Mutually Exclusive Uncertainty Relations

Abstract

The uncertainty principle is one of the characteristic properties of quantum theory based on incompatibility. Apart from the incompatible relation of quantum states, mutually exclusiveness is another remarkable phenomenon in the information- theoretic foundation of quantum theory. We investigate the role of mutual exclusive physical states in the recent work of stronger uncertainty relations for all incompatible observables by Mccone and Pati and generalize the weighted uncertainty relation to the product form as well as their multi-observable analogues. The new bounds capture both incompatibility and mutually exclusiveness, and are tighter compared with the existing bounds.

Introduction

Heisenberg’s uncertainty principle¹ is one of the fundamental notions in quantum theory. The original form is a result of noncommutativity of the position and momentum operators. Robertson’s formulation of Heisenberg’s uncertainty principle in matrix mechanics² states that for any pair of observables A and B with bounded spectrums, the product of standard deviations of A and B is no less than half of the modulus of the expectation value of their commutator:

where is the standard deviation of the self-adjoint operator A. Here the expectation value is over the state for any observable . In fact, Robertson’s uncertainty relation can be derived from a slightly strengthened Schrödinger uncertainty inequality³

where .

Besides their importance in quantum mechanics, uncertainty relations play a significant role in quantum information theory as well^{4,5,6,7,8,9,10}. The variance-based uncertainty relations possess clear physical meanings and have a variety of applications in quantum information processings such as quantum spin squeezing^{11,12,13,14,15}, quantum metrology^16,17,18, and quantum nonlocality^19,20.

While the early forms of variance-based uncertainty relations are vital to the foundation of quantum theory, there are two problems still need to be addressed: (1) Homogeneous product of variances may not fully capture the concept of incompatibility. In other words, a weighted relation may produce a better approximation (e.g., the uncertainty relation with Rényi entropy²¹ and variance-based uncertainty relation for a weighted sum), for more details and examples, see ref. 22; (2) The existing variance-based uncertainty relations are far from being tight, and improvement is needed. One also needs to know how to generalize the product form to the case of multiple observables for practical applications.

In ref. 22, the authors and collaborators have proposed weighted uncertainty relations to answer the first question and succeeded in improving the uncertainty relation. Let’s recall the weighted uncertainty relation for the sum of variances. For arbitrary two incompatible observables A, B and any real number λ, the following inequality holds

with

and

where , , and are orthogonal to |ψ〉. In information-theoretic context, it is also natural to quantify the uncertainty by weighted products of variances, which also help to estimate individual variance as in ref. 22.

Recently, Maccone and Pati obtained an amended Heisenberg-Robertson inequality²³:

which is reduced to Heisenberg-Robertson’s uncertainty relation when minimizing the lower bound over , and the equality holds at the maximum. This amended inequality gives rise to a stronger uncertainty relation for almost all incompatible observables, and the improvement is due to the special vector perpendicular to the quantum state |ψ〉. We notice that this can be further improved by using the mutually exclusive relation between and |ψ〉. Moreover, this idea can be generalized to the case of multi-observables. For this reason the strengthen uncertainty relation thus obtained will be called a mutually exclusive uncertainty relation.

The goal of this paper is to answer the aforementioned questions to derive the product form of the weighted uncertainty relation, and investigate the physical meaning and applications of the mutual exclusive physical states in variance-based uncertainty relations. Moreover, we will generalize the product form to multi-observables to give tighter lower bounds.

Results

We first generalize the weighted uncertainty relations from the sum form²² to the product form, and then introduce mutually exclusive uncertainty relations (MEUR). After that we derive a couple of lower bounds based on Mutually exclusive physical states (MEPS), and we show that our results outperform the bound in ref. 23, which has been experimentally tested recently²⁴. Finally, generalization to multi-observables is also given.

We start with the sum form of the uncertainty relation, which takes equal contribution of the variance from each observable. However, almost all variance-based uncertainty relations do not work for the general situation of incompatible observables, and they often exclude important cases. In ref. 22, the authors and collaborators solved this degeneracy problem by considering weighted uncertainty relations to measure the uncertainty in all cases of incompatible observables. Using the same idea, we will study the product form of weighted uncertainty relations to give new and alternative uncertainty relations in the general situation. The corresponding mathematical tool is the famous Young’s inequality. The new weighted uncertainty is expected to reveal the lopsided influence from observables. They contain the usual homogeneous relation of ΔA²ΔB² as a special case.

Theorem 1. Let A, B be two observables such that ΔAΔB > 0, and p, q two real numbers such that . Then the following weighted uncertainty relation for the product of variances holds.

where p < 1, and the equality holds if and only if ΔA = ΔB. If p > 1, then becomes a upper bound for the weighted product.

See Methods for a proof of Theorem 1.

The weighted uncertainty relations for the product of variances have a desirable feature: our measurement of incompatibility is weighted, which fits well with the reality that observables usually don’t always reach equilibrium, i.e., in physical experiments their contributions may not be the same (cf. ref. 22). As an illustration, let us consider the relative error function between the uncertainty and weighted bound, which is defined by

In general f is a function of both p and |ψ〉. It is hard to find its extremal points as it involves in partial differential equations. Also the extremal points hardly occur at homogeneous weights, so incompatible observables usually don’t contribute equally to the uncertainty relation, which explains the need for a weighted uncertainty relation in the product form.

In what follows, we show how to tighten Maccone and Pati’s amended Heisenberg-Robertson uncertainty relation²³ by regarding mutually exclusive physical states as another information resource, and then generalize the variance-based uncertainty relation to the case of multi-observables.

We will refer to (6) as a mutually exclusive uncertainty relation since the states |ψ〉 and represent two mutual exclusive states in quantum mechanics, which is the main reason for improving the tightness of the bound. Next we move further to improve the bound by combining mutually exclusive relations and weighted relations.

Maccone and Pati’s uncertainty relation can be viewed as a singular case in a family of uncertainty relations parameterized by positive variable λ, which corresponds to our recent work on weighted sum of uncertainty relations²². We proceed similarly as the case of the amended Heisenberg-Robertson uncertainty relation by considering a modified square-modulus and Holevo inequalities in Hilbert space²⁵ in the following result.

Theorem 2. Let A and B be two incompatible observables and |ψ〉 a fixed quantum state. Then the mutually exclusive uncertainty relation holds:

for any unit vector perpendicular to |ψ〉 and arbitrary parameter λ > 0.

See Methods for a proof of Theorem 2.

The obtained variance-based uncertainty relation is stronger than Maccone and Pati’s amended uncertainty relation. In fact, when the maximal value is reached at a point λ₀ ≠ 1, the new bound is stronger than that of Maccone-Pati’s amended uncertainty relation. Let (i = 1, 2) be two lower bounds given in the RHS of (8), define the tropical sum

This gives a tighter lower bound when the maximal value of is reached at different direction in H_ψ (hyperplane orthogonal to |ψ〉) for . In other words, the new lower bound is a piecewise defined function of MEPS taking the maximum of the two bounds. In particular, for λ₀ ≠ 1, the tropical sum offers a better lower bound than , the Maccone-Pati’s lower bound. Note that may have a smaller minimum value than when λ ≠ 1, as , while the minimum value of is just the bound for Heisenberg-Robertson’s uncertainty relation. Because we only consider the maximum, it does not affect our result.

For example, consider a 4-dimensional system with state , and take the following observables

Direct calculation gives

and

For and , set

and

both of them have modulus one, then

meanwhile

so

Both the lower bounds and are functions of MEPS . However, for each , gives a better approximation of ΔAΔB than . Figure 1 is a schematic diagram of these two lower bounds. It is clear that provides a closer estimate to ΔAΔB:

Figure 1

Schematic comparison of bounds: Top and middle lines are ΔAΔB and resp.

Tropical sum is the upper boundary above the shadow, and the red and green ones are and Maccone-Pati’s bound resp.

for any unit MEPS orthogonal to |ψ〉. This is due to the fact that the bound is continuous on both MEPS and λ, which shows the advantage of our mutually exclusive uncertainty principle. The shadow region in Fig. 2. illustrates the outline of ΔAΔB and our bound .

Figure 2

Schematic comparison: Top line is ΔAΔB.

Blue curve is our bound , the shadow region is the difference between ΔAΔB and our bound . Other bounds are shown in different colors.

In Fig. 3, we illustrate our results, showing how the obtained bound outperforms the recent work of ref. 26 as well as the Schrödinger uncertainty relation. We consider the angular momenta L_x and L_y for spin-1 particle with state |ψ〉 = cos θ |1〉 − sin θ |0〉 and  = sin θ |1〉 + cos θ |0〉, where |0〉 and |1〉 are eigenstates of the angular momentum L_z.

Figure 3

Lower bounds of ΔA²ΔB² for a family of spin-1 particles.

ΔA²ΔB², our bound , the bound given by Eq. (3) in ref. 26 and the Schrödinger bound are respectively shown in blue, orange, green and yellow.

Mutually exclusive physical states with different directions in H_ψ offer different kinds of mutually exclusive information and improvement of the uncertainty relation. When such an experiment of the mutually exclusive uncertainty relation is performed, one is expected to have infinitely many strong lower bounds of the variance-based uncertainty relation.

Now we further generalize the uncertainty relations to multi-observables. For simplicity, write

So

is continuous on both MEPS and λ. Repeatedly using (8) for and λ_jk, we obtain the following relation.

Theorem 3. Let A₁, A₂, …, A_n be n incompatible observables, |ψ〉 a fixed quantum state and λ_jk positive real numbers, we have

for any MEPS orthogonal to |ψ〉 with modulus one. If some is negative, a negative sign is inserted into the RHS of (14) to ensure positivity. The equality holds if and only if MEPS for all j > k.

As a corollary, Theorem 3 leads to a simple bound of the uncertainty relation for multi-observables.

Corollary 1. Let A₁, A₂, …, A_n be n incompatible observables, then the following uncertainty relation holds

See Methods for a proof of Corollary 1.

Next, we provide yet another mutually exclusive uncertainty relation.

Theorem 4. Let A and B be two incompatible observables and |ψ〉 a fixed quantum state. Then

for any unit MEPS orthogonal to |ψ〉, with

where MEPS are unit vectors in H_ψ.

See Methods for a proof of Theorem 4.

Obviously, (16) can be seen as an amended Schrödinger inequality and also offers a better bound than (2) and Maccone-Pati’s relation (6). Figure 4 illustrates the schematic comparison.

Figure 4

Schematic comparison of uncertainty relations.

Top, middle and bottom lines are ΔA²ΔB², Schrödinger’s and the square of Heisenberg’s bounds resp. Orange and purple curves are the square of Maccone-Pati’s amended Heisenberg bound and our amended Schrödinger bound resp.

In general, if there exists an operator M for A and B such that 〈M〉 = 0, , then we have the following:

Remark 1. Let A and B be two incompatible observables and |ψ〉 a fixed quantum state. We claim the following mutually exclusive uncertainty relation holds:

Eq. (18) also gives a generalized Schrödinger uncertainty relation. Here as usual MEPS is any unit vector perpendicular to |ψ〉. The proof of Theorem 4 and Remark 1 are similar to that of Theorem 2, so we sketch it here. It is easy to see that the RHS of (18) reduces to the lower bound of Schrödinger’s uncertainty relation (2) when minimizing over , and the equality holds at the maximum. The corresponding uncertainty relation for arbitrary n observables is the following result.

Theorem 5. Let A₁, A₂, …, A_n be n incompatible observables, |ψ〉 a fixed quantum state and λ_jk positive real numbers. Then we have that

where M_jk satisfy 〈M_jk〉 = 0, and MEPS orthogonal to |ψ〉 with modulus one.

The RHS of (19) has the minimum value

and the equality holds at the maximum. Therefore one obtains the following corollary.

Corollary 2. Let A₁, A₂, …, A_n be n incompatible observables, then the following uncertainty relation holds

See Methods for a proof of Corollary 2.

We note that our enhanced Schrödinger uncertainty relations offer significantly tighter lower bounds than that of Maccone-Pati’s uncertainty relations for multi-observables, as our lower bound contains an extra term of (compare (1) with (2)).

Finally, we remark that we can also replace the non-hermitian operator in (6) by a hermitian one. A natural consideration is the amended uncertainty relation

for any unit MEPS perpendicular to |ψ〉. The corresponding uncertainty relation for multi-observables can also be generalized.

The minimum of Maccone and Pati’s amended bound in the RHS of (6) agrees with the bound in Heisenberg-Robertson’s uncertainty relation, which is weaker than Schrödinger’s bound in (2). We point out that the bound given as a continuous function of MEPS’s will always produce a better lower bound. In fact, the continuity of in MEPS implies that there exists suitable such that is tighter than the bound of Heisenberg-Robertson’s uncertainty relation. Similarly our lower bound given in (27) or more generally in (18) provides a tighter lower bound than the enhanced Schrödinger’s uncertainty relation (2). This shows the advantage of lower bounds with MEPS’s. Furthermore, lower bounds with more variables give better estimates for the product of variances of observables, as in (19).

Conclusions

The Heisenberg-Robertson uncertainty relation is a fundamental principle of quantum theory. It has been recently generalized by Maccone and Pati to an enhanced uncertainty relation for two observables via mutually exclusive physical states. Based on these and weighted uncertainty relations²², we have derived uncertainty relations for the product of variances from mutually exclusive physical states (MEPS) and offered tighter bounds.

In summary, we have proposed generalization of variance-based uncertainty relations. By virtue of MEPS, we have introduced a family of infinitely many Schrödinger-like uncertainty relations with tighter lower bounds for the product of variances. Indeed, our mutually exclusive uncertainty relations can be degenerated to the classical variance-based uncertainty relations by fixing MEPS and the weight. Also, our study further shows that the mutually exclusiveness between states is a promising information resource.

Methods

Proof of Theorem 1. To prove the theorem, we recall Young’s inequality²⁷: for , p < 1 one has that

Note that the right-hand side (RHS) may be negative if p < 1. But this can be avoided by using the symmetry of Young’s inequality to get

Thus our bound is nontrivial. We remark that if p > 1, it is directly from the Young’s inequality²⁷

and equality holds in (22) and (23) only when ΔA = ΔB.                      ■

Proof of Theorem 2. Here we provide two proofs of the proposed mutually exclusive uncertainty relation (8). The first one, based on weighted relations²², is a natural deformation of ref. 23 and is sketched as follows. By maximizing the RHS of (8), we see that the maximum ΔAΔB is achieved when the mutually exclusive physical state (MEPS) . Clearly our uncertainty relation contains (6) as a special case of λ = 1.

The second proof uses geometric property and is preferred because of its mathematical simplicity and also working for the amended Heisenberg-Robertson uncertainty relation²³. In fact, the RHS of (6), denoted by , is a continuous function of λ and the unit MEPS . By the vector projection, the maximum value ΔAΔB of over the hyperplane of is attained when . Therefore for any λ > 0

where is the RHS of (6). Similarly

for any λ > 0 and the equality holds if λ = 1, which implies (8) and completes the second proof.       ■

Proof of Corollary 1. Obviously, taking the minimum of (14) over MEPS implies that

When λ_jk = 1 for all j > k, the minimum is . Meanwhile if λ_jk and MEPS vary, Eq. (14) provides a family of mutually exclusive uncertainty relations for arbitrary n observables with (24) as the lower bound.               ■

Proof of Theorem 4. By the same method used in deriving (8) it follows that , and is

which equals to the lower bound of the Schrödinger uncertainty (2). We can modify g into a function with the same maximum and lower bound as Schrödinger’s uncertainty relation. Note that s ≤ ΔA²ΔB², then

which is equivalent to (by solving ΔA²ΔB²)

for any unit MEPS orthogonal to |ψ〉. In fact, let be the RHS of (27). It is easy to see that and

Hence we have the mutually exclusive uncertainty relation appeared in (27).            ■

Proof of Corollary 2. Apparently, taking the minimum of (19) over MEPS implies that

with the minimum is . Meanwhile if the MEPS vary, Eq. (19) provides a family of mutually exclusive uncertainty relations for arbitrary n observables with (28) as the lower bound.               ■