On Random Unitary Channels

In this article we provide necessary and sufficient conditions for a completely positive trace-preserving (CPT) map to be decomposable into a convex combination of unitary maps. Additionally, we set out to define a proper distance measure between a given CPT map and the set of random unitary maps, and methods for calculating it. In this way one could determine whether non-classical error mechanisms such as spontaneous decay or photon loss dominate over classical uncertainties, for example in a phase parameter. The present paper is a step towards achieving this goal.


I. INTRODUCTION
In this paper we answer two questions about quantum maps. The first question is, given any completely positive tracepreserving (CPT) map, how can one determine whether this map can be decomposed as a convex combination of unitary maps? In more formal terminology we ask for necessary and sufficient conditions such that a CPT map Φ can be written as where the scalars p i form a probability distribution (i.e. they are non-negative and add up to 1) and where the U i are unitaries. Preferably, the method for doing so should be constructive and should as a bonus supply the p i and U i . CPT maps obeying this condition are called Random Unitary maps. The second, and very much related, question we answer here is about finding a proper distance measure between a CPT map and the set of Random Unitary maps, and methods for calculating it, be it numerical or (preferably) analytical ones. The physical motivation behind these questions is the desire to distinguish various error mechanisms afflicting the preparation and processing of quantum states. For example, errors occur in the realisation of quantum gates in quantum information processing. If the only error mechanism occurring is a classical uncertainty, for instance in a phase parameter, then the resulting "gate" will not be described by a particular unitary, but rather by a mixture of such unitaries; the mathematical description of such a mixture is effectively a Random Unitary map. If on the other hand, other mechanism can occur, such as spontaneous decay or photon loss, then the resulting gate can no longer be described by such a Random Unitary map. In a sense, the distance between this particular map and the set of Random Unitary maps determines and quantifies the presence of these non-classical error mechanisms.
The paper is structured as follows. In Section II, we present notations and definitions for a number of basic concepts that will be needed in the rest of the paper. Section III is devoted to the determination of whether a unital CPT map is a random unitary map. The techniques introduced in this Section are then generalised in Section IV to construct a genuine distance measure D between a map and the set of Random Unitary maps. A number of properties of D are subsequently derived. Finally, in Section V, a connection is made to the entanglement of assistance of bipartite states.

II. PRELIMINARIES
It is quite obvious that any Random Unitary map should, apart from being trace-preserving, also be unital, meaning that Φ(1 1) = 1 1. Maps like this, trace-preserving and unital ones, are also called doubly stochastic. This necessary condition of double stochasticity implies that the input and output dimensions of the map should be identical. We shall henceforth assume that Φ is a unital CPT map taking states on a d-dimensional Hilbert space H to states on that same space.
At this point it is of course very tempting to check whether the condition of double stochasticity may even be necessary. In fact, for qubit CP maps (d = 2) this is the case. By definition, the set of doubly stochastic CP maps is convex. In [1], Landau and Streater proved that for d = 2 the extremal points of this convex set are precisely the unitary maps. This is just a reformulation of the statement that the set of Random Unitary qubit-maps is precisely the set of doubly stochastic CP qubit-maps.
For higher dimensions this is no longer true, as was first shown by Tregub and by Kummerer and Maassen [1,2,3]. That is, for higher d there are extremal doubly stochastic CP maps that are not unitary. An example of such a map in odd dimensions (taken from Ref. [1]) is the following: where d = 2j + 1 and the J k are the three well-known generators of SU (2) in its d-dimensional irreducible representation [4]. Consequently, the set of all convex combinations of unitary maps is a proper subset of the set of all doubly stochastic CP maps.

A. Jamiołkowski Isomorphism
To proceed, we will next exploit the Jamiołkowski isomorphism between CP maps and states. For the purposes of this paper it is not strictly necessary to do so, but it has the benefit of widening the perspective. The Jamiołkowski isomorphism assigns to each CP map from H in to H out a (not necessarily normalised) state on H out ⊗ H in ; this state will be called here the state representative of the CP map, and will be denoted by Φ as well. The context will make clear whether the map is meant or its state representative. To avoid confusion, we will always assume that the state representative is normalised (i.e. it has unit trace). The density matrix representing the state representative is sometimes called the Choi matrix, after M.-D. Choi who proved that under the Jamiołkowski isomorphism CP maps are mapped to positive semidefinite matrices [10], which is not true for any non-CP map.
In our case, the state representative is a d × d state. To explain how the assignment is done, we first introduce the symbol |I for the maximally entangled state vector Then the following formula defines the Jamiołkowski isomorphism: where 1 1 stands for the identity map; so the map Φ operates on the first copy of H, while the second copy is left untouched (is "operated" upon by the identity map). As our map is trace-preserving, its state representative is automatically normalised. Moreover, the reduction of the state representative obtained by tracing out the first copy of H yields the maximally mixed state 1 1/d. Furthermore, unitality of the map shows itself through the fact that the second reduction (tracing out the second copy of H) is also the maximally mixed state.
Applying this game to the question under consideration yields that a unital CPT map is a Random Unitary map iff its state representative is of the form This actually says that the state representative should be a mixture of maximally entangled (ME) pure states. Indeed, any such pure state can be obtained from the "vintage" ME state |I by applying a local unitary on either party. The set of such mixtures forms, by its very definition, a convex set with the ME pure states as extremal points. We will denote this set by the symbol M. Furthermore, we will denote the set of states with maximally mixed reductions by N : Obviously, we have M ⊂ N . We will see below that for 2 × 2 states, M is actually equal to N , while this is no longer the case for higher-dimensional states. Our questions have therefore been reduced to determining whether a state is a mixture of ME pure states or, if that is not possible, how far the state is from the convex set M. What almost immediately comes to mind is the resemblance between these questions and the notorious questions of determining separability and entanglement, where the convex set under consideration is the convex hull of all pure product states. To wit, what we do in this paper is nothing but adapting mutatis mutandis various methods from entanglement theory to the problem at hand.
In the next few paragraphs we introduce some more notations that will simplify the rest of the presentation.

B. Matrixification
An important linear operation one can perform on bipartite state vectors is the "matrixification" operation [5]. For a given bipartite d × d state vector |ψ , we define its matrixification as a d × d matrix, denoted byψ, such that the following holds: Put less abstractly, the matrixψ consists of all the vector entries of ψ placed in a matrix frame. That is, if |ψ is given by |ψ = i,j a ij |i, j thenψ is just the matrix a = i,j a ij |i j|.
In passing, we remark that the Schmidt decomposition of a bipartite pure state is nothing but the singular value decomposition (SVD) of the matrixified state vector, and the Schmidt coefficients are just the singular values.
For every linear map Φ, one defines the dual map Φ ′ as that map for which Tr[AΦ(B)] = Tr[Φ ′ (A)B] for all A, B. If Φ is CP, then so is Φ ′ . Furthermore, if Φ is trace-preserving then Φ ′ is unital, and if Φ is unital then Φ ′ is trace-preserving.

C. Bloch Vector Formalism
We will also have the opportunity to use the well-known Bloch vector formalism for representing states. The Bloch vector formalism is based on the observation that the density matrices on H are themselves embedded in a Hilbert space, a space of dimension d 2 . Because density matrices are Hermitian, this Hilbert space is real. The inner product in that space is given by the functional ρ, σ := Tr[ρσ]. The Bloch vector of a density matrix is just the vector representing that matrix in this Hilbert space, the entries of which of course depend on the choice of basis.
The standard choice of basis elements are the Pauli matrices σ i (for d = 2) and generalisations thereof to higher dimensions; e.g., in d = 3 one has the Gell-Mann matrices. We use the notation τ i for these generalisations. They can be grouped into x, y and z groups [4]: Here, the notation a ×k stands for k entries equal to a. It is standard to set τ 0 = 2/d1 1. The inner products between these matrices are given by Tr[τ i τ j ] = 2δ i,j (the factor 2 is conventional). The entries of the Bloch vector in this basis are then given by ρ := (ρ 0 , ρ 1 , . . . , ρ d 2 −1 ) where ρ i := Tr[ρτ i ]/2; conversely, ρ = i ρ i τ i . One easily checks the relation Tr[ρσ] = 2 ρ, σ .
For normalised states, ρ 0 = 1/ √ 2d, which is a constant. It is therefore meaningful to employ the reduced Bloch vector instead, which is˜ ρ := (ρ 1 , . . . , ρ d 2 −1 ). For normalised states, we have Tr[ρσ]= 2 ˜ ρ,˜ σ + 1/d. When the state ρ is subjected to a unitary conjugation, ρ → U ρU † (U U † = 1 1), the corresponding Bloch vector will be rotated according to a certain orthogonal matrix O (OO T = 1 1). Let ρ = i ρ i τ i , then ρ ′ = U ρU † = i ρ i U τ i U † and This defines a real matrix O (because Bloch vectors are real), and we need to show that O is orthogonal. That is easily done as follows: so that, indeed, OO T = 1 1. Furthermore, as the unitary conjugation leaves τ 0 invariant (it is a multiple of the identity), O decomposes as [1] ⊕Õ, whereÕ operates on the reduced Bloch vector only. Positivity of the density matrix translates to certain conditions on the Bloch vector. It implies that the purity Tr[ρ 2 ] of states lies between 1/d and 1, so that the length of the Bloch vector is bounded from above by 1/ √ 2. Likewise, the maximal length of the reduced Bloch vector is (d − 1)/2d. States whose reduced Bloch vector is this long are automatically pure. This means that the reduced Bloch vectors of states all lie in a ball of radius (d − 1)/2d, the so-called Bloch ball.
For d = 2, all points in the Bloch ball turn out to correspond to states. For higher d this is no longer the case, and the set of Bloch vectors of states is more like a "dimpled" ball, the conditions on positivity leading to slices being cut away of the original ball. It is easy to see that this has to be so: the vector of diagonal entries of a density matrix is a probability vector and thus lies on a simplex, which means that there must be a linear projection under which the set of all Bloch vectors turns into that simplex. This is impossible for a ball, unless that simplex is 1-dimensional, as it is in the d = 2 case.

III. DETERMINING WHETHER A UNITAL CPT MAP IS A RANDOM UNITARY MAP
To answer the question of whether a unital CPT map is a Random Unitary map, we freely borrow the methods described in [5] and adapt it to the problem at hand.

A. Condition for Random Unitary-ness: pure case
The first step in answering the question is to find a quadratic criterion for judging whether a pure state is maximally entangled (ME). This is quite simple, and the answer is that a state vector |ψ is ME iff √ dψ is unitary. Indeed, we noted that ME states are characterised as |ψ = (U ⊗ 1 1)|I for some unitary U , so we trivially getψ = U .
The left-hand side is easily seen to be the reduction of the state |ψ ψ| to the first subsystem; thus the condition can also be phrased as It is immediate from the purity of the state that the reduction to the other subsystem will automatically be 1 1/d as well. For normalised ψ, this leaves us with a system of d(d + 1)/2 − 1 quadratic equations. In terms of CPT maps, pure states are the state representatives of rank-1 CP maps, i.e. maps with a single element in their Kraus decomposition. The above discussion then leads to the rather obvious fact that rank-1 CP maps are Random Unitary maps if and only if they are unital and trace preserving.
For higher-rank CP maps, this is no longer the case. Nevertheless, the condition Eq. (8) will play a central role for such maps as well. It will turn out to be convenient to rephrase that condition in terms of the Bloch vector of the reduced state. One easily sees that it is equivalent to the condition that the reduced state has the zero-vector as reduced Bloch vector; this also holds when the state is non-normalised. Thus, the condition is given by the system of equalities or

B. Decompositions of Mixed States
Now we must use the pure-state criterion to see whether a mixed state can be decomposed as a mixture of ME pure states. This brings us to the second step of the method: determine all possible convex decompositions of a state. The answer to that is well-known indeed and has been discovered many times. Let the eigenvalue decomposition (EVD) of the state ρ be given by ρ = U ΛU † , where U is the unitary whose columns are the eigenvectors u i , and Λ is a diagonal matrix, the diagonal elements of which are the eigenvalues λ i ; these are non-negative and add up to 1. Then any other convex decomposition of ρ is given by ρ = W M W † , where W is no longer unitary and need not even be square; its columns are the (normalised) vectors of which the decomposition is built. The matrix M , though, is still diagonal with non-negative diagonal elements adding up to 1; these are the convex weights of the decomposition. Alternatively, these weights can be absorbed in their corresponding vectors so that the norm squared of each vector then equals its weight. This gives ρ = ZZ † , with Z = W M 1/2 . In this way, each convex decomposition of ρ is related to a certain "square-root" of ρ.
Characterising all possible square roots of a matrix is a simple problem in matrix analysis. Starting from the eigenvalue square root, ρ 1/2 = U Λ 1/2 U † , one generates all others by right-multiplying it with a right-unitary matrix T . A rightunitary matrix T is a non-necessarily square matrix for which T T † = 1 1 holds (but not T † T = 1 1, unless T is square). Thus one has The number of columns in T is the number of vectors in the decomposition, and is called the cardinality of the decomposition; this number should be at least as large as the rank of ρ, i.e. the number of non-zero eigenvalues.
In quantum physics circles, the most recent and probably best-known incarnation of this result is the famous Hughston-Jozsa-Wootters (HJW) theorem [6]. The earliest occurrence is actually in Schrödinger's work on quantum steering [7], and it has been rediscovered many times by various physicists (see Ref. [8] and references therein).
We now combine the HJW theorem with the quadratic characterisation of pure ME states to obtain a method for determining Random Unitary-ness.

C. Criterion for Random Unitary-ness: mixed case
Let us consider a particular convex decomposition of our map, or of its state representative ρ, that is described by the matrix Z. As ρ is a d × d state, Z has d 2 rows. The cardinality K of the decomposition, being the number of columns of Z, cannot be smaller than the rank R of ρ. We want to check if all the vectors in this decomposition are ME state vectors. Thus we have to apply the criterion (10) to every column vector of Z. This yields the system of equalities Inserting Eq. (11) gives For succinctness we will introduce the d 2 − 1 matrices A i defined by The problem is thus reduced to the following: Necessary conditions for such a T to exist are that all A i should be traceless. For the A i of Eq. (14) this means that Tr[τ i Tr 2 [ρ]] = 0, i.e. Tr 2 [ρ] must be proportional to the identity, which is a condition we have already encountered.
In general, we do not know how to solve this problem analytically, and we have to resort to numerical methods, just as in [5]. This will be described below in Section IV on distance measures.
The off-diagonalisation problem can easily be solved for a single matrix A. From the lemma below it follows that the necessary condition Tr[A] = 0 is also sufficient. Here, K = d 2 suffices, and T is a genuine unitary matrix.

Lemma 1 For a Hermitian
Proof. Necessity is obvious as the trace is unitarily invariant. To show sufficience, consider Schur's majorisation theorem [9], which says that for any Hermitian matrix X, and equality holding for k = n. Horn's Lemma [9] adds to this that for any specified set of diagonal entries and eigenvalues obeying this majorisation relation a Hermitian matrix exists exhibiting those diagonal entries and eigenvalues. For fixed Hermitian X the mapping j → λ ↓ j (X) is non-increasing by definition, so that the mapping k → k j=1 λ ↓ j (X) is concave. Now note that if Tr[X] = 0 then n j=1 λ ↓ j (X) = 0, while 0 j=1 λ ↓ j (X) = 0 trivially. By the above concavity statement we then find that k j=1 λ ↓ j (X) ≥ 0 for all k. As the zero vector (0, . . . , 0) is majorised by any non-negative vector, Horn's Lemma then implies the existence of a Hermitian matrix with zero diagonal and any prescribed set of eigenvalues that add up to 0.
Consider now the eigenvalues of A, which add up to 0 by assumption. By the above, another matrix X must exist exhibiting the same eigenvalues, hence unitarily equivalent to A, and with zero diagonal. Therefore, a unitary T exists such that X jj = (T † AT ) jj = 0 for all j.
For qubit maps there are three A i to cope with. However, in the light of Landau and Streater's result that the qubit Random Unitary maps are exactly the doubly stochastic maps, the condition Tr[A i ] = 0 should also be sufficient for d = 2 and A i of the form (14). Indeed, the following Theorem holds, which therefore supplies an alternative proof of Landau and Streater's Theorem. In other words, in this case we do not even have to search for the matrix T .
Proof. The first thing to note is that the diagonal elements of W † (σ i ⊗ 1 1)W are 0 if and only if the diagonal elements of U † (σ i ⊗ 1 1)U are 0. As pure states in N are automatically in M, an equivalent statement of the Theorem is that a 2-qubit state is in N if and only if all its eigenvectors are.
A 2 × 2 density matrix can be represented by a 2 × 2 block matrix: where every block is a 2 × 2 matrix itself. The conditions on ρ then translate to for some real number k.
We will now show that if ρ satisfies these conditions, then its square also does. Without loss of generality we can apply a local unitary, so that B can be diagonalised. Let us set with a, b non-negative real numbers. The conditions on ρ then demand that D is given by Let us also put where x, y, z are complex numbers. The square of ρ is now given by Let us now test the required conditions: Straightforward calculations reveal so that the first condition is satisfied. Since CC † and C † C have the same trace, and Tr[B 2 ] = Tr[D 2 ] = a 2 + b 2 , the second condition is also satisfied. Finally, Tr[BC + CD] = (a+b)z −(a+b)z = 0, which shows that ρ 2 is of the required form.
We can now repeat the process of squaring and find that any 2 n -th power is of that same form. Let us now invoke the power method for finding the dominating eigenvector ψ of ρ: for ρ with non-degenerate spectra, its m-th power, after normalisation, tends to |ψ ψ| when m tends to infinity. By the above, we thus find that the projector on the dominating eigenvector of ρ is also in N . Therefore, if we "deflate" ρ by subtracting λ ↓ 1 |ψ ψ| from it and renormalise, we again obtain a state in N . Continuing in this way, we thus find that every eigenvector of ρ is in N , and hence in M.
For ρ with degenerate spectra, continuity considerations lead to the conclusion that one can always find vectors in its eigenspaces that are in M.

D. Extremal CPT and UCPT maps
The following theorem by Choi [10] characterises the extremal CPT maps. For CPT maps, the requirement is Φ ′ (1 1) = 1 1. As a simple consequence of this Theorem we note that the extremal ddimensional CPT maps have rank at most d. This is because at most d 2 matrices of size d × d can be linearly independent. In other words, the convex set of CPT maps on a d-dimensional Hilbert space is the convex hull of the set of CPT maps of rank at most d. Numerical experiments for d up to 6 reveal that in d dimensions one can find d matrices A k satisfying this condition, so that the constraint R ≤ d is saturated. Furthermore, the condition is generically satisfied for a randomly generated set of d matrices A k . Let us now proceed to the study of extremal unital CPT maps. The relevant generalisation of Choi's theorem is ([1], Theorem 2): is linearly independent.
This directly implies ( [1], Remark 3) that the extremal maps have rank not higher than √ 2d. Again, numerical experiments for d up to 6 reveal that in d dimensions one can find ⌊ √ 2d⌋ matrices A k satisfying the condition of the Theorem, so that the constraint R ≤ √ 2d is saturated; moreover, the condition is generically satisfied for a randomly generated set of ⌊ √ 2d⌋ matrices A k .
This remains particularly true for unital CPT maps, up to one exception: for d = 2 one cannot find more than 1 matrix A k obeying the independence condition. This is in accordance with the statement ([1], Theorem 1) that for d = 2 there are only rank-1 extremal maps. The existence of rank-2 extremal unital CPT maps is prevented by the conditions for double stochasticity, A 1 A † 1 + A 2 A † 2 = 1 1 and A † 1 A 1 + A † 2 A 2 = 1 1, which imply that the singular value decompositions of A 1 and A 2 must be A 1 = U Σ 1 V † and A 2 = U Σ 2 V † , (with the same U and V !) with Σ 2 1 + Σ 2 2 = 1 1. Therefore, the set {A † k A l ⊕ A l A † k ; k, l = 1, 2} is not independent. Indeed, two of its elements are equal: In this context, the following conjecture is of relevance [11]. The conjecture is supported by numerical evidence.

Conjecture 1 (Audenaert-Ruskai) Every d × d state ρ can be written as an equal-weight average of d states ρ i (not necessarily different) that are of rank at most d and have partial
traces Tr A ρ i and Tr B ρ i identical to those of ρ.

Remark.
In numerical experiments one is confronted with the question of how to generate random CPT maps, unital CP maps, and doubly stochastic CP maps. The first two questions are readily solved: one generates a random CP map, and then projects it onto the set of CPT maps or unital CP maps, respectively. Here, the two respective projections are the operations (performed at the level of the map's Choi matrix): Note that these projections preserve CP-ness and do not increase the rank of Φ.
The question of how to generate random unital CPT maps is slightly harder, as one has to satisfy the two constraints of TP-ness and unitality at once. Fortunately, this can also be done using a projection method. The method, called "projections on convex sets" (POCS), consists of an iterative scheme whereby the two projections G 1 and G 2 are alternatingly applied to an initial CP map. It turns out that, due to the convexity of the two sets, this process converges very quickly [12] to a CP map in their intersection, i.e. to a CP map that is both CPT and unital.

IV. DISTANCE MEASURES
While up to this point we have been looking at conditions under which a unital CPT map is a Random Unitary map, we can modify our method slightly to calculate a kind of distance between a given map and the set of Random Unitary maps. In general, one can define distances between a point and a set as the minimal distance between that point and any point from the set. Choosing different point-to-point distance measures thus induces different point-to-set distance measures. The exact choice of distance measure may be guided by considerations of physical relevance or just of mathematical convenience. An example of a distance measure with a clear physical meaning and relevance is the gate fidelity [13]. However, the Hilbert-Schmidt norm distance between the state representatives is arguably the simplest one when it comes to actually performing the minimisation.
To calculate the chosen point-to-set distance, the point-topoint distance has to be minimised over all points of the set. The problem we encounter in our situation is that the set is defined in terms of its extremal points, of which there is an infinite number. The minimisation has thus to be performed over all possible convex combinations of an infinite number of points. By Caratheodory's theorem [14], only a finite number of points can have a non-zero contribution to the convex combination. If the set is embedded in a d-dimensional real space, the maximal number of points required is d. In the present case (maps on C d ), we are dealing with d 2 × d 2 PSD matrices, which are embeddable in a d 4 -dimensional real space. Therefore, we need at most d 4 points to make up the convex combination. Nevertheless, the minimisation consists of varying d 4 − 1 real convex weights and d 2 − 1 real parameters (to make up an SU (d)-unitary) for each of the d 4 extremal points, hence of the order of d 6 parameters in total.
In the following we take a different approach, by modifying the treatment from the previous Sections such that a quantity emerges that is more easy to calculate than induced point-toset distance measures but also has an interpretation as a distance measure. To do so, we take the vectors of diagonal elements Diag(T † A i T ), concatenate them into a single vector and then find the T that minimises a well-chosen norm of that vector. The minimal norm then quantifies how far the given map is from the set of Random Unitary maps. At this point, we cannot yet say that what we get in this way is a genuine distance measure. What we do get already is that the given map is Random Unitary if and only if this minimal norm is 0.
First of all, we have to properly choose a vector norm. We can, for example, choose a norm of the form and its minimal value which is to serve as distance quantification between the given map and the set of Random Unitary maps. In what follows, we will restrict our attention to D 2,1 , which we denote by D without subscripts, because it has a number of desirable mathematical properties. For instance, it can be redefined as the convex hull of a simple function on pure states (see Property 4 below).
Calculating D now requires minimisation over T , which is a right-unitary matrix of dimension d 2 ×K with K at most d 4 ; again there are of the order of d 6 parameters, but in this case they are all contained in a single mathematical object, a rightunitary matrix, rather than in several unitaries and a number of convex weights. The minimisation therefore has a simpler mathematical structure, which leads to simplifications at the level of actual algorithms, but also concerning the derivation of its basic properties. In [5], a modified conjugated gradient method is described for minimising functionals over the manifold of right-unitary matrices. This method is directly applicable to the problem at hand, and we have implemented it in Matlab [15].
In the rest of this Section, we discuss various properties of D, including lower and upper bounds that are easy to calculate.

Property 1 D(ρ) is invariant under "local" unitaries, that is, unitaries operating on input or output space separately.
Proof. When ρ is subjected to local unitary rotations, ρ → W ρW † , where W = U ⊗ V , the A i matrices will transform according to The "outer" appearence of the unitary conjugation will of course be absorbed in the unitary T in (T † A i T ) and plays no further role for determining D(ρ). The unitary conjugation on the τ i corresponds to replacing τ i by k O ik τ k (1 ≤ i ≤ d 2 − 1), for some real orthogonal matrix O (see Section II C). The entries (T † A i T ) jj are replaced accordingly by k O ik (T † A k T ) jj , which amounts to rotating each one of the vectors ((T † A i T ) jj ) i (for all j). As rotations leave the length (ℓ 2 -norm) of a vector unchanged, this shows that D(ρ) is indeed invariant under local unitaries.

Property 2
The value of D for pure d × d states is given by The maximal possible value is 2(1 − 1/d), a value that is achieved for pure product states. The minimal possible value is 0, which is achieved for ME pure states.
Note that in the context of Random Unitary maps, we are not directly interested in D on all possible pure states. Nevertheless, it can be calculated and, moreover, it will be useful in what follows. Note also that (17) applied to mixed states in N always yields 0, irrespective of whether they are in M or not.
Proof. We already know that T will be of no influence, because a pure state can only be realised in one way. We can therefore put T = 1 1.
For ρ = |ψ ψ|, the matrices A i are given by Then the diagonal entries of A i are given by Thus, using σ as a shorthand notation for Tr B (|ψ ψ|),

Property 3
The maximal possible value of D for a state in N is 2(d − 1)/d.
An example of a state achieving this value is the state |0 0| ⊗ 1 1/d, which is the state representative of the CP map representing photon loss as well as spontaneous decay of excited atoms: Φ(·) = Tr[·]|0 0|.
Proof. For convenience, we consider the state |d d| ⊗ 1 1/d instead. The corresponding A i are given by The only τ i with non-vanishing (d, d)-entry is τ z;d−1 . Its (d, d)-entry is given by − 2(d − 1)/d. Thus, we get As this value is independent of T , this is also the value of D(ρ). By Property 2, this is the maximally achievable value of D throughout, and what we have just shown is that this value is achievable for states in N .

Property 4
The function D is the convex hull of its restriction to pure states. That is: As a direct consequence, D is a convex function.
Proof. We proceed in much the same way as we did in Section III C, where we obtained a criterion for Random Unitary-ness in the mixed case. Again we need to consider all possible ensembles realising the state ρ, which we can do by varying over all right-unitaries T in the expression Z = ρ 1/2 T (where ρ = ZZ † ). Recall that the j-th column of Z is then Z j := √ p j |ψ j . By this variation over T we minimise the quantity The minimisation over T appearing in the convex hull construction thus, indeed, yields D(ρ).
Remark. Since the convex hull construction has a dual, we can find an expression of D involving a maximisation. The convex hull conv(f ) of a function f can be expressed as the Legendre transform of the Legendre transform of f [14]. That is, conv(f ) = f * * , where the Legendre transform f * of f is, in the quantum context, defined as where the maximisation is over all states ρ, and the argument X is a Hermitian operator. In particular, we have where the last maximisation is over pure states because D is the convex hull of D restricted to the pure states.
Property 5 A lower bound on D(ρ) can be given in terms of the 2-norm distance of ρ to M: Namely, Proof. An upper bound on any norm distance of ρ to M is obtained by restricting σ to the extremal points of M, which are the ME pure states. Recall that these can be parameterised as |φ = (U ⊗ 1 1)|I for unitary U . Le us confine attention to pure ρ = |ψ ψ| first. Thus we are looking now at the minimisation Let us now consider the 2-norm distance D 2 . Taking its square yields Note that normalisation of ψ amounts to ψ 2 = 1. On the other hand, A simple application of Hölder's inequality yields the following: for X such that X 2 = 1, X 4 ≥ X −1/2 1 . Equality is achieved for X = 1 1 d / √ d. Thus we obtain, for pure states, It is easily checked that the RHS is a convex, increasing function of D 2 . Since D 2 (ρ) is by definition a convex function of ρ, it follows that the RHS is a convex function of ρ too. Now, D is the convex hull of the restriction of D to pure states, which means that the largest convex function that coincides with D on pure states is D itself. Therefore, the above inequality on pure states readily extends to mixed states.
For small D 2 , this lower bound simplifies to D(ρ) ≥ D 2 (ρ)/ √ d. An important consequence is that D has the desirable property of being approximately linear for states very close to M, just like the distance measures D 1 and D 2 . This tells us that D can indeed be interpreted as a distance (and not, for example, a power of a distance).

Property 6 A lower bound on D(ρ) is given by
Proof. This is a simple consequence of Property 4. By definition, the convex hull of a function f that is defined on the pure states is the largest convex function coinciding with f on the pure states. The two functions appearing in the maximum are both convex functions that coincide with D on the pure states, and must therefore be smaller than or equal to D.
For 2-qubit states that have rank 2, and hence for 2-element qubit maps, it turns out that equality holds so that we get a closed-form analytic expression for D: Property 7 For 2-qubit states ρ of rank 2, While we have not been able to prove this yet, numerical experiments indicate that the expression also holds for 2-qubit states of arbitrary rank. Furthermore, this would be a simple consequence of the Audenaert-Ruskai conjecture for d = 2, combined with convexity of D and the statement for rank-2 states.
Now K, G and H are 2 × 2 traceless Hermitian matrices, and are determined by three real parameters: the entries of their respective Bloch vectors. The previous equation can thus be put in vector form. In order for it to have a solution for θ, the three Bloch vectors of K, G and H must be linearly dependent, with real proportionality constants.
Expressing the conditions for linear dependence is the first step to solving our problem. Since the only non-zero component of K is its z-component, the three Bloch vectors are linearly dependent with real proportionality constants if and only if the (1,2)-entries of G and H (corresponding to their x and y Bloch vector entries) have the same argument (modulo π). That is: The only unknown here is χ, and it turns out that there always is a solution. Therefore, the condition of linear dependence fixes χ. To wit, the solution is e iχ = (sin(2α) − sin(2β) cos η − i sin(2β) sin η)/Q, Q = sin 2 (2α) + sin 2 (2β) − 2 sin(2α) sin(2β) cos η.
With this choice of χ, we thus have a solution to the equation b(K − G) = (K + G) + aH in a and b. Now we must make sure that this solution satisfies a = sin(2θ)/ cos γ, b = cos(2θ)/ cos γ for some γ, θ. This is so provided a 2 + b 2 ≥ 1. We have been able to show that the solution (a, b) of this system, with these rather formidable expressions for the coefficients, indeed satisfies a 2 + b 2 ≤ 1 for 0 ≤ β ≤ π/4, and a 2 + b 2 ≥ 1 for π/4 ≤ β ≤ π/2. This was done using a computer algebra system, and we refer the interested reader to the supplementary material [15]. This proves the Proposition for rank 2 states.

V. ENTANGLEMENT OF ASSISTANCE
In this final Section, we briefly touch upon another approach to characterise Random Unitary maps which is still similar in spirit to the one discussed before. Rather than using quadratic relations to characterise whether a pure state is maximally entangled, one can look at the entropy of entanglement of the state. Applying a similar procedure as above then yields the so-called entanglement of assistance. The entanglement of assistance, E A , of a state [16] is in some sense the converse of the entanglement of formation, E F . It is defined as whereas replacing the maximisation by a minimisation yields the entanglement of formation. In other words, while E F is the convex hull of the pure state entanglement functional E, E A is its concave hull.
The connection between this quantity and the problem of determining Random Unitaryness of a map is quite clear: a ddimensional CP map is a Random Unitary map if and only if its state representative is a convex combination of maximally entangled pure states, if and only if that state representative has the maximal possible E A of log d.
Fo qubit maps one has the alternative of using the concurrence of assistance, C A , defined in [17] as the concave hull of the pure state concurrence. Thus, a 2-dimensional CP map is a Random Unitary map if and only if its state representative has the maximal possible C A of 2. A closed-form expression for C A is given by [17] C A (ρ) = ρ 1/2 (σ y ⊗ σ y )ρ 1/2 1 . (23)

VI. CONCLUSIONS
The set of completely positive maps can be distinguished by their decomposability into a convex combination of unitary conjugations. We have shown that one can find necessary and sufficient conditions for determining whether a given CPT map belongs to the set of Random Unitary maps. The criterion requires the collective "off-diagonalization" of d 2 − 1 matrices of dimension d 2 × d 2 that are built from the d 2 − 1 basis elements of the Hilbert space into which the density matrices on H are embedded.
Based on this criterion, we have defined a proper distance measure to the set of Random Unitary matrices which can be used to quantify the extent to which non-classical error mechanisms have influenced the evolution of a quantum system. In this way, it would be possible to point towards the dominating error source in a specific physical realization of the CPT map. This means that tomographic reconstructions of physical processes can provide a host of valuable information about the process itself, of which the amount of non-classical error mechanisms is but one. We believe that a thorough investigation of tomographic process reconstruction can reveal a plethora of information about the underlying physical mechanisms that led to the realization of the CPT map under investigation, and which has hitherto not been fully appreciated.