A Stronger Multi-observable Uncertainty Relation

Uncertainty relation lies at the heart of quantum mechanics, characterizing the incompatibility of non-commuting observables in the preparation of quantum states. An important question is how to improve the lower bound of uncertainty relation. Here we present a variance-based sum uncertainty relation for N incompatible observables stronger than the simple generalization of an existing uncertainty relation for two observables. Further comparisons of our uncertainty relation with other related ones for spin- and spin-1 particles indicate that the obtained uncertainty relation gives a better lower bound.

Uncertainty relation is one of the fundamental building blocks of quantum theory, and now plays an important role in quantum mechanics and quantum information [1][2][3][4] . It is introduced by Heisenberg 5 in understanding how precisely the simultaneous values of conjugate observables could be in microspace, i.e., the position X and momentum P of an electron. Kennard 6

and Weyl 7 proved the uncertainty relation
 where the standard deviation of an operator X is defined by ψ ψ ψ ψ ∆ = − X X X 2 2 . Later, Robertson proposed the well-known formula of uncertainty relation 8 which is applicable to arbitrary incompatible observables, and the commutator is defined by [A, B] = AB − BA.
The uncertainty relation was further strengthed by Schrödinger 9 with the following form Here the commutator defined as {A, B} ≡ AB + BA. It is realized that the traditional uncertainty relations may not fully capture the concept of incompatible observables as the lower bound could be trivially zero while the variances are not. An important question in uncertainty relation is how to improve the lower bound and immune from triviality problem 10,11 . Various attempts have been made to find stronger uncertainty relations. One typical kind of relation is that of Maccone and Pati, who derived two stronger uncertainty relations , and the sign on the right-hand side of the inequality takes + (− ) while i〈 [A, B]〉 is positive (negative). The basic idea behind these two relations is adding additional terms to improve the lower bound. Along this line, more terms [12][13][14] and weighted form of different terms 15,16 have been put into the uncertainty relations. It is worth mentioning that state-independent uncertainty relations can immune from triviality problem [17][18][19][20] . Recent experiments have also been performed to verify the various uncertainty relations [21][22][23][24] .
Besides the conjugate observables of position and momentum, multiple observables also exist, e.g., three component vectors of spin and angular momentum. Hence, it is important to find uncertainty relation for multiple incompatible observables. Recently, some three observables uncertainty relations were studied, such as Heisenberg uncertainty relation for three canonical observables 25 , uncertainty relations for three angular momentum components 26 , uncertainty relation for three arbitrary observables 14 . Furthermore, some multiple observables uncertainty relations were proposed, which include multi-observable uncertainty relation in product 27,28 and sum 29,30 form of variances. It is worth noting that Chen and Fei derived an variance-based uncertainty relation 30 for arbitrary N incompatible observables, which is stronger than the one such as derived from the uncertainty inequality for two observables 10 .
In this paper, we investigate variance-based uncertainty relation for multiple incompatible observables. We present a new variance-based sum uncertainty relation for multiple incompatible observables, which is stronger than an uncertainty relation from summing over all the inequalities for pairs of observables 10 . Furthermore, we compare the uncertainty relation with existing ones for a spin-1 2 and spin-1 particle, which shows our uncertainty relation can give a tighter bound than other ones.

Results
Theorem 1. For arbitrary N observables A 1 , A 2 , … , A N , the following variance-based uncertainty relation holds The bound becomes nontrivial as long as the state is not common eigenstate of all the N observables.
Proof: To derive (7), start from the equality we obtain the uncertainty relation (7) QED.
When N = 2 we have the following corollary

Corollary 1.1. For two incompatible observables A and B, we have
which is derived from Theorem 1 for N = 2, and stronger than uncertainty relation (5).
To show that our relation (7) has a stronger bound, we consider the result in ref. 10, the relation (5) is derived from the uncertainty equality Using the above uncertainty equality, one can obtain two inequalities for arbitrary N observables, namely The bound in (6) is tighter than the one in (12) 30 . However, the lower bound in (6) is not always tighter than the one in (13) (see Fig. 1).
, the relation (12) becomes Simplify the above inequality, we obtain which is equal to the relation (12).
Similarly, by using , our relation (7) becomes Simplify the above inequality, we get  which is equal to the relation (7). It is easy to see that the right-hand side of (17) is greater than the right-hand side of (15). Hence, the relation (7) is stronger than the relation (12).
Comparison between the lower bound of our uncertainty relation (7) with that of inequalities (6) and (13). Here, we will show the uncertainty relation (7) is stronger than inequalities (13) and (6) for a spin-1 2 particle and measurement of Pauli-spin operators σ x , σ y , σ z . Then the uncertainty relation (7) has the form We consider a qubit state and its Bloch sphere representation where σ σ σ σ = ( , , ) x y z are Pauli matrices and the Bloch vector = 2 . The relation (18) has the form

. And the relation (19) becomes
Let us compare the lower bound of (22) with that of (23). The difference of these two bounds is x y x z y z x y z x y x z y z 1 9 for all ∈ − x y z , , [ 1,1]. When = = = ± x y z 1/ 3, the above inequality becomes equality, then the Eq. (24) has the minimum value > 1/2 0. This illustrates that the uncertainty relation (7) is stronger that the one (13) for a spin-1 2 particle and measurement of Pauli-spin operators σ x , σ y , σ z . Let us compare the uncertainty relation (18) with (20). The relation (20) has the form where we define β = − + 2 . Then the difference of these two bounds of relation (22) and (25)   . This illustrates that the uncertainty relation (7) is stronger that the one (6) for a spin-1 2 particle and measurement of Pauli-spin operators σ x , σ y , σ z .

Conclusion
We have provided a variance-based sum uncertainty relation for N incompatible observables, which is stronger than the simple generalizations of the uncertainty relation for two observables derived by Maccone and Pati [Phys. Rev. Lett. 113, 260401 (2014)]. Furthermore, our uncertainty relation gives a tighter bound than the others by comparison for a spin-1 2 particle with the measurements of spin observables σ x , σ y , σ z . And also, in the case of spin-1 with measurement of angular momentum operators L x , L y , L z , our uncertainty relation predicts a tighter bound than other ones.