The Optimal Uncertainty Relation
Employing the lattice theory on majorization, we obtain the optimal bound for the universal quantum uncertainty relation of any number observables and general measurement. It is found that the majorization lattice can induce one type of metric about the incompatibility of different observables, which provides a systematic optimizing procedure for the entropic uncertainty relation. We find this procedure is in fact correlated with the entanglement transformation under local quantum operations and classical communication. Interestingly, the optimality of the universal uncertainty relation is found can be depicted by the Lorenz curve, initially introduced in economics.
The uncertainty principle is one of the few extraordinary features distinguishing quantum theory from classical ones. It reflects the limitation in acquiring the information of different physical properties of a system simultaneously. The idea of indeterminacy was first proposed by Heisenberg in the form of , where is the Planck constant, and represent the precisions in determining the canonical conjugate observables and . In the literature, whereas the most representative uncertainty relation is the Heisenberg-Robertson one :
Here the uncertainty is characterized in terms of variance ( for an observable ). Equation (1) asserts a fundamental limit to the uncertainties of incompatible observables expressed in form of commutator.
The essence of different forms of the uncertainty relations lies in the lower bound, whose optimization is generally a challenging task. A lasting criticism on variance based uncertainty relation is about its lower bound state dependence . In order to be state independent [4, 5], the variance based uncertainty relations have to involve complex variance functions . On the other hand, the entropic uncertainty relation was proposed with state independent lower bound , in the form of
where denotes the Shannon entropy of outcome probability distribution while is measured; quantifies the complementarity of observables with and being the eigenvectors of and . Studies indicate that these two different forms of uncertainty relations are in fact mutually convertible .
One main subject in the study of entropic uncertainty relation is about the lower bound optimization, which turns out to be difficult for general observables in high dimensional system . The majorization uncertainty relation has been called universal  and been exploited to refine the entropic uncertainty relation , of which the direct sum form usually has a better lower bound than the direct product ones , and both of them remain to be further optimized [13, 14, 15]. The majorization relation is a partial order on probability distribution vectors with descending order components, and has been shown to form a lattice . The majorization lattice has proper definitions on upper and lower bounds, and a recent development appears in its application to econometrics [17, 18]. Notice of these, naturally, one is tempted to think of formulating the uncertainty relation from the lattice theory, in order to get a properly defined and optimized uncertainty relation.
In this work, by virtue of the properties of Hermitian matrix we shall derive the optimal universal uncertainty relation in the form of direct-sum majorization relation, which is applicable to multiple observables and general positive operator-valued measurements (POVM). It indicates that the lattice theory can guarantee the optimality of the universal uncertainty relation and implies a metric to the probability distribution vectors , which may be employed to improve the entropic uncertainty relation further. We illustrate the optimality of the universal uncertainty relation by Lorenz curve that was originally introduced to describe the wealth concentration in a society .
2 The optimal universal uncertainty relation
2.1 Quantum measurements and the majorization lattice
In quantum mechanics (QM), physical observables are represented by Hermitian operators. And therefore in the -level system, an observable appears in the form of a -dimensional Hermitian matrix, whose spectrum decomposition goes as
Here, is the eigenvector that . The quantum state of the system is also a Hermitian matrix with nonnegative eigenvalues , which may be expressed as a vector , where the superscript T denotes the transpose of matrix. Moreover, the measurement postulate of QM tells that when measuring over a quantum state one can only get its eigenvalue with a probability of . Similar to , we can express the probability distribution in the form of a vector, .
We define a set of Hermitian operators
where means the cardinality of the set . For given , equals to , that means the operators in are composed of various distinct projection operators from the complete set, and evidently . The partial sum of the probability distribution now may be expressed as
Here denotes the matrix with particular . Equation (5) also applies to the general POVM, given the projection operators are replaced by positive semidefinite operators , which satisfy the normalization condition .
The majorization relation between two tuples of real numbers, say for instance, is defined as :
where the superscript means that the components of vectors and are arrayed in descending order, and the equality holds when . For set
the following Lemma exist .
For all , there exists a unique least upper bound such that the followings are satisfied:
1. and ;
2. For arbitrary , if and , then .
There also exists a unique greatest lower bound defined as , and hence together with the majorization relation form a lattice. Practical methods for constructing and were given in Ref. .
2.2 The optimal universal uncertainty relation
Evidently, the probability distribution of observable measurement outcomes may be expressed as a high dimensional vector in the form of direct sum. Hence for observables , , and , the corresponding vector turns out to be the -dimensional vector , with , , . If the vector components are rearranged in descending order, one can notice . Different quantum state corresponds to different , and
with gives the sum of components of . Here is the eigenvalue list of , and pure state has the largest value of which is the largest eigenvalue of . According to equation (4), (similarly the and ) has different choices, hence varies with the choices of , , and ,
Let be the vector that has the largest sum of the first components, i.e., where the maximization runs over different that and for choices of , , and . We have the following optimal universal uncertainty relation as our main result:
In -dimensional quantum system , the probability distributions of measurements on , , and satisfy the following relation:
Here is the unique least upper bound for over all quantum states.
Proof: First, from the definition of and the associative laws for operation of lattice, we have
Because has the largest possible value of the sum of the first components than any other quantum states, satisfies for all quantum states.
Second, for arbitrary , if for all quantum states, we should find for all . According to Lemma 1, then
Repeatedly applying equation (12) to will in the end lead to . Q.E.D.
Note that the number of observables can be arbitrary in Theorem 1, and the general POVM measurement is also applicable here. Most importantly, Theorem 1 applies equally well to mixed states with given according to equation (8), and is optimal for such mixed states by maximizing the corresponding . From equation (8), it is also clear that the least upper bound of equation (10) for mixed states are majorized by that of the pure states . Though may not be unique for a given , different with the same sum of the first components will not effect the vector . Applying Theorem 1 to Shannon entropy of probability distribution vector, , we immediately obtain the following entropic uncertainty relation:
For observables , , there exists the following entropic uncertainty relation
Here with being the probability distribution of the measurement of -th observable ; is defined in Theorem 1 satisfying .
Given that one has noticed the Shannon entropy is a Schur-concave function , the prove of equation (13) is quite straightforward and no need for further explanation. The Corollary 1 in fact can be further improved by adding a state-dependent term, i.e.,
For a given set of incompatible observables, e.g. , , and , quantum states and will result in two probability vectors . Without loss of generality, here we assume the components of are arranged in non-increasing order. Direct application of Corollary 1 predicts that for all quantum states. The property of the majorization lattice tells that there exists a distance measure on , that is
In account of this metric, we may get the following corollary:
For arbitrary different probability distribution vectors and , we have the entropic uncertainty relation
The while and are different vectors.
Proof: The lattice theory tells that, if , then for both , and hence
Corollary 2 exhibits an interesting phenomena of the majorization lattice, i.e., the summation of two independent uncertainty relations produces a stronger one because . We belive that the lattice theory provides a more appropriate formalism for the study of uncertainty relation. From theorem 1, the unique least upper bound in majorization lattice establishes an optimal bound for the universal uncertainty relation. The metric revealed by the lattice theory can be employed to distinguish the uncertainties of different quantum states, whereas entropy can not, say can be nonzero even if . In the following section, we give some examples to show the extraordinary functions and uses of the Theorem and Corollaries.
2.3 The optimality of the uncertainty relation and Lorenz curve
Consider following two observables in the general qubit system,
The probability distribution vectors are then four dimensional. For states with , has the largest component (see equation (8) for the definition of ) and may be obtained, for instance, by . With descending order in components, we have
Since has the largest sum of any components, . The probability vector with the largest sum of any two components reads
which can be obtained by . Here and are orthogonal bases.
Following the procedure of Ref. , we have
The probability distribution vectors , , and are depicted in the form of Lorenz curve in Figure 1(a) for pure states of and . The Lorenz curve for a probability distribution vector is with . For completely mixed state , it has the probability distribution of , whose Lorenz curve goes from (0,0) to (4,2), the dashed anti-diagonal line in Figure 1(a). The Lorenz curve of each lies below the curve of and above the anti-diagonal line . Clearly, the Lorenz curve of is the least possible envelope, red dashed line in Figure 1(a), enclosing the curves of , and is optimal for the universal uncertainty relation for any quantum states.
Similarly, for three observables of , , and in pure qubit system, we can find the optimal bound for . The vectors , which have the largest sum of first components, are
The corresponding states giving are
and can be obtained through
we can readily get the optimal bound for the universal uncertainty relation, i.e.,
corresponds to the optimized bound of in .
Though being optimal for universal uncertainty relation, in Theorem 1 is unattainable for single quantum state, since it contains components from different as per operation. Hence, in Corollary 1 will not be the optimal lower bound for entropic uncertainty relation. Nevertheless, we notice that outperforms most of the uncertainty lower bounds in entropic form, especially for mixed states. For example, in qubit system with obervables of and in equation (18), there exist the following strengthened entropic uncertainty relation [14, 23]
where . For and , however, we have , which is greater than the lower bound of (31), .
The procedure of optimizing entropic uncertainty relation is to find the minimum value of over all quantum states. Giving the minimum value, the vector must be incomparable with under the majorization relation, that is the Lorenz curves of intercross with that of . For incomparable vectors under majorization, there exists the catalytic phenomenon which has been observed in entanglement transformation under local quantum operations and classical communication . This phenomenon makes the comparison of different entropic measures more complicated. That is, for and , there may exist an unknown catalytic probability tensor that determines the relative size of and . The optimization of entropic uncertainty relation is now turned to finding the quantum state whose catalytically majorizes others, which is hard to be solved analytically . It is worth mentioning that majorization lattice has, and may have more, profound applications in the entanglement transformation [27, 28].
In this work we have explored the uncertainty relation by employing the lattice theory, and obtained the optimal bound for universal uncertainty relation, which is applicable to general measurement. The lattice theory can not only provide a unique least upper bounds for the universal quantum uncertainty relation, but also substantially enhances the entropic uncertainty lower bound. Moreover, we find the optimality of the uncertainty relation can be intuitively exhibited by the Lorenz curve, which was initially introduced in social science. Finally, the majorization lattice is found can give out an explicit explanation for the difficulties in optimizing the entropic uncertainty relation, in addition to its important application to entanglement transformation.
This work was supported in part by the Ministry of Science and Technology of the Peoples’ Republic of China(2015CB856703); by the Strategic Priority Research Program of the Chinese Academy of Sciences, Grant No.XDB23030100; and by the National Natural Science Foundation of China(NSFC) under the Grants 11375200 and 11635009.
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For the sake of integrity, here we present some basic properties of majorization lattice and the method for constructing the least upper bound for the majorization lattice.
Appendix S1 The majorization lattice
The majorization relation between two tuples of real numbers is defined as [S1]:
where the superscript means that the components of vectors and are arrayed in descending order, and the equality holds when . Let be the set of all -dimensional probability distributions with components in nonincreasing order
The quadruple form a lattice, where is a set, is a partial ordering on , and there is a unique greatest lower bound (meet) and a unique least upper bound (join). The demonstration that is a lattice can be found in [S2, S3, S4, S5].
Appendix S2 Construction of the least upper bound
The construction of for can be found in [S5]. Here we summarize their procedure as follows.
First, we define the vector whose components are and
While , may not be in the set .
Second, there exists the following Lemma (Lemma 3 of [S5])
Let , and let be the smallest integer in such that . Moreover, let be the greatest integer in such that
Let the probability distribution be defined as
Then for the probability distribution we have that
Moreover, for all such that
we also have
Finally, if , i.e., there is no such that , then . If , by iteratively applying the transformation described in Lemma S1 with no more than iterations, we eventually obtain a vector such that, , and for any vector such that and , it holds also that . And therefore .
In order to construct the least upper bound for more than two probability distribution vectors we need the following theorem for a lattice (Theorem 2.9 in [S6])
Let be a lattice. Then and satisfy, for all
In a lattice, associativity of join and meet allows us to write iterated joins and meets unambiguously.
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