lies in the span of the $r$ vectors $\{ v_j\} $, so the rank is at most $r$. For the converse, use the spectral decomposition of the positive semidefinite Choi matrix, discard the zero-eigenvalue terms, and rescale the remaining eigenvectors into Kraus operators.

18.5 Stinespring dilation

For a normalized Kraus family, the Stinespring construction realizes the channel through an isometry $V$ into a larger space, $T(\rho )=\operatorname{tr}_E[V\rho V^\dagger ]$; tracing out the environment $E$ returns the channel:

$\begin{tenkz}[ sandwich, east={cup=$\rho$}, west label=$T(\rho)$ ] \tn{V} \\ \tn*{V} \end{tenkz}$

Definition 18.5.1 Stinespring matrix

✓

Given possibly rectangular Kraus operators $K_j:\mathbb {C}^{d_{\mathrm{in}}}\to \mathbb {C}^{d_{\mathrm{out}}}$, the Stinespring matrix $V:\mathbb {C}^{d_{\mathrm{in}}}\to \mathbb {C}^{d_{\mathrm{out}}}\otimes \mathbb {C}^r$ is defined by

\begin{align} V_{(i,j),k} & =(K_j)_{ik}. \label{eq:representations_stinespring_entries} \end{align}

Thus $V=\sum _j K_j\otimes |j\rangle $. It is an isometry precisely when the Kraus family satisfies the trace-preserving normalization.

Lemma 18.5.2 Entry formula for the Stinespring matrix

✓

For output index $i$, input index $k$, and $j\in \{ 0,\ldots ,r-1\} $, $V_{(i,j),k}=(K_j)_{ik}$.

Definition 18.5.3 Stinespring matrix with a finite index set

✓

For a Kraus family $(K_j)_{j\in J}$ indexed by an arbitrary finite set $J$, define $(V_J)_{(i,j),k}=(K_j)_{ik}$.

Lemma 18.5.4 Entry formula for the finite-index Stinespring matrix

✓

The finite-index Stinespring matrix satisfies $(V_J)_{(i,j),k}=(K_j)_{ik}$.

Theorem 18.5.5 Stinespring Gram identity

✓

The Stinespring matrix satisfies

\begin{align} V^\dagger V & =\sum _{j=0}^{r-1}K_j^\dagger K_j. \label{eq:representations_stinespring_gram} \end{align}

Definition 18.5.6 Local Stinespring matrix

✓

If a Stinespring matrix $V:\mathbb {C}^{d_{\mathrm{in}}}\to \mathbb {C}^{d_{\mathrm{out}}}$ acts on the first factor while a finite-dimensional space $R$ is left unchanged, the corresponding local Stinespring matrix is $W=V\otimes \mathbb {1}_R$.

Theorem 18.5.7 Local Stinespring isometry

✓

If $V^\dagger V=\mathbb {1}$, then $(V\otimes \mathbb {1}_R)^\dagger (V\otimes \mathbb {1}_R)=\mathbb {1}$.

Proof ▶

Use $(A\otimes B)(C\otimes D)=AC\otimes BD$ and $\mathbb {1}\otimes \mathbb {1}=\mathbb {1}$.

Theorem 18.5.8 Stinespring dual representation

✓

For Kraus operators $\{ K_j\} $ and $A\in M_{D}(\mathbb {C})$,

\begin{align} V^\dagger (A\otimes \mathbb {1}_r)V & =\sum _j K_j^\dagger A K_j =T^*(A). \label{eq:representations_stinespring_dual} \end{align}

This is the Stinespring representation of the dual (Heisenberg picture) map $T^*$.

Proof ▶

Compute $(V^\dagger (A\otimes \mathbb {1}_r)V)_{ab}$ by summing over the product index $(i,j)$; the $\delta _{jl}$ from $\mathbb {1}_r$ collapses the sum over $l$, giving $\sum _j K_j^\dagger A K_j$.

Theorem 18.5.9 The Stinespring isometry condition iff trace preservation

✓

In the square specialization $d_{\mathrm{in}}=d_{\mathrm{out}}=D$, one has $V^\dagger V=\mathbb {1}_D$ if and only if $\sum _j K_j^\dagger K_j=\mathbb {1}_D$. In particular, $V$ is an isometry precisely when the Kraus map is trace-preserving.

Proof ▶

One has $(V^\dagger V)_{ab} =\sum _{(i,j)}\overline{V_{(i,j),a}}V_{(i,j),b} =\sum _j\sum _i\overline{(K_j)_{ia}}(K_j)_{ib} =(\sum _jK_j^\dagger K_j)_{ab}$.

Theorem 18.5.10 Stinespring Schrödinger representation

✓

The Kraus map $T(\rho )=\sum _jK_j\rho K_j^\dagger $ equals the partial trace over the dilation space:

\begin{align} T(\rho )_{ij} & =\sum _{k=0}^{r-1}(V\rho V^\dagger )_{(i,k),(j,k)} =\operatorname{tr}_r(V\rho V^\dagger ). \label{eq:representations_stinespring_schrodinger} \end{align}

This is the Schrödinger-picture Stinespring representation.

Proof ▶

Expand the middle expression in (??) and sum over $k$ to recover $\sum _kK_k\rho K_k^\dagger $.

Theorem 18.5.11 Existence of a Stinespring dilation

✓

Every completely positive map admits an ancilla dimension $r$, Kraus operators $\{ K_j\} _{j=0}^{r-1}$, and the concrete representation $\pi (A)=A\otimes \mathbb {1}_r$ such that $E(A)=V^\dagger \pi (A)V$.

Proof ▶

Choose a Kraus representation of $E$ and apply the explicit Heisenberg-picture Stinespring formula.

Theorem 18.5.12 CPTP maps admit isometric Stinespring dilations

✓

Every quantum channel admits a Stinespring dilation whose matrix $V$ is an isometry, equivalently $V^\dagger V=\mathbb {1}$.

Proof ▶

Choose Kraus operators for the channel and use trace preservation to obtain the Stinespring isometry condition $V^\dagger V=\mathbb {1}$.

18.6 Ordered CP maps, Radon–Nikodym, and open-system representation

The next three results form the structural part of [ Wol12 , Theorems 2.3–2.5 ] : the ordered CP-map theorem, the Radon–Nikodym theorem for completely positive maps, and the open-system representation. They refine the Stinespring dilation by tracking how two CP maps related by domination or decomposition sit inside a common dilation space.

Definition 18.6.1 CP partial order

✓

For linear maps $S,T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})$, write $T\le S$ (in the CP order) when $S-T$ is completely positive.

Definition 18.6.2 Block-top contraction

✓

For natural numbers $r,s$, let $C=C_{r,s}\in \mathbb {C}^{r\times (r+s)}$ be the rectangular $r\times (r+s)$ matrix whose rows are the first $r$ rows of the identity on $\mathbb {C}^{r+s}$: $C_{ij}=1$ if $j=i{\lt}r$ and $0$ otherwise.

Lemma 18.6.3 $C$ is a co-isometry

✓

The block-top matrix satisfies $CC^\dagger =\mathbb {1}_r$.

Proof ▶

Direct entrywise computation: $(CC^\dagger )_{ii'}=\sum _jC_{ij}\overline{C_{i'j}}$ collapses to $\delta _{ii'}$.

Lemma 18.6.4 $C$ is a contraction

✓

The block-top matrix satisfies $C^\dagger C\le \mathbb {1}_{r+s}$.

Proof ▶

The matrix $C^\dagger C$ is diagonal with a $1$ on the first $r$ coordinates and a $0$ on the last $s$, so $\mathbb {1}-C^\dagger C$ is positive semidefinite.

Lemma 18.6.5 Intertwining of Stinespring isometries

✓

Let $K:\{ 0,\ldots ,r-1\} \to M_{D}(\mathbb {C})$ and $L:\{ 0,\ldots ,s-1\} \to M_{D}(\mathbb {C})$ be two Kraus families. Write $K\mathbin {+\! +}L$ for their concatenation as a family indexed by $\{ 0,\ldots ,r+s-1\} $. Then

\begin{align} V_K & =(\mathbb {1}_D\otimes C_{r,s})V_{K+\! +L}. \label{eq:representations_stinespring_intertwining} \end{align}

Proof ▶

Entrywise expansion of both sides of (??): the Kronecker product with $C_{r,s}$ picks out the first $r$ coordinates of the dilation space, matching the action of $V_{K+\! +L}$ on those coordinates, which is precisely $V_K$.

Theorem 18.6.6 Ordered CP maps via Stinespring contractions

✓

Let $T_1,T_2:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})$ be CP maps with $T_1\le T_2$. Then there exist ancilla dimensions $r_1,m$, Heisenberg-form Stinespring matrices $V_1:\mathbb {C}^D\to \mathbb {C}^D\otimes \mathbb {C}^{r_1}$ and $V_2:\mathbb {C}^D\to \mathbb {C}^D\otimes \mathbb {C}^m$ realizing $T_1,T_2$ via $T_i(A)=V_i^\dagger (A\otimes \mathbb {1})V_i$, and a rectangular contraction $\widetilde C:\mathbb {C}^m\to \mathbb {C}^{r_1}$ with $\widetilde C^\dagger \widetilde C\le \mathbb {1}_m$ such that

\begin{align} V_1 & =(\mathbb {1}_D\otimes \widetilde C)V_2. \label{eq:representations_ordered_intertwining} \end{align}

This is [ Wol12 , Theorem 2.3 ] .

Proof ▶

Choose a Heisenberg-form Kraus family $K_1$ of $T_1$ with Stinespring matrix $V_1=V_{K_1}$, and Kraus operators $L$ of the CP map $T_2-T_1$ in Schrödinger orientation. Let $L^\dagger $ denote the conjugate-transposed family. Set $m=r_1+s$ and let $K_2=K_1\mathbin {+\! +}L^\dagger $. Then

\begin{align} T_2(A) & =T_1(A)+(T_2-T_1)(A) \notag \\ & =\sum _i(K_1^i)^\dagger A K_1^i+\sum _jL_j A L_j^\dagger \notag \\ & =\sum _i(K_1^i)^\dagger A K_1^i +\sum _j(L_j^\dagger )^\dagger A L_j^\dagger . \notag \end{align}

This is the Heisenberg form for $K_2$, yielding $T_2(A)=V_{K_2}^\dagger (A\otimes \mathbb {1})V_{K_2}$. Taking $\widetilde C=C_{r_1,s}$, the intertwining $V_{K_1}=(\mathbb {1}_D\otimes \widetilde C)V_{K_2}$ is Lemma 18.6.5, and $\widetilde C^\dagger \widetilde C\le \mathbb {1}$ is Lemma 18.6.4.

Definition 18.6.7 Block-diagonal projectors on the dilation space

✓

For $r,s$, let $P_{\mathrm{top}}=C_{r,s}^\dagger C_{r,s}$ and $P_{\mathrm{bot}}=\mathbb {1}-P_{\mathrm{top}}$. Both are PSD (for $P_{\mathrm{bot}}$, by Lemma 18.6.4) and $P_{\mathrm{top}}+P_{\mathrm{bot}}=\mathbb {1}$.

Theorem 18.6.8 Radon–Nikodym theorem for finite quantum instruments

✓

Let $\{ T_i\} _{i\in I}$ be a nonempty finite family of completely positive maps, let $T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})$ satisfy $\sum _{i\in I}T_i=T$, and suppose that $T$ has a supplied Stinespring representation

\begin{align} T(A) & =V^\dagger (A\otimes \mathbb {1}_r)V. \label{eq:representations_radon_dilation} \end{align}

Then there are positive semidefinite operators $P_i\in M_{r}(\mathbb {C})$ with

\begin{align} \sum _{i\in I}P_i & =\mathbb {1}_r. \label{eq:representations_radon_resolution} \end{align}

For every $i\in I$ and $A\in M_{D}(\mathbb {C})$,

\begin{align} T_i(A) & =V^\dagger (A\otimes P_i)V. \label{eq:representations_radon_components} \end{align}

This is [ Wol12 , Theorem 2.4 ] , with the nonempty-family condition made explicit; see docs/paper-gaps/wolf_radon_nikodym_nonempty_family.tex.

Proof ▶

Choose Kraus families for the component maps $T_i$ and group all their Kraus operators into one finite family. Add zero Kraus operators to one component so that this grouped family has at least as many rows as the supplied Stinespring dilation. Write the grouped Kraus operators as $B_\alpha $, with a label map $\ell (\alpha )\in I$, and write the Kraus operators obtained from the blocks of $V$ as $A_j$. Rectangular Kraus freedom gives a matrix $W$ with

\begin{align} W^\dagger W & =\mathbb {1}_r, \notag \\ B_\alpha & =\sum _{j=0}^{r-1}W_{\alpha j}A_j. \notag \end{align}

For the fibre over $i$, set

\begin{align} (C_i)_{\alpha j} & = \begin{cases} \overline{W_{\alpha j}}, & \ell (\alpha )=i,\\ 0, & \ell (\alpha )\ne i, \end{cases} \notag \\ P_i & =C_i^\dagger C_i. \label{eq:representations_radon_effects} \end{align}

Then $P_i\ge 0$, and (??) follows from $W^\dagger W=\mathbb {1}_r$. Finally,

\begin{align} V^\dagger (A\otimes P_i)V & =\sum _{\ell (\alpha )=i}B_\alpha A B_\alpha ^\dagger =T_i(A), \notag \end{align}

which proves the claim.

Theorem 18.6.9 Binary Radon–Nikodym theorem for CP maps

✓

Let $T_1,T_2:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})$ be completely positive. There exist an ancilla dimension $m$, a Kraus family $K$ with Stinespring matrix $V=V_K$, and positive semidefinite operators $P_1,P_2:\mathbb {C}^m\to \mathbb {C}^m$ with $P_1+P_2=\mathbb {1}_m$ such that, for $i=1,2$ and every $A\in M_{D}(\mathbb {C})$,

\begin{align} T_i(A) & =V^\dagger (A\otimes P_i)V. \label{eq:representations_binary_radon} \end{align}

Remark 18.6.10 Relation with Wolf’s Radon–Nikodym theorem

Theorem 18.6.8 treats a finite family relative to a supplied Stinespring dilation. The binary case in Theorem 18.6.9 instead constructs a common dilation from the Kraus families of $T_1$ and $T_2$.

Proof ▶

Take Heisenberg-form Kraus families $K_1$ for $T_1$ and Schrödinger-form $L$ for $T_2$, and form $K=K_1\mathbin {+\! +}L^\dagger $ on $\mathbb {C}^{r_1+s}$. Choose $P_1=P_{\mathrm{top}}$, $P_2=P_{\mathrm{bot}}$ as in Definition 18.6.7. The Kronecker identity $A\otimes (C^\dagger C) =(\mathbb {1}\otimes C)^\dagger (A\otimes \mathbb {1})(\mathbb {1}\otimes C)$ rewrites $V^\dagger (A\otimes P_{\mathrm{top}})V$ as $((\mathbb {1}\otimes C)V)^\dagger (A\otimes \mathbb {1})((\mathbb {1}\otimes C)V)$, which equals $V_{K_1}^\dagger (A\otimes \mathbb {1})V_{K_1}=T_1(A)$ by Lemma 18.6.5. For the complementary block, $A\otimes P_{\mathrm{bot}} =A\otimes \mathbb {1}-A\otimes P_{\mathrm{top}}$, and sandwiching by $V$ yields $(T_1+T_2)(A)-T_1(A)=T_2(A)$.

Theorem 18.6.11 Open-system representation of quantum channels

✓

Every quantum channel $T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})$ admits a Stinespring isometry $V$ on an ancilla space $\mathbb {C}^r$ such that

\begin{align} T(\rho )_{ij} & =\sum _{k=0}^{r-1}(V\rho V^\dagger )_{(i,k),(j,k)} =\operatorname{tr}_r(V\rho V^\dagger ). \label{eq:representations_open_system} \end{align}

Proof ▶

Apply the existence of an isometric Stinespring dilation (Theorem 18.5.12) and the Schrödinger-picture identity (??).

Theorem 18.6.12 Open-system representation, partial-trace form

✓

Every quantum channel $T$ admits an isometric Stinespring dilation $V$ such that $T(\rho )=\operatorname{tr}_E(V\rho V^\dagger )$.

Proof ▶

Rewrite the componentwise identity (??) as a partial trace over the ancilla factor.

Definition 18.6.13 First environment embedding

✓

For $r\geq 1$, let $W_0:\mathbb {C}^D\to \mathbb {C}^D\otimes \mathbb {C}^r$ be the isometry $W_0x=x\otimes e_0$.

Lemma 18.6.14 First environment embedding is an isometry

✓

The first-environment embedding satisfies $W_0^\dagger W_0=\mathbb {1}$.

Proof ▶

Theorem 18.6.15 Open-system representation, unitary form

✓

For $D\geq 1$, every quantum channel $T$ on $\mathbb {C}^D$ admits an environment dimension $r\geq 1$ and a unitary $U$ on $\mathbb {C}^D\otimes \mathbb {C}^r$ such that, for every matrix $\rho $ on $\mathbb {C}^D$,

\begin{align} T(\rho ) & =\operatorname{tr}_E\! \left(UW_0\rho W_0^\dagger U^\dagger \right). \label{eq:representations_open_unitary} \end{align}

Proof ▶

Take the isometric open-system representation $T(\rho )=\operatorname{tr}_E(V\rho V^\dagger )$. Since $V^\dagger V=\mathbb {1}=W_0^\dagger W_0$, Lemma J.4.1 gives a unitary $U$ with $V=UW_0$. Substituting this identity into the partial trace formula gives (??).

18.7 POVMs and Naimark dilation

Definition 18.7.1 Positive operator-valued measure

✓

A positive operator-valued measure with $n$ outcomes on $\mathbb {C}^D$ is a family $\{ E_i\} _{i=0}^{n-1}$ of positive semidefinite operators on $\mathbb {C}^D$ satisfying the resolution of identity

\begin{align} \sum _{i=0}^{n-1}E_i & =\mathbb {1}_D. \label{eq:representations_povm_resolution} \end{align}

Definition 18.7.2 Naimark Kraus square roots

✓

For a POVM $\{ E_i\} $, each effect $E_i\ge 0$ admits a square-root factorisation $E_i=M_i^\dagger M_i$ with $M_i\in M_{D}(\mathbb {C})$. The operators $M_i$ are the Naimark Kraus square roots.

Theorem 18.7.3 Naimark square roots satisfy the Kraus normalization

✓

For a POVM $\{ E_i\} $ with square roots $E_i=M_i^\dagger M_i$, one has $\sum _iM_i^\dagger M_i=\mathbb {1}$.

Proof ▶

Sum the identities $E_i=M_i^\dagger M_i$ and use the defining resolution of the identity for the POVM.

Definition 18.7.4 Naimark isometry

✓

The Naimark isometry $V:\mathbb {C}^D\to \mathbb {C}^D\otimes \mathbb {C}^n$ of a POVM is the Stinespring-type construction

\begin{align} V & =\sum _{i=0}^{n-1}M_i\otimes |i\rangle , \notag \\ V_{(a,i),k} & =(M_i)_{a,k}. \label{eq:representations_naimark_isometry} \end{align}

Definition 18.7.5 Naimark projective measurement

✓

The Naimark projectors on $\mathbb {C}^D\otimes \mathbb {C}^n$ are

\begin{align} P_i & =\mathbb {1}_D\otimes |i\rangle \! \langle i|, \notag \\ (P_i)_{(a,b),(c,d)} & =\delta _{a,c}\delta _{b,i}\delta _{d,i}. \label{eq:representations_naimark_projectors} \end{align}

Theorem 18.7.6 Naimark isometry condition

✓

The Naimark isometry satisfies $V^\dagger V=\mathbb {1}_D$.

Proof ▶

The Stinespring Gram identity gives $V^\dagger V=\sum _iM_i^\dagger M_i=\sum _iE_i=\mathbb {1}_D$ by the resolution of identity.

Theorem 18.7.7 Naimark projectors form a projective measurement

✓

The Naimark projectors satisfy

\begin{align} P_i^2 & =P_i, \notag \\ P_i^\dagger & =P_i, \notag \\ i\ne j & \Rightarrow P_iP_j=0, \notag \\ \sum _iP_i & =\mathbb {1}_{\mathbb {C}^D\otimes \mathbb {C}^n}. \notag \end{align}

Proof ▶

Each identity follows from direct entrywise computation using the delta-function form in (??).

Theorem 18.7.8 Naimark dilation

✓

Every POVM $\{ E_i\} _{i=0}^{n-1}$ arises as a projective measurement on a dilation: for the isometry $V$ and projectors $P_i$ above, $E_i=V^\dagger P_iV$.

Proof ▶

Computing entrywise,

\begin{align} (V^\dagger P_iV)_{k,\ell } & =\sum _a\overline{(M_i)_{a,k}}(M_i)_{a,\ell } =(M_i^\dagger M_i)_{k,\ell } =(E_i)_{k,\ell }. \notag \end{align}

Theorem 18.7.9 Existential Naimark dilation

✓

For every POVM $\{ E_i\} $ on $\mathbb {C}^D$ there exist a dilation dimension $r$, an isometry $V:\mathbb {C}^D\to \mathbb {C}^D\otimes \mathbb {C}^r$, and a projective measurement $\{ P_i\} $ on the dilation satisfying $E_i=V^\dagger P_iV$.

Proof ▶

Take $r=n$ and the explicit witnesses $V=\sum _iM_i\otimes |i\rangle $ and $P_i=\mathbb {1}_D\otimes |i\rangle \! \langle i|$.

Definition 18.7.10 Naimark dilation as a structure

✓

A Naimark dilation of a POVM $\{ E_i\} $ consists of an isometry $V$ into a larger Hilbert space together with a projective measurement $\{ P_i\} $ there such that $E_i=V^\dagger P_iV$ for all $i$.

Theorem 18.7.11 Canonical dilation satisfies the Naimark dilation axioms

✓

The explicit isometry $V=\sum _iM_i\otimes |i\rangle $ and projectors $P_i=\mathbb {1}_D\otimes |i\rangle \! \langle i|$ satisfy the defining axioms of a Naimark dilation.

Proof ▶

Combine the previously proved identities $V^\dagger V=\mathbb {1}_D$, $P_i^2=P_i$, $P_i^\dagger =P_i$, $P_iP_j=0$ for $i\ne j$, $\sum _iP_i=\mathbb {1}$, and $V^\dagger P_iV=E_i$.

Theorem 18.7.12 Concrete uniqueness for canonical projectors

✓

Let $V:\mathbb {C}^D\to \mathbb {C}^D\otimes \mathbb {C}^n$ satisfy $V^\dagger P_iV=E_i$ for the canonical projectors $P_i=\mathbb {1}_D\otimes |i\rangle \! \langle i|$. Then there exists an isometry $W$ on the dilated space such that $V=WV_0$, where $V_0$ is the canonical Naimark isometry of $\{ E_i\} $.

Proof ▶

For each outcome $i$, the $i$-th block of $V$ has Gram matrix $E_i$. Comparing with the canonical square root $M_i$ gives $V_i^\dagger V_i=M_i^\dagger M_i$, hence $V_i=U_iM_i$ for a unitary $U_i$ on $\mathbb {C}^D$. Assemble the $U_i$ block-diagonally to obtain an isometry $W=\bigoplus _iU_i$ on $\mathbb {C}^D\otimes \mathbb {C}^n$, and then $V=W\sum _iM_i\otimes |i\rangle =WV_0$.

Definition 18.7.13 POVM from a PSD resolution of identity on a dilation

✓

Given an isometry $V:\mathbb {C}^D\to \mathbb {C}^{d'}$ with $V^\dagger V=\mathbb {1}_D$ and a family $\{ P_i\} _{i=0}^{n-1}$ of positive semidefinite operators on $\mathbb {C}^{d'}$ summing to the identity, the pulled-back operators $E_i:=V^\dagger P_iV$ form a POVM. In particular, any projective measurement on the dilation (a special case of a PSD resolution of identity) pulls back to a POVM.

Definition 18.7.14 Quantum instrument

✓

A quantum instrument with $n$ outcomes is a family $\{ \Phi _i\} _{i=0}^{n-1}$ of completely positive maps on $M_{D}(\mathbb {C})$ whose sum $\sum _i\Phi _i$ is trace-preserving.

Definition 18.7.15 Instrument operations

✓

Associated to an instrument are the total channel $\sum _i\Phi _i$, the unnormalized update $\rho \mapsto \Phi _i(\rho )$ for each outcome $i$, the outcome probability $p_i(\rho )=\operatorname{tr}(\Phi _i(\rho ))$, and the normalized posterior state $\Phi _i(\rho )/p_i(\rho )$ whenever $p_i(\rho )\neq 0$.

Theorem 18.7.16 Instrument total map is a channel

✓

The total map $\sum _i\Phi _i$ of an instrument is a quantum channel.

Proof ▶

Complete positivity of the sum follows from closure of CP maps under finite addition; trace preservation is built into the definition.

Theorem 18.7.17 Conservation of probability for instruments

✓

For every state $\rho \in M_{D}(\mathbb {C})$, the outcome probabilities $p_i(\rho ):=\operatorname{tr}(\Phi _i(\rho ))$ sum to $\operatorname{tr}(\rho )$.

Proof ▶

Linearity of trace and trace preservation of $\sum _i\Phi _i$ give

\begin{align} \sum _i\operatorname{tr}(\Phi _i(\rho )) & =\operatorname{tr}\left(\left(\sum _i\Phi _i\right)(\rho )\right) =\operatorname{tr}(\rho ). \notag \end{align}

Theorem 18.7.18 Non-negativity of instrument probabilities

✓

If $\rho \in M_{D}(\mathbb {C})$ is positive semidefinite, then each outcome probability $p_i(\rho )=\operatorname{tr}(\Phi _i(\rho ))$ is a non-negative real number.

Proof ▶

Each $\Phi _i$ is completely positive, hence positive, so $\Phi _i(\rho )$ is positive semidefinite and its trace is a non-negative real.

18.8 Trace-pairing expansion in transfer-matrix form

Definition 18.8.1 Trace-self-dual basis

✓

A basis $\{ \sigma _i\} _i$ of $M_{D}(\mathbb {C})$ is trace-self-dual when its coordinate functionals are given by trace pairing:

\begin{align} X & = \sum _i \operatorname{tr}(\sigma _i X)\sigma _i. \label{eq:representations_self_dual} \end{align}

Definition 18.8.2 Trace-pairing coefficients

✓

For a linear map $T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})$ and a trace-self-dual basis $\{ \sigma _i\} $, define coefficients $t_{ij}:=\operatorname{tr}(\sigma _i T(\sigma _j))$.

Theorem 18.8.3 Trace-pairing expansion of a linear map

✓

If $\{ \sigma _i\} $ is trace-self-dual, then every linear map $T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})$ admits the expansion

\begin{align} T(\rho ) & = \sum _i \sum _j t_{ij}\operatorname{tr}(\sigma _j\rho )\sigma _i, \label{eq:representations_trace_expansion}\\ t_{ij} & = \operatorname{tr}(\sigma _i T(\sigma _j)). \notag \end{align}

Proof ▶

Expand $\rho $ and each $T(\sigma _j)$ in the chosen basis and substitute. The trace-self-dual identity (??) replaces coordinates by traces, giving the double-sum formula (??).

18.9 SVD normal form (existence)

[ Wol12 , Section 2.3 ] discusses two families of normal forms for the transfer-matrix representation of a quantum channel: the singular value decomposition (SVD) and, for qubit channels, the Lorentz normal form obtained by general invertible Kraus-rank-one CP filtering operations. The SVD normal form is proved below. The Lorentz normal form theorem (Theorem 18.10.13) uses general invertible Kraus-rank-one completely positive filterings. Its proof requires the minimisation argument and Lorentz-orbit analysis in [ Wol12 , Propositions 2.9 and 2.11 ] ; see Section 18.10.

Theorem 18.9.1 SVD for positive semidefinite matrices

✓

Every positive semidefinite matrix $M \in M_{D}(\mathbb {C})$ admits a decomposition $M=U\operatorname{diag}(\sigma )U^{\dagger }$, where $U \in \mathcal{U}(D)$ is unitary and $\sigma : \{ 0,\ldots ,D-1\} \to \mathbb {R}_{\ge 0}$ has non-negative entries.

Proof ▶

The spectral theorem for Hermitian matrices provides an orthonormal eigenbasis with real eigenvalues. Positive semidefiniteness forces those eigenvalues to be non-negative, and the decomposition is written in SVD form with $U$ the eigenvector unitary and $\sigma $ the eigenvalue sequence.

Theorem 18.9.2 SVD existence for invertible matrices

✓

Every invertible complex square matrix $M \in M_{D}(\mathbb {C})$ admits a singular value decomposition $M=U\operatorname{diag}(\sigma )V^{\dagger }$, with $U,V \in \mathcal{U}(D)$ unitary and $\sigma _i{\gt}0$ for all $i$.

Proof ▶

Apply the spectral theorem to the positive definite matrix $M^{\dagger }M$ to obtain $M^{\dagger }M=V\operatorname{diag}(\lambda )V^{\dagger }$ with $\lambda _i{\gt}0$. Set $\sigma _i:=\sqrt{\lambda _i}$ and $\Sigma :=\operatorname{diag}(\sigma )$; since $\Sigma $ is a real positive diagonal, it is self-adjoint and invertible. Define $U:=MV\Sigma ^{-1}$. Then

\begin{align} U^{\dagger }U & = \Sigma ^{-1}V^{\dagger }(M^{\dagger }M)V\Sigma ^{-1} = \Sigma ^{-1}\operatorname{diag}(\lambda )\Sigma ^{-1} = \mathbb {1}, \notag \\ U\Sigma V^{\dagger } & = MV(\Sigma ^{-1}\Sigma )V^{\dagger } = MVV^{\dagger } = M. \notag \end{align}

Thus $U$ is unitary and the stated decomposition holds.

Theorem 18.9.3 SVD representation of a transfer matrix

✓

Every invertible transfer matrix $\widehat{T}$ of a linear super-operator $T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})$ admits a singular value decomposition $\widehat{T}=U\operatorname{diag}(\sigma )V^{\dagger }$ with $U,V$ unitary on $\mathbb {C}^{D}\otimes \mathbb {C}^{D}$ and $\sigma _i{\gt}0$.

Proof ▶

Direct specialisation of Theorem 18.9.2 to the index type $\{ 0,\ldots ,D-1\} \times \{ 0,\ldots ,D-1\} $ used by the transfer-matrix representation.

Remark 18.9.4 Sorted / unique singular values

Theorem 18.9.2 produces the singular values as an unordered family, without the usual convention that $\sigma _i$ are sorted in non-increasing order or the statement that they are uniquely determined by $M$. Downstream applications in [ Wol12 , Section 2.3 ] , such as trace-norm identities, polar decomposition, and the Lorentz normal form, will typically require the sorted / unique variant; that refinement is future work.

18.10 Lorentz normal form

This section states the existence theorems from [ Wol12 , Section 2.3 ] .

Definition 18.10.1 SL-filtering operation

✓

An SL-filtering for $D \times D$ matrices is a completely positive map of the form $\Phi (X)=SXS^{\dagger }$, where $S \in M_{D}(\mathbb {C})$ satisfies $\det S=1$. Such maps are invertible and have Kraus rank 1.

Definition 18.10.2 Doubly-stochastic map

✓

A linear map $T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})$ is doubly-stochastic if $T(\mathbb {1})\propto \mathbb {1}$ and the reduced density matrix $\operatorname{tr}_{1}[\tau ]$ of its Choi matrix $\tau =(T\otimes \operatorname{id})(|\Omega \rangle \! \langle \Omega |)$ is proportional to the identity. This is the normal form in [ Wol12 , Proposition 2.9 ] .

Lemma 18.10.3 Positive-definite trace lower bound

✓

Let $D\geq 1$, let $M\in M_{D}(\mathbb {C})$ be positive-definite, and let $\lambda _{\min }(M)$ be its smallest Hermitian eigenvalue. For every $X\in M_{D}(\mathbb {C})$,

\begin{align} \lambda _{\min }(M)\operatorname{tr}(X^{\dagger }X) & \leq \operatorname{tr}(XMX^{\dagger }). \label{eq:representations_posdef_trace_bound} \end{align}

Proof ▶

The matrix $M-\lambda _{\min }(M)\mathbb {1}$ is positive semidefinite by the spectral theorem. Since $X^{\dagger }X$ is also positive semidefinite, the trace product $\operatorname{tr}\! \left(X^{\dagger }X(M-\lambda _{\min }(M)\mathbb {1})\right)$ is non-negative. Expanding this expression and cycling the trace gives (??).

Lemma 18.10.4 Determinant AM–GM Hilbert–Schmidt bound

✓

Let $A\in M_{D}(\mathbb {C})$ with $D\geq 1$. If $|\det A|=1$, then $D\leq \operatorname{tr}(A^{\dagger }A)$.

Proof ▶

The matrix $A^{\dagger }A$ is positive semidefinite and has determinant $|\det A|^{2}=1$. Thus the product of its Hermitian eigenvalues is one. Denoting the eigenvalues of $A^{\dagger }A$ by $\lambda _{1},\ldots ,\lambda _{D}\geq 0$, the arithmetic-geometric mean inequality gives

\begin{align} \frac{\lambda _{1}+\cdots +\lambda _{D}}{D} & \geq (\lambda _{1}\cdots \lambda _{D})^{1/D} =1. \notag \end{align}

Therefore $\operatorname{tr}(A^{\dagger }A)=\lambda _{1}+\cdots +\lambda _{D}\geq D$.

Lemma 18.10.5 Kronecker Hilbert–Schmidt trace multiplicativity

✓

For complex matrices $A$ and $B$ of compatible rectangular sizes,

\begin{align} \operatorname{tr}\! \left((A\otimes _{k}B)^{\dagger } (A\otimes _{k}B)\right) & =\operatorname{tr}(A^{\dagger }A)\operatorname{tr}(B^{\dagger }B). \label{eq:representations_kronecker_hs} \end{align}

Proof ▶

Use $(A\otimes _{k}B)^{\dagger }=A^{\dagger }\otimes _{k}B^{\dagger }$, the mixed product identity for Kronecker products, and $\operatorname{tr}(C\otimes _{k}D)=\operatorname{tr}(C)\operatorname{tr}(D)$. With $C=A^{\dagger }A$ and $D=B^{\dagger }B$, these identities give (??).

Lemma 18.10.6 Infimum attainment for SL-filterings

✓

For a positive-definite Choi matrix $\tau $, the infimum of

\begin{align} \operatorname{tr}\! \left[(S_{2}\otimes _{k}S_{1})\tau (S_{2}\otimes _{k}S_{1})^{\dagger }\right] \notag \end{align}

over $S_{1},S_{2}\in M_{D}(\mathbb {C})$ with $\det S_{1}=\det S_{2}=1$ is attained.

Proof ▶

Write $X=S_{2}\otimes _{k}S_{1}$. Lemma 18.10.3 gives the trace lower bound below. Lemma 18.10.5 factors the Hilbert–Schmidt trace. For each $i\in \{ 1,2\} $, Lemma 18.10.4 supplies the first row. Together,

\begin{align} \operatorname{tr}(S_i^\dagger S_i) & =\| S_i\| ^2\geq D, \notag \\ \operatorname{tr}(X\tau X^\dagger ) & \geq \lambda _{\min }(\tau )\operatorname{tr}(X^\dagger X) \notag \\ & =\lambda _{\min }(\tau ) \operatorname{tr}(S_2^\dagger S_2)\operatorname{tr}(S_1^\dagger S_1) \notag \\ & \geq \lambda _{\min }(\tau )D\| S_i\| ^2. \notag \end{align}

Hence any point whose value is at most the value at the identity lies in a fixed Frobenius ball $\{ \, \| S\| \leq C\, \} $. Intersecting with the closed set $\det S=1$ produces a compact set, on which the continuous trace functional attains its minimum by the extreme-value theorem. A point outside the sublevel set has value larger than the value at the identity, so the minimiser on the compact set is a global minimiser.

Lemma 18.10.7 Arithmetic–geometric-mean inequality, product/sum form

✓

For a non-negative family $f_{0},\ldots ,f_{D-1}$ of real numbers,

\begin{align} D^{D}\prod _{i}f_{i} & \leq \left(\sum _{i}f_{i}\right)^{D}. \label{eq:representations_amgm_prod_sum} \end{align}

Proof ▶

Apply the weighted arithmetic–geometric-mean inequality with uniform weights $1/D$:

\begin{align} \left(\prod _{i}f_{i}\right)^{1/D} & \leq \frac{1}{D}\sum _{i}f_{i}. \notag \end{align}

Raising both sides to the $D$-th power gives (??).

Lemma 18.10.8 Trace–determinant AM–GM for positive-semidefinite matrices

✓

For a positive-semidefinite $D \times D$ matrix $M$,

\begin{align} D^{D}\det M & \leq (\operatorname{tr}M)^{D}, \label{eq:representations_trace_det_amgm} \end{align}

and equality holds if and only if $M=(\operatorname{tr}M/D)\mathbb {1}$.

Proof ▶

Both $\det M=\prod _{i}\lambda _{i}$ and $\operatorname{tr}M=\sum _{i}\lambda _{i}$ are expressed through the non-negative eigenvalues $\lambda _i$ of $M$. The bound is the arithmetic–geometric-mean inequality with uniform weights $1/D$, raised to the $D$-th power; equality holds exactly when all eigenvalues coincide, that is, when $M$ is a scalar matrix.

Theorem 18.10.9 Generic normal form for CP maps

✓

Let $T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})$ be a completely positive map whose Choi matrix is positive-definite (equivalently, $T$ has full Kraus rank). Then there exist SL-filterings $\Phi _{1},\Phi _{2}$ such that $\Phi _{2}\circ T\circ \Phi _{1}$ is doubly-stochastic.

This is the equal-dimension (square) case $T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})$; the source [ Wol12 , Proposition 2.9 ] states the proposition for a general $T : \mathcal{M}_{d_1} \to \mathcal{M}_{d_2}$ with separate filterings on each factor.

Proof ▶

Let $\tau $ be the Choi matrix of $T$ and take the minimiser $(S_{1},S_{2})$ of

\begin{align} \operatorname{tr}\! \left[(S_{2}\otimes _{k}S_{1})\tau (S_{2}\otimes _{k}S_{1})^{\dagger }\right] \notag \end{align}

over $\det S_{1}=\det S_{2}=1$ from Lemma 18.10.6. Set $\Phi _{1},\Phi _{2}$ to be the filterings with matrices $S_{1}^{\mathsf T}$ and $S_{2}$, so that the Choi matrix of $\Phi _{2}\circ T\circ \Phi _{1}$ equals $(S_{2}\otimes _{k}S_{1})\tau (S_{2}\otimes _{k}S_{1})^{\dagger }$. Holding one filtering factor fixed, the minimiser is optimal in the other coordinate, so each partial trace minimises a functional $\operatorname{tr}(SMS^{\dagger })$ over $\det S=1$ for a positive-semidefinite $M$, where $\operatorname{tr}(SMS^{\dagger })\geq D(\det M)^{1/D}$, with equality at the minimiser; by the equality case of Lemma 18.10.8, this forces $SMS^{\dagger }\propto \mathbb {1}$, hence both partial traces are proportional to the identity, which is exactly the doubly-stochastic condition.

Definition 18.10.10 Diagonal Lorentz normal form

✓

For a completely positive, trace-preserving qubit channel $T' : M_{2}(\mathbb {C}) \to M_{2}(\mathbb {C})$, write $\widehat{T'}$ for its matrix in the normalized Pauli basis $\{ \sigma _{0}/\sqrt{2},\sigma _{1}/\sqrt{2}, \sigma _{2}/\sqrt{2},\sigma _{3}/\sqrt{2}\} $. The channel is in diagonal Lorentz normal form when it is unital and every off-diagonal entry of $\widehat{T'}$ is zero.

Definition 18.10.11 Non-diagonal Lorentz normal form

✓

For a completely positive, trace-preserving qubit channel $T' : M_{2}(\mathbb {C}) \to M_{2}(\mathbb {C})$, write $\widehat{T'}$ in the normalized Pauli basis. The channel is in non-diagonal Lorentz normal form when, for some $x \in [0,1]$,

\begin{align} \widehat{T'} & = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & x/\sqrt{3} & 0 & 0 \\ 0 & 0 & x/\sqrt{3} & 0 \\ 2/3 & 0 & 0 & 1/3 \end{pmatrix}. \label{eq:representations_lorentz_nondiagonal} \end{align}

Trace preservation supplies the first row of the displayed matrix.

Definition 18.10.12 Singular Lorentz normal form

✓

\begin{align} \widehat{T'} & = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{pmatrix}, \label{eq:representations_lorentz_singular} \end{align}

equivalently, every input state is mapped to $(1+\sigma _{3})/2$. Trace preservation supplies the first row of the displayed matrix.

Theorem 18.10.13 Lorentz normal form for qubit channels

For every qubit channel $T : M_{2}(\mathbb {C}) \to M_{2}(\mathbb {C})$, there exist invertible completely positive maps $\Phi _{1},\Phi _{2}$, both of Kraus rank one, such that the filtered channel $T'=\Phi _{2}\circ T\circ \Phi _{1}$ is in one of the three Lorentz normal forms: diagonal, non-diagonal, or singular. These general filters include scalar freedom and are not restricted to determinant-one $\operatorname{SL}(2,\mathbb {C})$ filterings. A proof requires the corresponding scalar normalization and the classification of Lorentz orbits.

18.11 Determinant of a quantum channel

Definition 18.11.1 Channel determinant

✓

For a linear map $T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})$, the channel determinant $\det T$ is the determinant of the matrix of $T$ with respect to the standard matrix-unit basis of $M_{D}(\mathbb {C})$. This is the quantity studied in [ Wol12 , Section 6.1 ] .

Remark 18.11.2 Channel determinant as an operator determinant

✓

The channel determinant agrees with the ordinary determinant of the linear endomorphism $T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})$.

Definition 18.11.3 Unitary channel

✓

For a unitary matrix $U \in \mathcal{U}(D)$, the unitary channel is $T(\rho )=U\rho U^\dagger $. It is automatically a quantum channel.

Theorem 18.11.4 Unitary channels have determinant one

✓

If $T(\rho )=U\rho U^\dagger $ is a unitary channel, then $\det T=1$ and hence $|\det T|=1$.

Proof ▶

Vectorization identifies $T$ with a Kronecker product $\overline{U}\otimes U$, whose determinant is $\overline{\det U}^{\, D}(\det U)^D=1$.

Theorem 18.11.5 Determinant bound for positive trace-preserving maps

✓

For any positive trace-preserving map $T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})$, $|\det T|\leq 1$. This is [ Wol12 , Theorem 6.1(1) ] .

Proof ▶

Theorem 18.11.6 Determinant one iff the channel is unitary

✓

For a CPTP map $T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})$,

\begin{align} |\det T|=1 & \iff \exists \, U \in \mathcal{U}(D),\quad T(\rho )=U\rho U^\dagger . \label{eq:representations_det_unitary_iff} \end{align}

This is [ Wol12 , Theorem 6.1(2) ] .

Proof ▶

Reverse direction is Theorem 18.11.4. Forward direction: $|\det T|=1$ together with $|\mu |\leq 1$ for every eigenvalue $\mu $ (from trace preservation and positivity) forces every eigenvalue to satisfy $|\mu |=1$. Determinant saturation transfers to the unital Heisenberg dual $T^*(Y)=\sum _iK_i^\dagger YK_i$. The determinant–Hilbert–Schmidt bound and a trace-summing argument then force equality in the Kadison–Schwarz inequality on the standard matrix basis. The resulting Kraus commutation relations make $T^*$ multiplicative on all matrices, hence a $*$-automorphism of $M_{D}(\mathbb {C})$. By the Skolem–Noether theorem it is inner, and therefore $T(\rho )=U\rho U^\dagger $ for some unitary $U$. Kraus freedom then gives $K_i=c_iU$ with $\sum _i|c_i|^2=1$.