On the Alberti-Uhlmann Condition for Unital Channels

We address the problem of existence of completely positive trace preserving (CPTP) maps between two sets of density matrices. We refine the result of Alberti and Uhlmann and derive a necessary and sufficient condition for the existence of a unital channel between two pairs of qubit states which ultimately boils down to three simple inequalities.

It is well known that every positive trace-preserving map (PTP) is a contraction in the trace-norm and hence Alberti-Uhlmann is necessary for the existence of a PTP map T . Interestingly, this condition is sufficient in two dimensions-even for the existence of a quantum channel-and fails to be sufficient for dimension three and larger [13,Ch. VII.B].
In [9], Chefles et al. generalized the problem to input and output sets {ρ 1 , . . . , ρ n } and {τ 1 , . . . , τ n }, respectively, with arbitrary dimension and arbitrary value of n, under the constraint that at least one of the two sets must be a set of pure states. They derived conditions for the existence of a CPTP map between the sets in terms of the Gram matrices of the two sets. A result for arbitrary (non-pure) states was derived by Huang et al. [17] where their characterization (of existence of a CPTP map) goes via the existence of some more abstract decomposition of the initial and target states.
More interestingly, they considered the case of qubit states {ρ 1 , ρ 2 , ρ 3 }, {τ 1 , τ 2 , τ 3 } under the generic assumption of the input states being pure (cf. footnote [26]). However the characterization they derived, while verifiable with standard software, seems to not generalize to a condition for arbitrary input states and, moreover, seems to not lead to much physical insight.
In [13], Heinosaari et al. considered a slightly different variant of the problem and studied the conditions for existence of only CP maps between two sets of quantum states. Moreover they gave a fidelity characterization for the existence of a CPTP transformation on pairs of qubit states, consisting of only a finite number of conditions.
In a more recent paper [11], Dall'Arno et al. derived a condition for the existence of a CPTP map when the input set is a collection of qubit states which can be, through a simultaneous unitary rotation, written as real matrices. They study the problem from the perspective of quantum statistical comparison and derive that if the testing region of the input real states include the testing region of the output states then there exists a CPTP map connecting them. For further analysis on the relation of this problem with quantum statistical comparison see [6,7].
In this paper we refine the original Alberti-Uhlmann problem asking about the existence of a unital channel, that is, T maps ρ k to τ k and additionally T (1 1) = 1 1. Original condition (1) guarantees the existence of a CPTP map but does say nothing whether T is unital. Clearly, condition (1) is again necessary but no longer sufficient. Note, that the map T is uniquely defined only on an at most 3-dimensional subspace M spanned by {1 1, ρ 1 , ρ 2 } and one asks whether this map can be extended to the whole algebra M 2 (C) such that the extended map is CPTP and unital. Extension problems such as this one were already considered by many authors before [20]. The classical result of Arveson [4] says that if M is an operator system in B(H), that is, M is a linear subspace closed under hermitian conjugation and containing 1 1, and if Φ : M → B(H) is completely positive unital map, then it can be extended to a unital completely positive map Φ : B(H) → B(H). Note, however, that this result says nothing about trace-preservation. Hence, even if the unital map Φ is trace-preserving the unital extension Φ need not be trace-preserving. Actually, unitality may be relaxed by assuming that the hermitian subspace M contains a strictly positive operator [13]. Interestingly, it was shown [18] that if M is spanned by positive operators and Φ is completely positive then there exists a completely positive extension Φ.
The main result of this paper reads as follows.
The following statements are equivalent.
(i) There exists a unital quantum channel mapping Here ≺ denotes classical matrix majorization which is usually defined via the comparison of eigenvalues and is well-known to be equivalent to the existence of a unital CPTP map which maps the right to the left-hand side (refer to, e.g., [2,Ch. 7]).
Clearly, the original Alberti-Uhlmann condition (1) provides now only the necessary condition corresponding to α = 0 in (4). This condition readily serves as the necessary condition for existence of the unital channel as the trace-norm is contractive under the action of any PTP map [21]. While the conditions in Theorem 1 give conceptional insight one can also reduce the problem three easy-to-verify conditions. Theorem 2. There exists a unital quantum channel mapping qubit states {ρ 1 , ρ 2 } into {τ 1 , τ 2 } if and only if det(τ j ) ≥ det(ρ j ) for j = 1, 2 as well as where (·) # denotes the adjugate [15, Ch. 0.8], [27].
In other words Proposition 1 guarantees the existence of unitary channels which rotate both domain and codomain of T into the subspace span{1 1, σ x , σ y } and, at the same time, diagonalize T . From the latter one has the obvious inference: is positive iff |a| ≤ 1 and |b| ≤ 1.
In abuse of notation we henceforth write M for span{1 1, σ x , σ y }. Now, to extend S from the 3-dim. subspace M to the full space L(H 2 ) one needs the action of S on σ z . Due to hermiticity and the trace preservation condition, one has in general S(σ z ) = xσ x + yσ y + zσ z , with x, y, z ∈ R. Proof. The map S : M → L(H 2 ) maps a density matrix represented by a Bloch vector r = (r x , r y , r z ) to S(1 1 + r x σ x + r y σ y + r z σ z ) = 1 1 + ar x σ x + br y σ y + r z (xσ x + yσ y + zσ z ) . (8) Therefore the action on the Bloch vector is realized as The map S is positive iff it maps Bloch ball to Bloch ball, that is, if the matrix M is contraction. Its supremum norm reads and the inequality is saturated for x = y = 0. Hence, whenever |a| ≤ 1 and |b| ≤ 1, there exist x, y, z defining the extension (7). Remark 1. Note, that the simplest extension corresponds to S(σ z ) = 0, i.e. x = y = z = 0. Actually, it was shown [8] that if M is a 3-dim. operator system, then there exists unital positive trace-preserving projector Π : L(H 2 ) → M. Hence, S • Π defines unital positive trace-preserving extension of S : M → L(H 2 ). Interestingly, it was proved [8] that there is no completely positive trace-preserving projector Π : L(H 2 ) → M. Proof. One has and hence the corresponding Choi matrix |i j| ⊗ S(|i j|), The extended map S is completely positive iff C ≥ 0 [10]. A necessary condition for its positivity will be given by positivity of its two main minors: which is equivalent to the following conditions for z: Clearly, there exists nontrivial solution for z iff 2 ≥ |a − b| + |a + b| = 2 max{|a|, |b|}.
Observe, that for x, y = 0 the matrix (15) becomes a direct sum of blocks (16), hence the necessary condition becomes also sufficient.
(ii) ⇔ (iii) is readily verified using homogeneity and continuity of the norm.
Proof of Thm. 2. From the eigenvalue formula for 2 × 2 matrices it is easy to see that if A, B ∈ M 2 (C) are hermitian and of same trace then A ≺ B iff det(A) ≥ det(B). Using (3) (iv) is equivalent to for all t ∈ R. But a parabola t → at 2 + bt + c is nonnegative iff a, c ≥ 0 and b 2 ≤ 4ac which concludes the proof.

III. EXAMPLE
We now present an example where for an input and output pair of qubit states, although a CPTP extension exists, there is no unital CPTP extension. Thus this example clearly emphasizes the necessity of Theorem 1. Consider the following map which is CPTP whenever 1 ≥ p ≥ κ 2 . Let p = κ = 1 2 , and consider two density matrices: which are mapped to The Alberti-Uhlmann condition is obviously satisfied since by constriction the τ k are related to the ρ k via a CPTP map. Now let us check whether a unital extension exists via one of the equivalent conditions. By considering α = −.5 and β = γ = 1 in (19) one finds and and their trace norms are √ 1.6 and 1 respectively. Hence there is no unital channel mapping ρ k to τ k . Alternatively one can check that condition (6) reads .16 = (.8 − .4) 2 ≤ 4 · 0 · .2 = 0 , a contradiction. Indeed this condition allows us to answer the following question: how does τ 2 have to be modified to guarantee a simultaneous unital state transformation? Because of det(ρ 1 ) − det(τ 1 ) = 0 the transition by (6) is possible in a unital manner iff tr(τ # 1 τ 2 ) = .8 so with |z| ≤ .4. Thus the only allowed channels in this scenario are those which relax ((21) with 1 = p ≥ κ 2 ) and rotate the off-diagonals (unitaries of the form diag(1, e iφ )).

IV. CONCLUSIONS & OUTLOOK
In this paper we derived necessary and sufficient conditions for the existence of a unital quantum channel mapping a pair of qubit states {ρ 1 , ρ 2 } into {τ 1 , τ 2 }. These conditions connect the problem to trace-norm inequalities (in the spirit of Alberti-Uhlmann (1) which is reproduced by setting α = 0 in (19)) and majorization on matrices. Moreover we reduced the infinite family of conditions to just three inequalities which are simple enough to be verified with pen and paper. We also provided an example of two pairs of qubit states which sat-isfy the Alberti-Uhlmann condition, that is, there exists a quantum channel mapping ρ k to τ k , but condition (4) is violated which implies that there is no unital channel between ρ k and τ k .
We expect that our result will encourage more research in this direction and shed light on finding more general closed form conditions for existence of channels between sets of quantum states. Possible next steps could focus on the case of the input set consisting of any three linearly independent qubit states or-in spirit of thermoand general D-majorization [24]-how to modify Theorem 1 & 2 if the fixed point of the channel is not the identity but an arbitrary Gibbs state (i.e. an arbitrary positive-definite state D).