Mathematik in den Naturwissenschaften Leipzig Multi-observable Uncertainty Relations in Product Form of Variances

We investigate the product form uncertainty relations of variances for n (n ≥ 3) quantum observables. In particular, tight uncertainty relations satisfied by three observables has been derived, which is shown to be better than the ones derived from the strengthened Heisenberg and the generalized Schrödinger uncertainty relations, and some existing uncertainty relation for three spin-half operators. Uncertainty relation of arbitrary number of observables is also derived. As an example, the uncertainty relation satisfied by the eight Gell-Mann matrices is presented. Uncertainty relations [1] are of profound significance in quantum mechanics and also in quantum information theory like quantum separability criteria and entanglement detection [2, 3], security analysis of quantum key distribution in quantum cryptography [4], and nonlocality [5]. The Heisenberg-Robertson uncertainty relation [1,6] presents a lower bound on the product of the standard deviations of two observables, and provides a trade-off relation of measurement errors of these two observables for any given quantum states. Since then different types of uncertainty relations have been studied. There are many ways to quantify the uncertainty of measurement outcomes. In [1, 6–11] the product uncertainty relations for the standard deviations of the distributions of observables is studied. In [12–14] the uncertainty relations related to the sum of varinces or standard deviations have been investigated. And in [15–27] entropic uncertainty relations with majorization technique are explored. Uncertainty relations are also described in terms of the noise and disturbance [28,29], and according to successive measurements [30–33]. Let ρ be a quantum state and A be a quantum mechanical observable. The variance of A with respect to the state

which is further improved by Schrödinger, where {A, B} is the anticommutator of A and B. However, till now one has no product form uncertainty relations for more than two observables. Since there is no relations like Schwartz inequality for three or more objects, generally it is difficult to have a nontrivial inequality satisfied by the quantity (Δ A) 2 (Δ B) 2 … (Δ C) 2 . In ref. 14 Kechrimparis and Weigert obtained a tight product form uncertainty relation for three canonical observables p, q and r,  where τ = 2 3 , q and p are the position and momentum respectively, and = − −ˆr p q. As τ > 1 the relation (3) is stronger than the one obtained directly from the commutation relations = = =ˆˆˆˆp q q r r p [ , ] [ , ] [ , ] triple. In fact, besides the dual observables like position and momentum, there are also triple physical observables like spin, isospin (isotopic spin) related to the strong interaction in particle physics, angular momentum that their components are pairwise noncommutative.
Generally speaking, uncertainty relations are equalities or inequalities satisfied by functions such as polynomials of the variances of a set of observables. In this paper, we investigate the product form uncertainty relations of multiple observables. We present a new uncertainty relation which gives better characterization of the uncertainty of variances.

Theorem 1
The product form uncertainty of three observables A, B, C satisfies the following relation, where Re{S} stands for the real part of S. See Methods for the proof of Theorem 1.
The right hand side of (4) contains terms like 〈 BC〉 and 〈 CA〉 . These terms can be expressed in terms of the usual form of commutators and anti-commutators. From the Hermitianity of observables and (〈 AB〉 { , } ) 1 2 . By using these relations formula (4) can be reexpressed as, Formulae (4) or (5) give a general relation satisfied by (Δ A) 2 , (Δ B) 2 and (Δ C) 2 . To show the advantages of this uncertainty inequality, let us consider the case of three Pauli matrices A = σ x , B = σ y , and C = σ z . Our Theorem says that Let the qubit state ρ to be measured be given in the Bloch representation with Bloch vector → = r r r r ( , , ) 1 2 3 , i . e . ρ σ = + → ⋅ → I r ( ) 1 2 , w h e r e σ σ σ σ → = ( , , ) x y And the uncertainty relation (6) has the form The difference between the right and left hand side of (7) is − ∑ = r (1 ) . Therefore, the uncertainty inequality is tight for all pure states. Usually, a lower bound on the product of variances implies a lower bound on the sum of variances 37 . Indeed in these cases the l owe r b ou nd i n ( 7 ) a ls o g ive s a t i g ht l owe r b ou nd of t he su m of v ar i anc e s , s i nc e σ σ σ , where 7  is the right hand side of (7).
In fact, from the Heisenberg and Robertson uncertainty relation, One has Let us compare the lower bound of (7) with that of (8). The difference of these two bounds satisfies the following inequality, r r r r r r r r r r r r r r r r r r r r r rr r r r r . This illustrates that the uncertainty relation of three Pauli operators from Theorem 1 is stronger than the tighten uncertainty relation (8), obtained from the Heisenberg and Robertson uncertainty relation.
From the generalized Schrödinger uncertainty relation (2), one can also get an uncertainty relation for three observables, { , } Comparing directly the right hand side of (17) with the right hand side of (9), we obtain  where the second inequality is obtained by (9). Hence our uncertainty relation is also stronger than the one obtained from the generalized Schrödinger uncertainty relation.  are the right hand sides of inequalities (7), (8) and (9), respectively, see Fig. 1.
We have presented a product form uncertainty relation for three observables. Our approach can be also used to derive product form uncertainty relations for multiple observables. Consider n observables . Let  be the set consisting of all the subsets of I, and  k the set consisting of the subsets of I with k elements. Then we , , , A i s are the variance operators of A i s. For instance, we calculate the product form uncertainty relation for the eight Gell-Mann matrices λ = { } n n 1 8 , . And each two of them are anticommute i.e. {λ m , λ n } = 0(m ≠ n). Let us consider a general qutrit state ρ 40 , where ∈  r R 8 is the Bloch vector of ρ and λ is a formal vector given by the Gell-Mann matrices. For pure qutrit states the Bloch vectors satisfy =  r 1, and for mixed states <  r 1. However, not all Bloch vectors with ≤  r 1 correspond to valid qutrit states. For simplicity, we set r 2 = r 3 = r 5 = r 7 = r 8 = 0, and r 1 = a cos α, r 4 = a sin α cos β, r 6 = a sin α sin β, |a| ≤ 1. In this case ρ has the form From (11) we have the lower bound of (15),  , see the Fig. 2 for the uncertainty relation of these observables. For explicity, we fix the parameter β such that sin 2β = 1, the uncertainty relation is shown by Fig. 3.

Conclusion
We have investigated the product form uncertainty relations of variances for n (n ≥ 3) quantum observables. Tight uncertainty relations satisfied by three observables has been derived explicitly, which is shown to be better than the ones derived from the strengthened Heisenberg and the generalized Schrödinger uncertainty relations, and some existing uncertainty relation for three spin-half operators. Moreover, we also presented a product form uncertainty relation for arbitrary number of observables. As an example, we first time calculated the uncertainty relation satisfied by the eight Gell-Mann matrices. Our results have been derived from a class of semi-definite positive matrices. Other approaches may be also applied to get different types of product form uncertainty relations for multiple quantum observables.

Proof of Theorem 1
To prove the theorem, we first consider the case that all observables are measured in a pure state |ψ〉 . Let us consider a matrix M defined by and its lower bound. The upper surface is λ Then for any given mixed state ρ with arbitrary pure state decomposition ρ ψ ψ = ∑ p i i i i , the corresponding matrix M satisfies Therefore M is semi-definite positive for all variance operators A, B, C and any state ρ. Hence, we have det(M) ≥ 0, namely, By substituting the variance operator X = X − 〈 X〉 I, X = A, B, C, into the above inequality, we obtain the uncertainty relation (4). This completes the proof. The dashed line is  × 3 8 16 .