Uncertainty Relations

Uncertainty relations are inequalities representing the impossibility of simultaneous measurement in quantum mechanics. The most well-known uncertainty relations were presented by Heisenberg and Schrödinger. In this chapter, we generalize and extend them to produce several types of uncertainty relations.


Introduction
Let M n  ð Þ (resp. M n,sa  ð Þ) be the set of all n Â n complex matrices (resp. all n Â n self-adjoint matrices), endowed with the Hilbert-Schmidt scalar product A, B h i¼ Tr A * B ½ . Let M n,þ  ð Þ be the set of strictly positive elements of M n  ð Þ and M n,þ,1  ð Þ⊂ M n,þ  ð Þ be the set of strictly positive density matrices, that is M n,þ,1  ð Þ ¼ ρ ∈ M n  ð ÞjTr ρ ½ ¼ 1, ρ > 0 f g . If not otherwise specified, hereafter, we address the case of faithful states, that is ρ > 0. It is known that the expectation of an observable A ∈ M n,sa  ð Þ in state ρ ∈ M n,þ,1  ð Þ is defined by and the variance of an observable A ∈ M n,sa  ð Þ in state ρ ∈ M n,þ,1  ð Þ is defined by In Section 2, we introduce the Heisenberg and Schrödinger uncertainty relations. In Section 3, we present uncertainty relations with respect to the Wigner-Yanase and Wigner-Yanase-Dyson skew information. To represent the degree of noncommutativity between ρ ∈ M n,þ,1  ð Þ and A ∈ M n,sa  ð Þ, the Wigner-Yanase skew information I ρ A ð Þ is defined by where X, Y ½ ¼XY À YX. Furthermore, the Wigner-Yanase-Dyson skew information I ρ,α A ð Þ is defined by The convexity of I ρ,α A ð Þ with respect to ρ was famously demonstrated by Lieb [1], and the relationship between the Wigner-Yanase skew information and the uncertainty relation was originally developed by Luo and Zhang [2]. Subsequently, the relationship between the Wigner-Yanase-Dyson skew information and the uncertainty relation was provided by Kosaki [3] and Yanagi-Furuichi-Kuriyama [4]. In Section 4, we discuss the metric adjusted skew information defined by Hansen [5], which is an extension of the Wigner-Yanase-Dyson skew information. The relationship between metric adjusted skew information and the uncertainty relation was provided by Yanagi [6] and generalized by Yanagi-Furuichi-Kuriyama [7] for generalized metric adjusted skew information and the generalized metric adjusted correlation measure. In Sections 5 and 6, we provide non-Hermitian extensions of Heisenberg-type and Schrödinger-type uncertainty relations related to generalized quasi-metric adjusted skew information and the generalized quasimetric adjusted correlation measure. As a result, we obtain results for non-Hermitian uncertainty relations provided by Dou and Du as corollaries of our results. Finally, in Section 7, we present the sum types of uncertainty relations.

Wigner-Yanase skew information
To represent the degree of noncommutativity between ρ ∈ M n,þ,1  ð Þ and A ∈ M n,sa  ð Þ, the Wigner-Yanase skew information I ρ A ð Þ and related quantity J ρ A ð Þ are defined as The quantity U ρ A ð Þ representing a quantum uncertainty excluding the classical mixture is defined as We note the following relation: Luo [8] then derived the uncertainty relation of U ρ A ð Þ.

Wigner-Yanase-Dyson skew information
Here, we introduce a one-parameter inequality extended from (3). For 0 ≤ α ≤ 1, A, B ∈ M n,sa  ð Þ and ρ ∈ M n,þ,1  ð Þ, we define the Wigner-Yanase-Dyson skew information as follows: We also define We note that however, we have We then have the following inequalities: because From (2), (4), and (5), we have We provide the following uncertainty relation with respect to U ρ,α A ð Þ as a direct generalization of (3).

Quantum Mechanics
: By (7) and the Schwarz inequality, Then, we have We also have Thus, we have the final result, (6). □ When α ¼ 1 2 , we obtain the result in Theorem 1.3.

Metric adjusted skew information and metric adjusted correlation measure 4.1 Operator monotone function
3. f is operator monotone.
Example 1. Examples of elements of F op are given by the following: We introduce the sets of regular and non-regular functions 10]). The correspondence f !f is a bijection between F r op and F n op .

Metric adjusted skew information
In the Kubo-Ando theory [11] of matrix means, a mean is associated with each operator monotone function f ∈ F op by the following formula: Using the notion of matrix means, the class of monotone metrics can be defined by the following formula: ð Þ, we define as follows: : Quantity I f ρ A ð Þ is referred to as the metric adjusted skew information, and A, B h i ρ,f is referred to as the metric adjusted correlation measure. Proposition 1. The following holds: then it holds that where A, B ∈ M n,sa  ð Þ. To prove Theorem 1.6, several lemmas are used. Lemma 1. If (8) holds, then the following inequality is satisfied: Proof of Lemma 1. By (8), we have Since we have Lemma 2. Let jϕ 1 i, jϕ 2 i, ⋯, jϕ n i f g be a basis of eigenvectors of ρ, corresponding to the eigenvalues λ 1 , λ 2 , ⋯, λ n f g . We set a jk ¼ ϕ j jA 0 jϕ k we have Proof of Theorem 1.6. Since We also have Thus, we have the final result (9). □

Generalized metric adjusted skew information
We assume that f ∈ F r op satisfies the following condition (A): Definition 4. For A, B ∈ M n,sa  ð Þ, ρ ∈ M n,þ,1  ð Þ we define the following: : A ð Þ is referred to as the generalized metric adjusted skew information, and Þis referred to as the generalized metric adjusted correlation measure. Theorem 1.7 ([7]). Under condition (A), the following holds: 2. (Heisenberg type) For A, B ∈ M n,sa  ð Þ, ρ ∈ M n,þ,1  ð Þ, we assume the following condition (B): Then,

Generalized quasi-metric adjusted skew information
In this section, we present general uncertainty relations for non-Hermitian observables X, Y ∈ M n  ð Þ. Definition 5. For X, Y ∈ M n  ð Þ, A, B ∈ M n,þ  ð Þ we define the following:  Theorem 1.8 ([12]). Under condition (A), the following holds: 2. (Heisenberg type) For X, Y ∈ M n  ð Þ, A, B ∈ M n,þ  ð Þ, we assume condition (B). Then, h In particular, where X ∈ M n  ð Þ and A, B ∈ M n,þ  ð Þ. Proof of 1 in Theorem 1.8. By the Schwarz inequality, we have Now, we prove the second inequality. Since Similarly, we have Then, Thus, We use the following lemma to prove 2: Lemma 3 Proof of Lemma 3. By conditions (A) and (B), we have We then have be the spectral decompositions of A and B, respectively. Then, we have Then, by Lemma 3, we have we obtain the result in Theorem 1.7.
We assume that We then obtain the following trace inequality by substituting X ¼ I in (11).
This is a generalization of the trace inequality provided in [13]. In addition, we produce the following new inequality by combining a Chernoff-type inequality with Theorem 1.8. Theorem 1.9 ( [14]). We have the following: The following lemma is necessary to prove Theorem 1.9.
f s ð Þ is convex in s. □ Proof of Theorem 1.9. The third and fourth inequalities follow from Lemma 4 and (12), respectively. Thus, we only prove the following inequality: Then, we have And since we have Then, we have Therefore, Then, Thus, Since 2x α y 1Àα À x þ yÀjx À yj ð Þ ≥ 0 for x, y > 0, 0 ≤ α ≤ 1 in general, we can obtain Theorem 1.9. □ Remark 2. We note the following 1, 2:

Sum type of uncertainty relations
Let A, B ∈ M n,sa  ð Þ have the following spectral decompositions: