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On the minimum number of unitaries needed to describe a random-unitary channel

We provide, in an extremely simple way, an upper bound to the minimum number of unitary operators describing a general random-unitary channel.


Introduction
A channel E-i. e. a completely positive trace-preserving map-acting on density matrices ρ defined on a finite dimensional input Hilbert space H (for sake of simplicity, we consider here channels with equal input and output Hilbert spaces; the generalization is straightforward) is called random-unitary if it admits a Kraus representation [1] as where p i are probabilities and U i are unitary operators. This definition involves an existential quantifier, and there is no known constructive algorithm to check whether a given channel is random-unitary or not. Only the necessary condition of being unital, that is, of preserving the identity matrix, E(1 1 1) = 1 1 1, holds 1 . Nonetheless, random-unitary channels play a very special physical role among all possible evolutions that an open quantum system can undergo [3]. In fact, Gregoratti and Werner [4] proved that they are the only irreversible channels that can be perfectly corrected using, as the only side-resource, classical information extracted from the environment. This property is actually sufficient and necessary for a channel to be random-unitary and one would prefer to adopt this one as the physical and operational definition of random-unitary channels. The idea of using the environment as a resource then initiated investigations about environment-assisted capacities for quantum channels [5,6,7]. In Ref. [8] the problem also of quantifying the amount of classical information needed to perfectly correct a random-unitary channel was raised for the first time in the case of decohering evolutions. In fact, for a given random-unitary channel, the form (1) is highly non-unique and the Shannon entropy H(p i ) of the probability distribution weighing the unitaries U i can be "artificially" made as large as desired. Consequently, in order to derive sensible information-theoretic relations regarding the information dynamics in a random-unitary evolution, one has to single out the random-unitary Kraus representation minimizing H(p i ).
In the present paper we derive an upper bound for the minimal number of unitary operators needed in Eq. (1), thus providing also a bound to the amount H(p i ) of classical information needed to be extracted from the environment in order to invert the random-unitary evolution. Our bound, proved for generic dimension, does not catch the peculiar geometry that bistochastic qubit channels enjoy: For d = 2, it is provably non tight. However, the qubit case is completely understood and all random-unitary qubit maps have already been explicitly characterized (see, e. g. Ref. [9]). In this sense, a bound for the qubit case is completely useless. On the contrary, as soon as one leaves the two-dimensional world, already for d = 3, the bound we provide is generally non trivial.

Properties of random-unitary channels
Let us given a channel E acting on density matrices ρ defined on the input Hilbert space H . As a consequence of the Stinespring theorem [10], we can write it as follows [11] namely, as a unitary interaction between the system and an ancilla (or environment, described by the Hilbert space H a ), followed by a trace over the ancillary degrees of freedom. If the ancilla input state is a pure one-like in Eq. (2)-Gregoratti and Werner [4] proved that, for all possible unitary interactions U in Eq. (2), and for all possible decompositions of the channel As an immediate consequence, if the channel E admits a random-unitary decomposition as E(ρ) = i p i U i ρU † i , with U i unitary operators, there exists a rank-one POVM on the ancilla, {|α i α i | a }, such that the probability distribution of its outcomes does not depend on the input state ρ, since It is now useful to introduce the channel E from density matrices on H to density matrices on H a defined as Since the unitary interaction U is unique up to local isometries on H a , we can consider such an ancillary (or complementary [12]) channel as a canonical one. In turn, the channel E acting on density matrices, induces a unique dual ancillary channel E ′ acting on operators O a on H a as follows This is nothing but the Heisenberg picture for the ancillary channel E. Using this somehow involved notation, we can translate the Gregoratti and Werner theorem stating that a channel E admits a random-unitary representation (1) if and only if there exists a rank-one POVM {|α i α i | a } such that for all i, and for some fixed probability distribution p i . In fact In other words, the POVM {|α i α i | a } is mapped to a classical dice, namely, the POVM {p i 1 1 1 a }. Notice that the cardinality N of the POVM {|α i α i | a } coincides with the cardinality in the random-unitary decomposition (1).

Extremal rank-one POVM's
Let us now suppose that N > (dim H a ) 2 . Then we know that such a POVM is non extremal [13,14,15,16,17] and it can be convexly decomposed into extremal components On the other hand, the normalization condition i |α i α i | a = 1 1 1 a rules out the possibility that N < dim H a . A von Neumann rank-one measurement, with α i |α j = δ ij , achieves the lower bound N = dim H a .

The result
By now, we showed that a random-unitary channel always admits a randomunitary decomposition (1) involving at most (dim H a ) 2 unitary operators.
We can now tighten this bound by choosing dim H a as small as possible. The smallest 2 achievable dim H a for a given channel E coincides with the number of Kraus elements in an orthogonal -or canonical -Kraus representation, that is, E(ρ) = j K j ρK j with Tr[K † j K l ] ∝ δ jl . Such a number is precisely the rank of the Choi-Jamio lkowski [19,20] positive operator R E in one-to-one correspondence with the channel E and defined as where I is the identity channel, and |Ω is a non normalized ( Ω 2 = d) maximally entangled vector in H ⊗H . An orthogonal Kraus representation of E corresponds then to a diagonalization of R E . Thus, we have the main result Theorem A random unitary channel E always admits a random-unitary Kraus representation The bound (12) holds regardless of the dimension d of the input Hilbert space H . It is then reasonable that it fails in accurately describing the peculiar case of qubits (d = 2). In fact, it is known that all bistochastic qubit channels are actually Pauli channels (see, for example, Ref. [9]), that is, they can always be written as (apart from an overall rotation of the whole Bloch sphere) where {σ 0 ≡ 1 1 1, σ x , σ y , σ z } are the usual 2 × 2 Pauli unitary matrices, and p i is a probability distribution. Notice that Tr[σ i σ j ] ∝ δ ij : This means that the Pauli form of qubit bistochastic channels is a diagonalization of the channel itself and the equality K = rankR E (14) holds in this case. However, already for d = 3 there exist bistochastic channels that cannot be diagonalized on unitary operators (for an explicit example, see Ref. [8]). This evidence clearly does not prove our bound to be tight. It nonetheless shows that things, already for d = 3, acquire highly non-trivial geometric properties and get much more complicated. In all these cases, the bound given in the Theorem could be tight.
As an immediate consequence of the Theorem, it stems the following Corollary The minimum amount of classical information needed to be extracted from the environment in order to perfectly correct a random-unitary channel E is upper bounded as Moreover, since rankR E ≤ d 2 , the following quite loose-yet independent of the particular channel-bound holds

Concluding remark
It is noteworthy that we need no more than (rankR E ) 2 rank-one POVM elements in order to extract all the "useful" classical information from the ancilla. This is analogous to what happens in the case of optimal accessible information extraction: as proved by Davies [21], one never needs more that d 2 rank-one POVM elements in order to extract the maximum achievable accessible information from a d-dimensional system. In the case of accessible information extraction, Davies' bound seems to be tight, in the sense that examples can be constructed in which the maximum information gathering is achieved only by a POVM with maximum number of elements [22]. If the analogy is correct, the bound in the Theorem could be proved to be tight as well, while we expect that the bound given in the Corollary can be refined.