Enhanced Information Exclusion Relations

In Hall’s reformulation of the uncertainty principle, the entropic uncertainty relation occupies a core position and provides the first nontrivial bound for the information exclusion principle. Based upon recent developments on the uncertainty relation, we present new bounds for the information exclusion relation using majorization theory and combinatoric techniques, which reveal further characteristic properties of the overlap matrix between the measurements.

(2) depicts that it is impossible to probe the register B to reach complete information about observables M 1 and M 2 if the maximal overlap c max between measurements is small. Unlike the entropic uncertainty relations, the bound r H is far from being tight. Grudka et al. 15 conjectured a stronger information exclusion relation based on numerical evidence (proved analytically only in some special cases) is a nontrivial quantum bound. Such a quantum bound is recently given by Zhang et al. 18 for the information exclusion principle of multi-measurements in the presence of the quantum memory. However, almost all available bounds are not tight even for the case of two observables. Our goal in this paper is to give a general approach for the information exclusion principle using new bounds for two and more observables of quantum systems of any finite dimension by generalizing Coles-Piani's uncertainty relation and using majorization techniques. In particular, all of our results can be reduced to the case without the presence of quantum memory.
The close relationship between the information exclusion relation and the uncertainty principle has promoted mutual developments. In the applications of the uncertainty relation to the former, there have been usually two available methods: either through subtraction of the uncertainty relation in the presence of quantum memory or utilizing the concavity property of the entropy together with combinatorial techniques or certain symmetry. Our second goal in this work is to analyze these two methods and in particular, we will show that the second method together with a special combinatorial scheme enables us to find tighter bounds for the information exclusion principle. The underlined reason for effectiveness is due to the special composition of the mutual information. We will take full advantage of this phenomenon and apply a distinguished symmetry of cyclic permutations to derive new bounds, which would have been difficult to obtain without consideration of mutual information.
We also remark that the recent result 19 for the sum of entropies is valid in the absence of quantum side information and cannot be extended to the cases with quantum memory by simply adding the conditional entropy between the measured particle and quantum memory. To resolve this difficulty, we use a different method in this paper to generalize the results of ref. 19 in Lemma 1 and Theorem 2 to allow for quantum memory.

Results
We first consider the information exclusion principle for two observables, and then generalize it to multi-observable cases. After that we will show that our information exclusion relation gives a tighter bound, and the bound not only involves the d largest c u u ( , ) The symbol ↓ means re-arranging the components in descending order. The majorization vector bound ω for probability tensor distributions p p ( ) The bound means that Scientific RepoRts | 6:30440 | DOI: 10.1038/srep30440 for any density matrix ρ and  is defined by comparing the corresponding partial sums of the decreasingly rearranged vectors. Therefore ω only depends on c i i 1 2 20 . We remark that the quantity H(A) − 2H(B) assumes a similar role as that of H(A|B), which will be clarified in Theorem 2. As for more general case of N measurements, this quantity is replaced by (N − 1)H(A) − NH(B) in the place of NH(A|B). A proof of this relation will be given in the section of Methods. The following is our first improved information exclusion relation in a new form.

Theorem 1.
For any bipartite state ρ AB , let M 1 and M 2 be two measurements on system A, and B the quantum memory correlated to A, then 1 2 where ω is the majorization bound and B is defined in the paragraph under Eq. (7). See Methods for a proof of Theorem 1. Eq. (8) gives an implicit bound for the information exclusion relation, and it is tighter than log 2 (4c max ) + 2H(B) − H(A) as our bound not only involves the maximal overlap between M 1 and M 2 , but also the second largest element based on the construction of the universal uncertainty relation ω 21,22 . Majorization approach 21,22 has been widely used in improving the lower bound of entropic uncertainty relation. The application in the information exclusion relation offers a new aspect of the majorization method. The new lower bound not only can be used for arbitrary nonnegative Schur-concave function 23 such as Rényi entropy and Tsallis entropy 24 , but also provides insights to the relation among all the overlaps between measurements, which explains why it offers a better bound for both entropic uncertainty relations and information exclusion relations. We also remark that the new bound is still weaker than the one based on the optimal entropic uncertainty relation for qubits 25 .
As an example, we consider the measurements Our bound and log 2 4c max for φ = π/2 with respect to a are shown in Fig. 1. Figure 1 shows that our bound for qubit is better than the previous bounds r H = r G = r CP almost everywhere. Using symmetry we only consider a in       , 1 1 2 . The common term 2H(B) − H(A) is omitted in the comparison. Further analysis of the bounds is given in Fig. 2.
Theorem 1 holds for any bipartite system and can be used for arbitrary two measurements M i (i = 1, 2). For example, consider the qutrit state and a family of unitary matrices Upon the same matrix U(θ), comparison between our bound ω + B 2 1 2 and Coles-Piani's bound r CP is depicted in Fig. 3.
In order to generalize the information exclusion relation to multi-measurements, we recall that the universal bound of tensor products of two probability distribution vectors can be computed by optimization over minors of the overlap matrix 21,22 . More generally for the multi-tensor product corresponding to measurement M m on a fixed quantum state, there exists similarly a universal upper bound ω: . Then we have the following lemma, which generalizes Eq. (7).
See Methods for a proof of Lemma 1. We remark that the admixture bound introduced in ref. 19 was based upon the majorization theory with the help of the action of the symmetric group, and it was shown that the bound outperforms previous results. However, the admixture bound cannot be extended to the entropic uncertainty relations in the presence of quantum memory for multiple measurements directly. Here we first use a new method to generalize the results of ref. 19 to allow for the quantum side information by mixing properties of the conditional entropy and Holevo inequality in Lemma 1. Moreover, by combining Lemma 1 with properties of the entropy we are able to give an enhanced information exclusion relation (see Theorem 2 for details).
The following theorem is obtained by subtracting the entropic uncertainty relation from the above result.  Our new bound for multi-measurements offers an improvement than the bound recently given in ref. 18. Let us recall the information exclusion relation bound 18 for multi-measurements (state-independent):  Now we compare ∼ 1  with r x . As an example in three-dimensional space, one chooses three measurements as follows 26 : (0, 1, 0), (0, 0, 1);  Figure 4 shows the comparison when a changes and φ = π/2, where it is clear that r x is better than  ∼ 1 . The relationship between r opt and r x is sketched in Fig. 5. In this case r x is better than r opt for three measurements of dimension three, therefore min{r opt , r x } = min{r x }. Rigorous proof that r x is always better than r opt is nontrivial, since all the possible combinations of measurements less than N must be considered.
On the other hand, we can give a bound better than ∼ 2  . Recall that the concavity has been utilized in the formation of  ∼ 2 , together with all possible combinations we will get following lemma (in order to simplify the process, we first consider three measurements, then generalize it to multiple measurements).

Lemma 2.
For any bipartite state ρ AB , let M 1 , M 2 , M 3 be three measurements on system A in the presence of quantum memory B, then        and all figures have shown our newly construct bound min {r x , r y } is tighter. Note that there is no clear relation between r x and r y , while the bound r y cannot be obtained by simply subtracting the bound of entropic uncertainty relations in the presence of quantum memory. Moreover, if r y outperforms r x , then we can utilize r y to achieve new bound for entropic uncertainty relations stronger than ω − B N 1 .

Conclusions
We have derived new bounds of the information exclusion relation for multi-measurements in the presence of quantum memory. The bounds are shown to be tighter than recently available bounds by detailed illustrations. Our bound is obtained by utilizing the concavity of the entropy function. The procedure has taken into account of all possible permutations of the measurements, thus offers a significant improvement than previous results which had only considered part of 2-permutations or combinations. Moreover, we have shown that majorization of the probability distributions for multi-measurements offers better bounds. In summary, we have formulated a systematic method of finding tighter bounds by combining the symmetry principle with majorization theory, all of which have been made easier in the context of mutual information. We remark that the new bounds can be easily computed by numerical computation.

Methods
Proof of Theorem 1. Recall that the quantum relative entropy ρ σ ) Tr ( log ) Tr ( log ) 2 2 satisfies that ρ σ τρ τσ    H(A) is another quantity appearing on the right-hand side that describes the amount of entanglement between the measured particle and quantum memory besides − H(A|B).
We now derive the information exclusion relation for qubits in the form of + I M B I M B ( : ) ( : ) 1 2 , and this completes the proof.  Then consider the action of the cyclic group of N permutations on indices 1, 2,..., N, and take the average can obtain the following inequality:  we have