Conditions for optimal input states for discrimination of quantum channels
We find optimality conditions for testers in discrimination of quantum channels. These conditions are obtained using semidefinite programming and are similar to optimality conditions for discrimination of quantum states. We get a simple condition for existence of an optimal tester with any given input state with maximal Schmidt rank, in particular with a maximally entangled input state. In case when maximally entangled state is not optimal an upper bound on the optimal success probability is obtained. The results for discrimination of two channels are applied to covariant channels, qubit channels, unitary channels and simple projective measurements.
The problem of multiple hypothesis testing in the setting of quantum channels can be formulated as follows. Assume that is an unknown quantum channel, but some a priori information is available, in the sense that is one of given channels , with probabilities . The task is to find a procedure that determines the true channel, with the greatest possible probability of success.
For quantum states, this problem was formulated by Helstrom  and since then has been the subject of active research, see e.g.  for an overview and further references. Here, an ensemble is given, where are quantum states with prior probabilities , with a similar interpretation as above. A testing procedure, or a measurement, for this problem is described by a positive operator valued measure (POVM) , defined as a collection of positive operators summing up to the identity operator . The value is interpreted as the probability that the procedure chooses while the true state is . The task is to maximize the average success probability
over all POVMs. In the case , it is well known that the optimal POVM is projection valued, given by the projections onto the positive and negative parts of the operator , . For , there is no explicit expression for the optimal POVM in general, but it is known that a POVM is optimal if and only if it satisfies
In the case of quantum channels, a most general measurement scheme is described by a triple , where is an ancilla, a (pure) state on and is a POVM on . For , the value
is interpreted as the probability that is chosen when the true channel is . The average success probability is then
The task is to maximize this value over all triples .
It was observed [17, 5, 21, 20] that using entangled input states may give greater success probability and it was shown in  that every entangled state is useful for some channel discrimination problem. However, there are situations when e.g. the maximally entangled input state does not give an optimal success probability. It is therefore important to find out whether an optimal scheme with a given input state exists.
In the broader context of generalized decision problems, conditions for existence of an optimal scheme with an input state having maximal Schmidt rank were found in , a related problem was studied in . In the present paper, we show that these conditions can be obtained using the methods of semidefinite programming. Such methods were already applied before in the context of discrimination of quantum channels, see [7, 25, 12]. Compared to these works, we are more concerned with the choice of an optimal input state. It is an easy observation that if a given scheme is optimal, then must be an optimal measurement for the ensemble . We show that, at least in the case that the input state is assumed to have maximal Schmidt rank, the optimality condition for a channel measurement can be divided into the condition (1) for this ensemble and an additional condition that ensures optimality of the input state. If the Schmidt rank of the input state is not maximal we obtain a comparably weaker result, but show an example where the use of such an input state is required.
As an important special case, we get a necessary and sufficient condition for existence of an optimal scheme with a maximally entangled input state. If this condition is not fulfilled, we give an upper bound on the optimal success probability. For discrimination of two channels, we use the known form of an optimal POVM for two states to obtain a relatively simple condition in terms of Choi matrices of the involved channels, which we call the (MEI) condition. We also derive an upper bound on the diamond norm, which is tighter than the previously known bound given e.g. in , see Remark 2 below. The results are applied to discrimination of covariant channels, qubit channels, unitary channels and simple projective measurements.
The paper is organized as follows: in the next section we rewrite the problem as a problem of SDP from which we obtain necessary and sufficient conditions for optimal solution and derive an upper bound on the optimal success probability. In Section 3 we investigate the (MEI) condition and the related bounds. It the last two sections, we study special cases of channels and present some examples demonstrating the results.
2. Optimality conditions
Let be a finite dimensional Hilbert space. We denote by the set of positive operators and by the set of states, that is, positive operators of unit trace. A completely positive trace preserving map is called a channel, we will denote the set of all channels by . Any linear map is represented by its Choi matrix , defined in  as
here is a fixed orthonormal basis of . Note that if and only if is positive and .
An alternative description of a channel measurement is given in terms of process POVMs  (or testers , see also ). A process POVM is a collection of positive operators in with for some state . For any triple , there is a process POVM such that for all and ,
Conversely, for any process POVM , one can find some ancilla , a pure state and a POVM such that (3) holds . To see this, let , for and observe that by Schmidt decomposition, we have for some unitary operator and for every . Denoting we get
Since the channel in (3) acts only on the first part of the system we get
Using the description by process POVMs, we will show that the maximization of the success probability can be written as a problem of semidefinite programming:
Here , is the canonical basis of and is any basis of the (real) linear subspace
Put . Then:
and the problem of maximizing can be understood as the problem of maximizing . We have and from it follows . Note also that since is block-diagonal, we may extend the maximization over all positive elements with , (and not only over block-diagonal ones).
To rewrite this to the more usable form stated above, we need to note that with if and only if for all and . To prove this statement, let us first assume that , then for any ,
Conversely, assume that for all and . Consider as a Hilbert space with Hilbert-Schmidt inner product, then there is an orthonormal basis in , where each is a self-adjoint operator such that . With respect to this basis, each can be expressed as:
with some . From the condition we obtain . Expressing and using the condition we get
Since there is no restriction on for , we must have for all , and hence . To conclude the proof, from the condition we get . Moreover, it is worth realizing that from the condition we get , hence .
Let be a process POVM. Then is optimal if and only if there is some and some , such that for all ,
Moreover, in this case, the maximal success probability is
As first, we will formulate the dual problem. Let , be some basis of and let , then dual problem is:
Let be dual feasible, then since and , we must have . If we denote , then
and from we obtain . Hence there is some channel , such that . From the condition we obtain
for all . From here we see, that the dual problem may be formulated as:
Now let , then is a primal feasible plan. Moreover belongs to the interior of the cone of positive operators, therefore by Slater’s condition we obtain that the duality gap is zero, in other words or , where by we denote the primal optimal plan and by we denote the dual optimal plan. Since is feasible, we have and we get:
As and , the sum may be zero if and only if all summands are zero. Moreover, trace of the product of two positive matrices is zero if and only if their product is zero. To see this let and . We have and hence and .
By the above argumentation, we get from (7)
Using this result, we can characterize optimality of measurement schemes with input states of maximal Schmidt rank.
Let be a pure state such that is invertible. Then a measurement scheme is optimal if and only if
majorizes for all
Let and let be the operator such that , so that the process POVM corresponding to is given by , see (4). Note that by our assumptions, is invertible, and .
Assume that is optimal, then by Theorem 1, there must be some and such that and
Summing up over , we obtain
Multiplying the above equality by from the left and by from the right, we get using the above expression for ,
The two conditions follow easily from this equality.
Assume conversely that the conditions (i), (ii) are satisfied. Put , then (i) and (ii) imply that and . It follows that there is some positive number and such that . Moreover, (i) implies that for all and
It follows that and this implies the optimality condition of Theorem 1, exactly as in its proof. ∎
the majorization is satisfied. It follows that the existence of an optimal scheme with the given input state is equivalent to the condition (ii). Clearly, in this case, is the optimal scheme and the optimal success probability is .
Next, we show that the conditions of Corollary 1 are necessary for a general pure input state.
We will show that the measurement scheme is optimal for some problem with reduced input space. Let us denote by the support of . Since is pure, it must be of the form for some Schmidt decomposition of . From here we see that , where is a subspace isomorphic to . Let be the restriction of to and let be the projection onto , then it is clear that , moreover, defines an optimal measurement scheme for the reduced channels, with full Schmidt rank input state. The rest follows from the previous corollary. ∎
In general, the opposite implication does not hold. That is because if we limit the problem to some subspace of the original Hilbert space , then in general we don’t have a guarantee that the optimal input state will be supported on a subspace of the form , or in other words we would have to maximize the average success probability over all choices of the subspace . We demonstrate this by the following simple example.
Let , where and let , where is the optimal POVM for discrimination of the ensemble . By (1) we have
and . It is easy to see that both conditions (i) and (ii) are satisfied, but as argued in , there are cases when entangled input states give strictly larger probability of success than any separable state, so that a scheme of the form cannot be optimal.
It seems that optimality of input states strongly depends on the structure of the channels. In some cases it is even necessary to use an input state with lower Schmidt rank, because using maximal Schmidt rank input state would ”waste” some normalization of the input state on parts of the channels where it is unnecessary, as will be demonstrated in Example 4. It is an open question whether some stronger conditions for general input states can be obtained. See also [22, 20] for a discussion of a similar problem in the case of qubit Pauli channels.
We will next present an upper bound for in the case that condition (ii) is violated. We assume that the input state is maximally entangled, but a similar bound can be obtained for any input state having a maximal Schmidt rank.
Let be an optimal POVM for discrimination of the ensemble and let , . Let denote the operator norm. Then the optimal success probability satisfies
Note that is the largest success probability that can be obtained by the maximally entangled input state, this implies the first inequality. Further, note that we have by optimality of the POVM . If now and are such that , then , correspond to a dual feasible plan, hence by (6). To obtain the tightest upper bound in this way, we put
By the Choi isomorphism, there is some completely positive map , such that . As it was shown in  (see Corollary 2 and Section 3.1), , where the diamond norm is defined as
Moreover, since is completely positive, this norm simplifies to
In general, the bound that we obtain in this way does not have to be meaningful, that is, it may happen that . But, as will be demonstrated by the examples in the last section, there are cases when the bound is meaningful, or even tight.
Note that if , then and the value of indicates how much the condition (ii) is violated. It is easy to see that , this shows that if is small, the maximally entangled state is close to optimal.
3. Discrimination of two channels by maximally entangled input states
Let and . The following notation will be used throughout. Let , then we put
Let be the maximally entangled state and consider any two-outcome POVM on , given by for some operator on . The average success probability for the triple as defined by equation (2) is:
The optimal POVM is obtained if is the projection onto the support of the positive part of . In this case,
An optimal measurement scheme with a pure maximally entangled input state exists if and only if the Choi operators satisfy
By the remarks below Corollary 1, such a scheme exists if and only if , equivalently, . Since we always have and
the condition can be rewritten as stated. ∎
The following corollary describes the upper bound of the optimal probability.
We have the following bounds
If the condition (MEI) is satisfied, the inequalities become equalities.
We only have to note that if the (MEI) condition is satisfied, then . ∎
It is well known that is related to the diamond norm as . To our knowledge, the only known bounds on the diamond norm in terms of the Choi matrices are the following
where . This shows that if (MEI) is nearly satisfied, the above bounds are quite precise.
To show that the upper bound given by (11) is better than the bound we will show that in general
since is a state. This implies the above inequality. We also see that this inequality is strict unless is of rank 1.
We apply the results of the previous section to the problem of discrimination of covariant channels, unitary channels, qubit channels and measurements. In the case of covariant channels and unital qubit channels, similar results were obtained in  for more general decision problems on families of quantum channels.
4.1. Covariant channels
Let denote the unitary group of . For , let
Let be a group and let and be unitary representations. Assume that and are covariant channels, that is,
Irreducibility of plays a strong role, as we will see. In this case, the only non-zero projection that commutes with all is , see e.g. .
Let denote the transpose of with respect to the fixed basis . Let and let be as in (9). We will prove the proposition by showing that is invariant under and by the discussion above this implies that . For every we have
In case the representation is reducible, let us sketch an upper bound of the optimal probability. By the previous proof, we have , hence
where and are projections onto the subspaces of the irreducible representation, orthogonal sum of which is . Let then and we have
4.2. Qubit channels
Let and let us denote . Let be the Werner-Holevo channel, where denotes the transpose map with respect to the canonical basis . Then is a unitary channel, given by the unitary such that
It can be easily checked that and . If is a linear map such that there is some , satisfying
where . Moreover, for a self-adjoint , if and only if .
Let and be two qubit channels and let for as before. By the previous remarks, the condition (MEI) is equivalent to . We are now going to investigate this equality.
Note that satisfies (13) with . Since is a unitary channel, we have
Using further properties of and , we get
The last equality follows from the fact that , where denotes transpose with respect to the product basis , and that for any . Thus we have proved:
For a pair of qubit channels, the condition (MEI) holds if and only if
In particular, this is true if . If both channels are unital, this holds for any , hence maximally entangled input state is optimal, as it was already observed in  and in  in the case of qubit Pauli channels. If , then the condition (MEI) is satisfied if , even if the channels are not unital.
4.3. Unitary channels
Let and let , be the corresponding unitary channels. As it was proved in , it is not necessary to use entangled inputs for optimal discrimination of two unitary channels. Nevertheless, it is an interesting question whether a maximally entangled state is also optimal, this will be addressed in this paragraph.
Let . Since any input state may be replaced by , it is clear that discrimination of and is equivalent to discrimination of and the identity channel, and that a maximally entangled input state is optimal for one problem if and only if it is optimal for the other. We may therefore assume that and . Since the unitaries are given only up to a phase, we may also assume that . Put
so that , are the Choi matrices of the unitary channel and identity. By the results of the Appendix it is clear that if and only if , where
Since the transpose is a linear map and , we see that (MEI) is equivalent to