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. 2016 Aug 8;6:31192. doi: 10.1038/srep31192

Multi-observable Uncertainty Relations in Product Form of Variances

Hui-Hui Qin 1, Shao-Ming Fei 2,3,a, Xianqing Li-Jost 3
PMCID: PMC4976374  PMID: 27498851

Abstract

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 relations1 are of profound significance in quantum mechanics and also in quantum information theory like quantum separability criteria and entanglement detection2,3,4, security analysis of quantum key distribution in quantum cryptography5, and nonlocality6. The Heisenberg-Robertson uncertainty relation1,7,8 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 refs 1,7, 8, 9, 10, 11, 12, 13, 14 the product uncertainty relations for the standard deviations of the distributions of observables is studied. In refs 15, 16, 17 the uncertainty relations related to the sum of varinces or standard deviations have been investigated. And in refs 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30 entropic uncertainty relations with majorization technique are explored. Uncertainty relations are also described in terms of the noise and disturbance31,32, and according to successive measurements33,34,35,36. Let ρ be a quantum state and A be a quantum mechanical observable. The variance of A with respect to the state ρ is defined by (ΔA)2 = 〈A2〉 − 〈A〉2, where 〈A〉 = tr(Aρ) is the mean value of A. From Heisenberg and Robertson1,7, the product form uncertainty relation of two observables A and B is expressed as

graphic file with name srep31192-m1.jpg

which is further improved by Schrödinger,

graphic file with name srep31192-m2.jpg

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 (ΔA)2 (ΔB)2 … (ΔC)2. In ref. 14 Kechrimparis and Weigert obtained a tight product form uncertainty relation for three canonical observables Inline graphic, Inline graphic and Inline graphic,

graphic file with name srep31192-m6.jpg

where Inline graphic, Inline graphic and Inline graphic are the position and momentum respectively, and Inline graphic. As τ > 1 the relation (3) is stronger than the one obtained directly from the commutation relations Inline graphic and the uncertainty relation (1). Here the ‘observable’ Inline graphic is not a physical quantity, neither independent in this 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.

Results

Theorem 1 The product form uncertainty of three observables A, B, C satisfies the following relation,

graphic file with name srep31192-m13.jpg

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〉)* = 〈BA〉, one has Inline graphic. By using these relations formula (4) can be reexpressed as,

graphic file with name srep31192-m15.jpg

Formulae (4) or (5) give a general relation satisfied by (ΔA)2, (ΔB)2 and (ΔC)2. To show the advantages of this uncertainty inequality, let us consider the case of three Pauli matrices A = σx, B = σy, and C = σz. Our Theorem says that

graphic file with name srep31192-m16.jpg

Let the qubit state ρ to be measured be given in the Bloch representation with Bloch vector Inline graphic, i.e. Inline graphic, where Inline graphic, Inline graphic. Then one has Inline graphic. And the uncertainty relation (6) has the form

graphic file with name srep31192-m22.jpg

The difference between the right and left hand side of (7) is Inline graphic. That is, the equality holds iff Inline graphic. 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 variances37. Indeed in these cases the lower bound in (7) also gives a tight lower bound of the sum of variances, since Inline graphic Inline graphic , where Inline graphic is the right hand side of (7).

In fact, from the Heisenberg and Robertson uncertainty relation, One has Inline graphic Inline graphic. However, this inequality is not tight. In ref. 38 the inequality is made tight by multiplying a constant factor Inline graphic on the right hand side, and the tighten uncertainty relation reads,

graphic file with name srep31192-m31.jpg

Let us compare the lower bound of (7) with that of (8). The difference of these two bounds satisfies the following inequality,

graphic file with name srep31192-m32.jpg

for all r1r2r3 ∈ [−1, 1], where in the first inequality we have used the fact that Inline graphic. 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,

graphic file with name srep31192-m34.jpg

where Inline graphic for X, Y = A, B, C, and A, B and C are the variance operators of A, B and C, respectively, defined by O = O − 〈O〉I for any operator O. Comparing directly the right hand side of (17) with the right hand side of (9), we obtain

graphic file with name srep31192-m36.jpg

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.

As an example, let us take the Bloch vector of the state ρ to be Inline graphic. Then we get Inline graphic, where Inline graphic, Inline graphic and Inline graphic are the right hand sides of inequalities (7), (8) and (9), respectively, see Fig. 1.

Figure 1. Uncertainty relations satisfied by observables σx, σy and σz with state ρ parameterized by Bloch vector Inline graphic: solid line for (Δσx)2 (Δσy)2 (Δσz)2, dot-dashed line for lower bound in (7), dashed line for the lower bound in (8), dotted line for lower bound in (9).

Figure 1

Open in a new tab

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 Inline graphic. Denote I = In = {1, 2, …, n}, Ik = {i1, i2, …, ik} ⊆ I with k elements of I, k = 1, 2, …, n, Inline graphic. Let Inline graphic be the set consisting of all the subsets of I, and Inline graphic the set consisting of the subsets of I with k elements. Then we have Inline graphic, Inline graphic. We have

Theorem 2

graphic file with name srep31192-m48.jpg

where

graphic file with name srep31192-m49.jpg

when n − k is even, and

graphic file with name srep31192-m50.jpg

when n − k is odd, Inline graphic, Ais are the variance operators of Ais.

For instance, we calculate the product form uncertainty relation for the eight Gell-Mann matrices Inline graphic,

graphic file with name srep31192-m53.jpg

which are the standard su(3) generators39 and obey the commutation relations: Inline graphic, where the structure constants fmns are completely antisymmetric, f 123 = 1, Inline graphic, Inline graphic. And each two of them are anticommute i.e. {λm, λn} = 0(m ≠ n).

Let us consider a general qutrit state ρ40,

graphic file with name srep31192-m57.jpg

where Inline graphic is the Bloch vector of ρ and Inline graphic is a formal vector given by the Gell-Mann matrices. For pure qutrit states the Bloch vectors satisfy Inline graphic, and for mixed states Inline graphic. However, not all Bloch vectors with Inline graphic correspond to valid qutrit states. For simplicity, we set r2 = r3 = r5 = r7 = r8 = 0, and r1 = a cos α, r4 = a sin α cos β, r6 = a sin α sin β, |a| ≤ 1. In this case ρ has the form

graphic file with name srep31192-m63.jpg

Then the uncertainty related to the set of observables Inline graphic has the form,

graphic file with name srep31192-m65.jpg

From (11) we have the lower bound of (15),

graphic file with name srep31192-m66.jpg

When Inline graphic, the equality (16) holds for all parameters α and β corresponding to valid qutrit density matrices. This means that the lower bound (16) is tight for Inline graphic. For Inline graphic, 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.

Figure 2. The uncertainty of observables Inline graphic in state ρ parameterized by the Bloch vector Inline graphic and its lower bound.

Figure 2

Open in a new tab

The upper surface is Inline graphic. The lower surface is Inline graphic, where Inline graphic is the lower bound (right side hand of the inequality (16)).

Figure 3. The uncertainty of observables Inline graphic in state ρ parameterized by the Bloch vector Inline graphic and its lower bound.

Figure 3

Open in a new tab

The solid line is Inline graphic. The dashed line is Inline graphic.

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.

Methods

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

graphic file with name srep31192-m70.jpg

where 〈XY〉 = 〈ψ|XY|ψ〉 for X, Y = A, B, C, respectively. For an arbitrary three dimensional complex vector Inline graphic, we have

graphic file with name srep31192-m72.jpg

Then for any given mixed state ρ with arbitrary pure state decomposition Inline graphic, the corresponding matrix M satisfies

graphic file with name srep31192-m74.jpg

Therefore M is semi-definite positive for all variance operators A, B, C and any state ρ. Hence, we have det(M) ≥ 0, namely,

graphic file with name srep31192-m75.jpg

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.

Additional Information

How to cite this article: Qin, H.-H. et al. Multi-observable Uncertainty Relations in Product Form of Variances. Sci. Rep. 6, 31192; doi: 10.1038/srep31192 (2016).

Acknowledgments

The work is supported by the NSFC under number 11275131. Qin acknowledges the fellowship support from the China scholarship council.

Footnotes

Author Contributions H.-H.Q., S.-M.F. and X.L.-J. wrote the main manuscript text. All of the authors reviewed the manuscript.

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