A Continuity Theorem for Stinespring's Dilation

Introduction and Overview

Completely positive maps (cp maps, for short) describe the dynamics of open quantum systems. Stinespring’s dilation theorem is the basic structure theorem for such maps. It states that any cp map $T\mathpunct{:}\mathcal{A}\rightarrow\mathcal{B}$ between two $C^{*}$-algebras $\mathcal{A}$ and $\mathcal{B}\subset\mathcal{B(H)}$ can be written as a concatenation of two basic cp maps: a ∗-homomorphism $\pi\mathpunct{:}\mathcal{A}\rightarrow\mathcal{B(K)}$ into a larger (dilated) algebra $\mathcal{B(K)}$ (the bounded operators on some Hilbert space $\mathcal{K}$), followed by a compression $V^{*}(\cdot)V$ into the range algebra $\mathcal{B}$:

Stinespring’s theorem provides a neat characterization of the set of permissible quantum operations and is also a most useful tool in the theory of open quantum systems and quantum information . In a way, the increased system size is the price one has to pay for a simpler description of the map $T$ in terms of just two basic operations.

A triple $(\pi,V,\mathcal{K})$ such that Eq. (1) holds is called a Stinespring representation for $T$. Stinespring’s representation is unique up to partial isometries on the dilation spaces: given two representations $(\pi_{1},V_{1},\mathcal{K}_{1})$ and $(\pi_{2},V_{2},\mathcal{K}_{2})$ for a completely positive map $T\mathpunct{:}\mathcal{A}\rightarrow\mathcal{B(H)}$, there exists a partial isometry $U\mathpunct{:}\mathcal{K}_{1}\rightarrow\mathcal{K}_{2}$ such that

for all $a\in\mathcal{A}$. A Stinespring representation $(\pi,V,\mathcal{K})$ of a cp map $T\mathpunct{:}\mathcal{A}\rightarrow\mathcal{B(H)}$ is called minimal iff the set $\{\pi(a)\,V|\psi\rangle\mid a\in\mathcal{A},|\psi\rangle\in\mathcal{H}\}$ is dense in $\mathcal{K}$. If $(\pi_{1},V_{1},\mathcal{K}_{1})$ and $(\pi_{2},V_{2},\mathcal{K}_{2})$ are two minimal dilations for the cp map $T$, then $U$ in Eq. (2) is unitary. Hence, any two minimal dilations are unitarily equivalent.

Our contribution is a continuity theorem for Stinespring’s dilation: two cp maps, $T_{1}$ and $T_{2}$, are close in cb-norm iff there exist corresponding dilations, $V_{1}$ and $V_{2}$, that are close in operator norm:

This result generalizes the uniqueness clause in Stinespring’s theorem to cp maps that differ by a finite amount. As we have seen, uniqueness holds only up to partial isometries on the dilation spaces. So we cannot expect that any two dilations satisfy such a norm bound, only that they can be chosen in a suitable way. Hence the infimum in Eq. (3).

The continuity bound Eq. (3) shows that the distance between two cp maps can equivalently be evaluated in terms of their dilations. We call this distance measure the Bures distance, since it generalizes Bures’s metric from positive functionals to general cp maps. In Sec. 2 we will formally introduce the Bures distance between general cp maps and state the continuity theorem. The remainder of the article is devoted to the proof of the theorem. Sec. 3 gives the lower bound on the Bures distance in terms of the cb-norm, which is elementary. The upper bound is established in Sec. 4; it relies on Bures’s corresponding result for positive functionals and on Ky Fan’s minimax theorem. We first discuss cp maps with range $\mathcal{B(H)}$, and then extend the results to cp maps with injective range in Sec. 5. We conclude with a pair of appendices: In App. A we show that the Bures distance is indeed a metric on the set of completely positive maps, and in App. B for completeness we reproduce Bures’s proof of the upper bound for positive functionals.

Building on earlier work by Belavkin et al. , the continuity theorem has appeared in for the special case of unital cp maps (i. e., quantum channels) between finite-dimensional matrix algebras and has been applied to derive bounds on the tradeoff between information gain and disturbance in quantum physics, to establish a continuity bound for the no-broadcasting theorem, and to improve security bounds for quantum key distribution with faulty devices. A generalization to channels between direct sums of finite-dimensional matrix algebras has been used to derive a strengthened impossibility proof for quantum bit commitment .

Main Results

The Bures distance evaluates the distance between two cp maps in terms of their dilations. We first discuss maps with range algebra $\mathcal{B}=\mathcal{B(H)}$, the bounded operators on some Hilbert space $\mathcal{H}$.

(Bures Distance) Assume a $C^{*}$-algebra $\mathcal{A}$, a Hilbert space $\mathcal{H}$, and two cp maps $T_{i}\mathpunct{:}\mathcal{A}\rightarrow\mathcal{B(H)}$.

The $\pi$-distance between $T_{1}$ and $T_{2}$ is defined as

where the $\pi$-fiber $S(T,\pi)$ of a cp map $T\mathpunct{:}\mathcal{A}\rightarrow\mathcal{B(H)}$ and a representation $\pi\mathpunct{:}\mathcal{A}\rightarrow\mathcal{B(K)}$ is defined as the set of all operators $V\mathpunct{:}\mathcal{H}\rightarrow\mathcal{K}$ such that $(\pi,V,\mathcal{K})$ dilates $T$. If one or both of the fibers are empty, we set $\beta_{\pi}(T_{1},T_{2}):=2$.

The Bures distance between $T_{1}$ and $T_{2}$ is the smallest such $\pi$-distance:

For cp maps with one-dimensional range algebra, i.e. positive functionals, $\beta$ coincides with Bures’s distance function, as introduced in his seminal $1969$ paper . Our definition is the natural generalization to arbitrary cp maps; we hence choose the same name. The statement of the continuity theorem amounts to showing that the cb-norm and the Bures distance are equivalent distance measures for cp maps.

(Continuity of Stinespring’s Dilation) Let $\mathcal{A}$ be a C*-algebra, and let $T_{i}\mathpunct{:}\mathcal{A}\to\mathcal{B(H)}$ be completely positive maps such that $T_{i}\neq 0$ for at least one $i\in\{1,2\}$. With $\beta(T_{1},T_{2})$ defined as in Eq. (5), we then have the following inequality:

Moreover, there exist a common representation $\pi\mathpunct{:}\mathcal{A}\to\mathcal{B(K)}$ for $T_{1}$ and $T_{2}$ and two corresponding Stinespring dilations $V_{i}\mathpunct{:}\mathcal{H}\to\mathcal{K}$ such that

If $(\hat{\pi}_{i},\hat{V}_{i},\hat{\mathcal{K}}_{i})$ is the minimal Stinespring dilation for the cp map $T_{i}$, we can choose $\pi:=\hat{\pi}_{1}\oplus\hat{\pi}_{2}$ as the common representation in Th. 1. Even more is known for positive functionals: in that case the Bures distance can be evaluated in any common representation . We do not yet know whether this result extends to general cp maps.

What about general range algebras $\mathcal{B}\neq\mathcal{B(H)}$? Since any $C^{*}$-algebra $\mathcal{B}$ can be faithfully embedded into a norm-closed self-adjoint algebra $\mathcal{B(H)}$ with a suitably chosen Hilbert space $\mathcal{H}$, it may appear natural to define the Bures distance for cp maps $T_{i}\mathpunct{:}\mathcal{A}\rightarrow\mathcal{B}$ in terms of the concatenated maps $\sigma\circ T_{i}$, with a faithful representation $\sigma\mathpunct{:}\mathcal{B}\rightarrow\mathcal{B(H)}$. However, $\beta(\sigma\circ T_{1},\sigma\circ T_{2})$ might possibly depend on the embedding representation $\sigma$. We instead choose an intrinsic definition of the Bures distance — and show that it reduces to Def. 1 if $\mathcal{B}=\mathcal{B(H)}$.

(Bures Distance for General Range Algebras) Given two $C^{*}$-algebras $\mathcal{A}$ and $\mathcal{B}$ and two cp maps $T_{i}\mathpunct{:}\mathcal{A}\rightarrow\mathcal{B}$, the Bures distance is defined as

with completely bounded maps $\hat{T}_{ij}\mathpunct{:}\mathcal{A}\rightarrow\mathcal{B}$ satisfying $\hat{T}_{ii}=T_{i}$.

While this definition of the Bures distance admittedly looks quite different from Def. 1, we will show in Sec. 5.1 that the definitions coincide if $\mathcal{B}=\mathcal{B(H)}$, and hence it is justified to use the same symbol for both.

With this definition of the Bures distance, Th. 1 can now be generalized to cp maps with injective range algebras. Recall that a $C^{*}$-algebra $\mathcal{B}$ is called injective if for every $C^{*}$-algebra $\mathcal{A}$ and operator system $\mathcal{S}$ contained in $\mathcal{A}$, every completely positive map $R\mathpunct{:}\mathcal{S}\rightarrow\mathcal{B}$ can be extended to a completely positive map on all of $\mathcal{A}$ (cf. , Ch. 7). In fact, in order to show that $\mathcal{B}\subset\mathcal{B(H)}$ is injective it is enough to find a completely positive map $P\mathpunct{:}\mathcal{B(H)}\rightarrow\mathcal{B}$ such that $P(b)=b$ for all $b\in\mathcal{B}$. $P$ is usually called a completely positive conditional expectation. Connes has shown that a von Neumann algebra $\mathcal{B}$ is injective iff it is hyperfinite, which means that $\mathcal{B}$ contains an ascending sequence of finite dimensional subalgebras with dense union. We refer to Ch. XVI in Takesaki’s textbook for this and further equivalent conditions for injectivity of von Neumann algebras. A characterization of injective $C^{*}$-algebras has been given by Robertson et al. . For cp maps with non-injective range, we only have a lower bound on $\beta(T_{1},T_{2})$, though we could always apply Th. 1 to the concatenated maps $\sigma\circ T_{1}$ with some faithful embedding $\sigma$.

(Continuity for General Range Algebras) Let $\mathcal{A}$ and $\mathcal{B}$ be $C^{*}$-algebras, and let $T_{i}\mathpunct{:}\mathcal{A}\rightarrow\mathcal{B}$ be completely positive. With $\beta(T_{1},T_{2})$ defined as in Def. 2 above, we have

If in addition $\mathcal{B}$ is injective, we also have

and $\beta(T_{1},T_{2})=\beta(\sigma\circ T_{1},\sigma\circ T_{2})$ for any faithful representation $\sigma\mathpunct{:}\mathcal{B}\rightarrow\mathcal{B(H)}$.

The remainder of the article is devoted to the proof of Theorems 1 and 2. We start in Sec. 3 with a lower bound on the Bures distance in terms of the cb-norm.

Lower Bound

A lower bound on the Bures distance $\beta(T_{1},T_{2})$ in terms of the cb-norm distance $\|T_{1}-T_{2}\|_{cb}$ easily follows from the standard properties of the operator norm.

(Lower Bound) Let $\mathcal{A}$ be a $C^{*}$-algebra, and $T_{1},T_{2}\mathpunct{:}\mathcal{A}\rightarrow\mathcal{B(H)}$ be completely positive maps. We then have

Upper Bound

In this Section we will complement Prop. 3 with an upper bound on the Bures distance $\beta(T_{1},T_{2})$ in terms of the cb-norm $\|T_{1}-T_{2}\|_{cb}$. We start by investigating several alternative ways to evaluate the Bures distance — a useful tool for our proof but also a result of independent interest.

Given two cp maps $T_{i}\mathpunct{:}\mathcal{A}\rightarrow\mathcal{B(H)}$ and a representation $\pi\mathpunct{:}\mathcal{A}\rightarrow\mathcal{B(K)}$, we set

The $\pi$-distance $\beta_{\pi}(T_{1},T_{2})$ can now be calculated in terms of $\mathcal{N}_{\pi}(T_{1},T_{2})$ as follows:

For cp maps $T_{i}\mathpunct{:}\mathcal{A}\rightarrow\mathcal{B(H)}$ and a representation $\pi\mathpunct{:}\mathcal{A}\rightarrow\mathcal{B(K)}$, we have

where $\mathcal{B}_{*,1}^{+}\mathcal{(H)}$ denotes the positive trace class operators of unit trace on the Hilbert space $\mathcal{H}$.

Proof: The map $x\mapsto{\rm tr}((\cdot)x)$ defines an isometric isomorphism from $\mathcal{B(H)}$ to the normalized trace class operators $\mathcal{B}_{*,1}\mathcal{(\mathcal{H})}$ (cf. Sec. VI.6 in ). Since in addition $(V_{1}-V_{2})^{*}(V_{1}-V_{2})$ is positive, we can write

for $V_{i}\in S(T_{i},\pi)$ and any given representation $\pi$. The result then immediately follows from the definition of $\beta_{\pi}(T_{1},T_{2})$ in Eq. (4) and $\mathcal{N}_{\pi}(T_{1},T_{2})$ in Eq. (15). $\blacksquare$

The following lemma allows to replace the infimum over representations $\pi$ and corresponding $N\in\mathcal{N}_{\pi}$ in Lemma 4 with an infimum over intertwiners $W\mathpunct{:}\mathcal{K}_{2}\rightarrow\mathcal{K}_{1}$ between any two fixed Stinespring representations. As advertised in Sec. 2, we will also show how to find a common representation $\pi$ such that $\beta(T_{1},T_{2})=\beta_{\pi}(T_{1},T_{2})$.

(Evaluation of the Bures Distance) Let $\mathcal{A}$ be a $C^{*}$-algebra, $\mathcal{H}$ a Hilbert space, and $T_{1},T_{2}\mathpunct{:}\mathcal{A}\rightarrow\mathcal{B(H)}$ be two completely positive maps.

Assuming Stinespring dilations $(\pi_{i},V_{i},\mathcal{K}_{i})$ for $T_{i}$, we define

The set $\mathcal{M}(T_{1},T_{2})\subset\mathcal{B(H)}$ depends only on the cp maps $T_{i}$, not on the dilations $(\pi_{i},V_{i},\mathcal{K}_{i})$.

The set $\mathcal{M}(T_{1},T_{2})$ can be represented alternatively as

where the union is over all representations $\pi$ admitting a common Stinespring representation for $T_{1}$ and $T_{2}$, and $\mathcal{N}_{\pi}(T_{1},T_{2})$ is defined in Eq. (15).

There exists a representation $\pi$ such that $\beta(T_{1},T_{2})=\beta_{\pi}(T_{1},T_{2})$. We can choose $\pi:=\hat{\pi}_{1}\oplus\hat{\pi}_{2}$, where $\hat{\pi}_{i}$ is a minimal representation for $T_{i}$.

Proof: $(1)$ For the first part, our strategy is to show that $\mathcal{M}(T_{1},T_{2})$, defined via some dilations $(\pi_{i},V_{i},\mathcal{K}_{i})$, coincides with $\hat{\mathcal{M}}(T_{1},T_{2})$ defined via the minimal dilations $(\hat{\pi}_{i},\hat{V}_{i},\hat{\mathcal{K}}_{i})$. Given two dilations $(\pi_{i},V_{i},\mathcal{K}_{i})$ for $T_{1}$ and $T_{2}$, respectively, we know from the uniqueness clause in Stinespring’s theorem that there exist isometries $U_{i}\mathpunct{:}\hat{\mathcal{K}}_{i}\rightarrow\mathcal{K}_{i}$ such that $U_{i}\hat{V}_{i}=V_{i}$ and $U_{i}^{*}V_{i}=\hat{V}_{i}$. Since $U_{i}U_{i}^{*}$ is a projector onto the closed linear span of $\{\pi_{i}(a)V_{i}|\psi\rangle\}$, we have $U_{i}U_{i}^{*}V_{i}=V_{i}$, and hence

for all $W\mathpunct{:}\mathcal{K}_{2}\rightarrow\mathcal{K}_{1}$, where we have set $\hat{W}:=U_{1}^{*}WU_{2}\mathpunct{:}\hat{\mathcal{K}}_{2}\rightarrow\hat{\mathcal{K}}_{1}$. The intertwining relations $U_{i}\,\hat{\pi}_{i}(a)=\pi_{i}(a)\,U_{i}$ and $W\,\pi_{2}(a)=\pi_{1}(a)\,W$ imply that

for all $a\in\mathcal{A}$. Moreover, $\|\hat{W}\|=\|U_{1}^{*}WU_{2}\|\leq\|W\|\leq 1$, since the $U_{i}$ are isometric. Hence, $\mathcal{M}(T_{1},T_{2})\subset\hat{\mathcal{M}}(T_{1},T_{2})$. The converse is completely analogous, starting with $\hat{W}$ and setting $W:=U_{1}\hat{W}U_{2}^{*}$. $\blacktriangle$

$(2)$ In order to show that $\mathcal{M}(T_{1},T_{2})\subset\mathcal{N}(T_{1},T_{2})$, it is sufficient to find a common representation $\pi$ such that $\mathcal{M}(T_{1},T_{2})\subset\mathcal{N}_{\pi}(T_{1},T_{2})$. Since $\mathcal{M}(T_{1},T_{2})$ is independent of the dilations according to part $(1)$, we can assume it to be defined via the minimal dilations $(\hat{\pi}_{i},\hat{V}_{i},\hat{\mathcal{K}}_{i})$. Given $\hat{W}\mathpunct{:}\hat{\mathcal{K}}_{2}\rightarrow\hat{\mathcal{K}}_{1}$ such that $\|\hat{W}\|\leq 1$, we define the bounded operators $V_{i}\mathpunct{:}\mathcal{H}\rightarrow\hat{\mathcal{K}}_{1}\oplus\hat{\mathcal{K}}_{2}$ by setting

$(3)$ From the proof of part $(2)$ we have $\mathcal{N}_{\pi}(T_{1},T_{2})\subset\mathcal{M}(T_{1},T_{2})\subset\mathcal{N}_{\hat{\pi}_{1}\oplus\hat{\pi}_{2}}(T_{1},T_{2})$ for any common representation $\pi$. We can then immediately conclude from Lemma 4 that

implying $\beta(T_{1},T_{2})=\beta_{\hat{\pi}_{1}\oplus\hat{\pi}_{2}}(T_{1},T_{2})$. Consequently, the Bures distance can always be evaluated in the direct sum representation of the minimal representations. $\blacksquare$

Lemma 4 and Lemma 5 can now be applied to derive the desired upper bound on the Bures distance in terms of the cb-norm. For the special case of positive functionals, this result was obtained by Bures (cf. Prop. 11 in App. B), and will now be lifted to cp maps with the help of Ky Fan’s minimax theorem .

(Upper Bound) Let $\mathcal{A}$ be a $C^{*}$-algebra, and $T_{1},T_{2}\mathpunct{:}\mathcal{A}\rightarrow\mathcal{B(H)}$ be completely positive maps. We can then find a common representation $\pi\mathpunct{:}\mathcal{A}\rightarrow\mathcal{B(K)}$ and corresponding dilations $(\pi,V_{i},\mathcal{K})$ for $T_{i}$ such that

Proof: Spelling out $\beta_{\pi}(T_{1},T_{2})$ as in Lemma 4 and then making use of the relation $\mathcal{N}(T_{1},T_{2})=\mathcal{M}(T_{1},T_{2})$ from Lemma 5, we have

with $W\in\mathcal{B}(\mathcal{K}_{2},\mathcal{K}_{1})$, where $(\pi_{1},V_{1},\mathcal{K}_{1})$ and $(\pi_{2},V_{2},\mathcal{K}_{2})$ are now any two fixed dilations for the cp maps $T_{1}$ and $T_{2}$, respectively. The target functional in Eq. (26) is affine in both inputs. Since the state $\varrho\in\mathcal{B}_{*,1}^{+}\mathcal{(H)}$ is trace-class, so is $V_{2}\varrho V_{1}^{*}$, and hence the functional is weakly continuous in $W$. Moreover, we know from the Banach-Alaoglu theorem (cf. Sec. IV.5 in ) that the unit ball $\|W\|\leq 1$ is weakly compact, and hence the infimum is attained. In addition, both optimizations in Eq. (26) are performed over convex sets. Under these conditions, Ky Fan’s minimax theorem guarantees that the order of the optimizations in Eq. (26) can be interchanged to yield

which is the desired result. For the cb-norm bound in the last step we have used that the finite rank operators are dense in $\mathcal{B_{*}(H)}$. We have seen above that there exists an intertwiner $W\mathpunct{:}\mathcal{K}_{2}\rightarrow\mathcal{K}_{1}$ which attains the infima in Eq. (26) and Eq. (27). Lemma 5 then by construction yields a common representation $\pi$ and corresponding dilations $(\pi,V_{i},\mathcal{K})$ such that

Th. 1 now immediately follows by combining the bounds from Prop. 3 and Prop. 6.

Bures Distance for General Range Algebras

So far our discussion has focused on channels with range algebra $\mathcal{B(H)}$. In this Section we will investigate completely positive maps $T_{i}\mathpunct{:}\mathcal{A}\rightarrow\mathcal{B}$ with general range algebra $\mathcal{B}$. Our results are twofold: in Sec. 5.1 we will justify the intrinsic definition of the Bures distance $\beta(T_{1},T_{2})$ by showing that it indeed coincides with Def. 1 if $\mathcal{B}=\mathcal{B(H)}$. For general range algebras $\mathcal{B}$, we will then show in Sec. 5.3 that $\beta(T_{1},T_{2})\geq\beta(\sigma\circ T_{1},\sigma\circ T_{2})$ for any representation $\sigma$. If $\mathcal{B}$ is injective and $\sigma$ is faithful, we even have equality, and hence the Bures distance does not depend on the details of the embedding and can then be shown to be completely equivalent to the cb-norm distance. For the proof we need a monotonicity result for the Bures distance, which we will present in Sec. 5.2.

For the moment, we will denote the Bures distance for cp maps $T_{i}\mathpunct{:}\mathcal{A}\rightarrow\mathcal{B}$ with general range algebra $\mathcal{B}$, as introduced in Def. 2, by $\beta^{\prime}(T_{1},T_{2})$. We will show in this Section that indeed $\beta^{\prime}(T_{1},T_{2})=\beta(T_{1},T_{2})$ if $\mathcal{B}=\mathcal{B(H)}$. Thus, Def. 2 is a consistent generalization of Def. 1 to general range algebras, and we may henceforth drop the prime.

Let $\mathcal{A}$ be a $C^{*}$-algebra, and let $T_{i}\mathpunct{:}\mathcal{A}\rightarrow\mathcal{B(H)}$ be completely positive. With $\beta(T_{1},T_{2})$ defined as in Def. 1 and $\beta^{\prime}(T_{1},T_{2})$ defined as in Def. 2, we then have

holds independently of $\hat{T}$, and hence $\beta(T_{1},T_{2})\leq\beta^{\prime}(T_{1},T_{2})$ follows immediately from Def. 2. $\blacktriangle$

implying that $\beta^{\prime}(T_{1},T_{2})\leq\beta(T_{1},T_{2})$. $\blacksquare$

2 Monotonicity of the Bures Distance under Cp Maps

We will now show that the Bures distance $\beta(T_{1},T_{2})$ decreases under quantum operations. Only Eq. (33) is needed in the proof of Th. 2 below, but we include Eq. (34) for completeness.

(Monotonicity) Given three $C^{*}$-algebras $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{D}$ and cp maps $T_{1},T_{2}\mathpunct{:}\mathcal{A}\rightarrow\mathcal{B}$ and $S\mathpunct{:}\mathcal{B}\rightarrow\mathcal{D}$, we have

For cp maps $T_{i}$ as above and $S\mathpunct{:}\mathcal{D}\rightarrow\mathcal{A}$ we have

Since Eq. (35) holds for all extensions $\hat{T}$, Eq. (33) is proven. The proof of Eq. (34) is completely analogous. $\blacksquare$

3 Equivalence of Bures Distance and Cb-Norm for Injective Range Algebras

The following proposition shows that for cp maps $T_{i}\mathpunct{:}\mathcal{A}\rightarrow\mathcal{B}$ with injective range algebra $\mathcal{B}$, the Bures distance $\beta(T_{1},T_{2})$ may be evaluated in any faithful representation $\sigma\mathpunct{:}\mathcal{B}\rightarrow\mathcal{B(H)}$.

Let $\mathcal{A}$ and $\mathcal{B}$ be $C^{*}$-algebras, and $T_{1},T_{2}\mathpunct{:}\mathcal{A}\rightarrow\mathcal{B}$ be completely positive maps. We then have

for any representation $\sigma\mathpunct{:}\mathcal{B}\rightarrow\mathcal{B(H)}$. Moreover, if $\mathcal{B}$ is injective and the representation $\sigma$ is faithful equality holds in Eq. (36).

where in the last step we have used that both $\sigma^{-1}$ and $P$ are completely positive with norm $\leq 1$. Since Eq. (37) holds for all extensions of $\sigma\circ T_{i}$, we conclude that

for any faithful representation $\sigma$, as suggested. $\blacksquare$

With the help of Prop. 9, the proof of Th. 2 can now be obtained directly from Th. 1.

Proof of Th. 2: Since the cb-norm is invariant under faithful representations, Eq. (11) immediately follows by choosing the representation $\sigma$ to be faithful in Eq. (36) and applying the corresponding bound from Th. 1. If in addition the range algebra $\mathcal{B}$ is injective, Eq. (12) follows in the same way from Prop. 9 and Th. 1. $\blacksquare$

Appendix A: Properties of the Bures Distance

In this Section we will show that the Bures distance $\beta(T_{1},T_{2})$ defined in Eq. (5) indeed has all the properties of a distance measure.

(Bures Distance) The functional $(T_{1},T_{2})\mapsto\beta(T_{1},T_{2})$ is a metric on the set of cp maps $T_{i}\mathpunct{:}\mathcal{A}\rightarrow\mathcal{B(H)}$.

Proof: Positivity and symmetry are immediate from the definition of $\beta(T_{1},T_{2})$. Obviously, $\beta(T_{1},T_{1})=0$. Conversely, Prop. 3 shows that $\beta(T_{1},T_{2})=0$ entails $\|T_{1}-T_{2}\|_{cb}=0$, and hence $T_{1}=T_{2}$. Thus, it only remains to establish the triangle inequality, $\beta(T_{1},T_{3})\leq\beta(T_{1},T_{2})+\beta(T_{2},T_{3})$ for all triples of cp maps $T_{i}$. To this end, let $(\pi,V_{i},\mathcal{K})$ be dilations for the cp maps $T_{1}$ and $T_{2}$ with a common representation $\pi$. Further assume that $(\check{\pi},\check{V}_{j},\check{\mathcal{K}})$ are dilations for the pair $T_{2},T_{3}$ with a common representation $\check{\pi}$. As before, $(\hat{\pi}_{i},\hat{V}_{i},\hat{\mathcal{K}}_{i})$ will denote the corresponding minimal dilations, with intertwiners $U_{i}\mathpunct{:}\hat{\mathcal{K}}_{i}\rightarrow\mathcal{K}$ for $i\in\{1,2\}$ and $\check{U}_{j}\mathpunct{:}\hat{\mathcal{K}}_{j}\rightarrow\check{\mathcal{K}}$ for $j\in\{2,3\}$. We now set

Now assume that $(\pi,V_{i},\mathcal{K})$ and $(\check{\pi},\check{V}_{j},\check{\mathcal{K}})$ are chosen as in Prop. 6 such that

Hence, Eq. (43) and the triangle inequality for the operator norm imply that

Appendix B: Bures’s Upper Bound for Positive Functionals

The proof of the upper bound $\beta(T_{1},T_{2})\leq\sqrt{\|T_{1}-T_{2}\|}$ for cp maps $T_{i}$ that we present in Sec. 4 relies on the corresponding result for positive functionals. In his original paper Bures assumed (normalized) states on von Neumann algebras. The generalization to arbitrary bounded positive functionals on $C^{*}$-algebras is straightforward. We nevertheless include it here for completeness and reference.

(Bures’s Bound for Positive Functionals) Let $\mathcal{A}$ be a $C^{*}$-algebra, and let $\omega_{0},\omega_{1}\in\mathcal{A}^{*}$ be positive functionals. We then have

The following lemma will establish Prop. 11 under an additional dominance condition. This extra condition will then be removed with the help of Lemma 13, which proves the continuity of the Bures distance with respect to convex mixtures.

Let $h=\int\lambda\,p(d\!\lambda)$ denote the spectral decomposition of $h$, and set $p:=\int_{\lambda=0}^{1}p(d\!\lambda)$. We then find

Adding Eqs. (49) and (50), we see from Eq. (48) that

Let $\mathcal{A}$ be a $C^{*}$-algebra, and let $\omega_{0},\omega_{1}\in\mathcal{A}^{*}$ be two positive functionals. Then the inequality

Proof of Lemma 13: Again, the proof proceeds via a direct sum construction. For $|\psi_{i}\rangle\in S(\omega_{i},\pi)$ we have $|\psi_{0}\rangle\oplus 0\in S(\omega_{0},\pi\oplus\pi)$ and $|\psi_{0}\rangle\oplus|\psi_{1}\rangle\in S(\omega_{0}+\omega_{1},\pi\oplus\pi)$, and thus

We know from Prop. 10 that the Bures distance is indeed a metric, and hence we can use the triangle inequality and then Eq. (53) to conclude that

We now have all the necessary tools at hand for the

Proof of Prop. 11: Given a parameter $s\in(0,1]$, we define the convex mixture $\omega_{s}:=(1-s)\,\omega_{0}+s\,\omega_{1}$. Choosing a positive integer $n>s^{-1}$, we have $n\,\omega_{s}-\omega_{1}>0$, and hence $\beta(\omega_{s},\omega_{1})\leq\sqrt{\|\omega_{s}-\omega_{1}\|}$ follows from Lemma 12. We can then conclude from Lemma 13 that the estimate

holds for all $s\in(0,1]$. The limit $s\rightarrow 0$ yields the desired result. $\blacksquare$

Acknowledgments

We would like to thank Mauro D’Ariano and Vern Paulsen for fruitful and stimulating discussions.

DS acknowledges financial support from Consorzio Nazionale Interuniversitario per le Scienze della Materia (CNISM). DK is grateful for generous support from Deutscher Akademischer Austauschdienst (DAAD).

References