Sum type of uncertainty relations

However, since we have A^α¼X

λ^α_i∣ϕ_ii ϕ_ij¼X

i, j

λ^α_ijϕ_i

* +

ϕ_ijψ_j

D E

hψ_j∣,

B^1�α¼X

μ^1�α_j ∣ψ_ji ψ_jj¼X

i, j

μ^1�α_j jϕ_i

* +

ϕ_ijψ_j

D E

hψ_j∣, A^αB^1�^α¼X

i, j

λ^α_iμ^1�_j ^α∣ϕ_iiDϕ_ijψ_jE hψ_j∣:

Then,

Tr A� ^αB^1�^α�

¼X

i, j

λ^α_iμ^1�_j ^α∣hϕ_i ψ_ji

��

�

��

�²:

Thus, 2Tr A� ^αB¹^�^α�

�Tr A½ þB�jLA�RBjI� ¼X

i, j

2λ^α_iμ¹_j^�^α��λiþμ_j�jλi�μ_jj�

n o

∣hϕ_i��ψ_ji

��

�²: Since 2x^αy^1�^α�ðxþy�jx�yjÞ≥0 for x, y>0, 0≤ α ≤1 in general, we can

obtain Theorem 1.9. □

Remark 2. We note the following 1, 2:

1.¹₂Tr A½ þB�jA�Bj�≤ inf

0≤ α ≤1Tr A� ^1�^αB^α�

≤Tr Ah ¹⁼²B¹⁼²i

≤

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1

2Tr A½ þB�

� �2

�1

4ðTr½jA�Bj�Þ² s

2. There is no relationship between Tr½jLA�RBjI�and Tr½jA�Bj�. When

A¼ 3 2

1 1 2 2

3 2 0 B@

CA, B¼ 4 0 0 1

� �

we have Tr½jLA�RBjI� ¼3, Tr½jA�Bj� ¼ ffiffiffiffiffiffi p10

. When

A¼ 13

2 7 7 2 2

13 2 0 B@

CA, B¼ 2 0 0 5

� �

we have Tr½jLA�RBjI� ¼8, Tr½jA�Bj� ¼ ffiffiffiffiffi p58

However, since we have A^α¼X

λ^α_i∣ϕ_ii ϕ_ij¼X

i, j

λ^α_ijϕ_i

* +

ϕ_ijψ_j

D E

hψ_j∣,

B^1�α¼X

μ^1�α_j ∣ψ_ji ψ_jj¼X

i, j

μ^1�α_j jϕ_i

* +

ϕ_ijψ_j

D E

hψ_j∣, A^αB^1�^α¼X

i, j

λ^α_iμ^1�_j ^α∣ϕ_iiDϕ_ijψ_jE hψ_j∣:

Then,

Tr A� ^αB^1�^α�

¼X

i, j

λ^α_iμ^1�_j ^α∣hϕ_iψ_ji

��

�

��

�²:

Thus, 2Tr A� ^αB¹^�^α�

�Tr A½ þB�jLA�RBjI� ¼X

i, j

2λ^α_iμ¹_j^�^α��λiþμ_j�jλi�μ_jj�

n o

∣hϕ_i��ψ_ji

��

�²: Since 2x^αy^1�^α�ðxþy�jx�yjÞ≥0 for x, y>0, 0≤ α ≤1 in general, we can

obtain Theorem 1.9. □

Remark 2. We note the following 1, 2:

1.¹₂Tr A½ þB�jA�Bj�≤ inf

0≤ α ≤1Tr A� ^1�^αB^α�

≤Tr Ah ¹⁼²B¹⁼²i

≤

2Tr A½ þB�

� �2

�1

4ðTr½jA�Bj�Þ² s

2. There is no relationship between Tr½jLA�RBjI�and Tr½jA�Bj�. When

A¼ 3 2

1 1 2 2

3 2 0 B@

CA, B¼ 4 0 0 1

� �

we have Tr½jLA�RBjI� ¼3, Tr½jA�Bj� ¼ ffiffiffiffiffiffi p10

. When

A¼ 13

2 7 7 2 2

13 2 0 B@

CA, B¼ 2 0 0 5

� �

we have Tr½jLA�RBjI� ¼8, Tr½jA�Bj� ¼ ffiffiffiffiffi p58

7. Sum type of uncertainty relations

Let A, B∈Mn,sað Þ have the following spectral decompositions:

A¼Xⁿ

i¼1

λ_i∣ϕ_ii ϕ_ij, B¼Xⁿ

i¼1

μ_ijψ_i

* +

hψ_i∣:

Quantum Mechanics

For any quantum state∣ϕi, we define the two probability distributions P¼�p₁, p₂,⋯, p_n�

, Q ¼�q_i, q₂,⋯, q_n� , where p_i¼jhϕ_ijϕij², q_j¼∣hψ_jj jϕi². Let

H Pð Þ ¼ �Xⁿ

i¼1

p_ilog p_i, H Qð Þ ¼ �Xⁿ

j¼1

q_jlog q_j

be the Shannon entropies of P and Q, respectively.

Theorem 1.10. The following uncertainty relation holds:

H Pð Þ þH Qð Þ≥ �2 log c, where c¼ maxi,j∣ ϕD _ijψ_jE

∣.

For details, see [15, 16].

Definition 6. The Fourier transformation ofψ∈L²ð Þ is defined as ψ ω^ð Þ ¼

ð_∞

�∞

ψð Þet ^�2πiωtdt:

We also define

Qð Þ ¼ f∈L²ð Þ; ð_∞

�∞t²jf tð Þj²dt<∞

� �

Proposition 2. Ifψ∈L²ð Þ, ∥ψ∥²¼1 satisfiesψ,ψ^∈Qð Þ, then Sð Þ þψ Sð Þψ^ ≥log e

2, where

Sð Þ ¼ �ψ ð_∞

�∞

ψð Þt

j j²logjψð Þtj²dt, Sð Þ ¼ �ψ^ ð_∞

�∞

ψ^ð Þt

j j²logjψ^ð Þtj²dt:

For details, see [17].

Theorem 1.11 ([18]). For any X, Y∈Mnð Þ, A, B ∈Mn,þð Þ, the following holds: 1.I^ð_A,B^g,f^ÞðX, YÞ þI^ð_A,B^g,f^Þð ÞY ≥₂¹max In^ð_A,B^g,f^ÞðXþYÞ, I^ð_A,B^g,f^ÞðX�YÞo

2. ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I^ð_A,B^g,f^Þð ÞX q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I^ð_A,B^g,f^Þð ÞY

q ≥max ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I^ð_A,B^g,f^ÞðXþYÞ

q , ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I^ð_A,B^g,f^ÞðX�YÞ

� q �

3. ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I^ð_A,B^g,f^Þð ÞX q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I^ð_A,B^g,f^Þð ÞY

q ≤2 max ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I^ð_A,B^g,f^ÞðXþYÞ

q , ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I^ð_A:B^g,f^ÞðX�YÞ

� q �

Proof 1. The Hilbert-Schmidt norm∥�∥satisfies

∥X∥²þ∥Y∥²¼1

2�∥XþY∥²þ∥X�Y∥²�

≥1

2max�∥XþY∥²,∥X�Y∥²� : (13) Uncertainty Relations

DOI: http://dx.doi.org/10.5772/intechopen.92137

Since I^ð_A,B^g,f^ÞðX, XÞis the second power of the Hilbert-Schmidt norm,∥X∥¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

I^ð_A,B^g,f^Þð ÞX

q . We then obtain the result by substituting (13),

2. By the triangle inequality of a general norm, we apply the triangle inequality for∥X∥¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I^ð_A,B^g,f^Þð ÞX q

3. We prove the following norm inequality:

∥X∥þ∥Y∥ ≤ ∥XþY∥þ∥X�Y∥: (14) Since

∥X∥¼∥1

2ðXþYÞ þ1

2ðX�YÞ∥ ≤ 1

2∥XþY∥þ1

2∥X�Y∥

and

∥Y∥¼∥1

2ðYþXÞ þ1

2ðY�XÞ∥ ≤1

2∥YþX∥þ1

2∥Y�X∥,

we add two inequalities and obtain (14). □

Author details

Kenjiro Yanagi^1,2

1 Faculty of Science, Department of Mathematics, Josai University, Japan 2 Yamaguchi University, Japan

*Address all correspondence to: [email protected]

by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Quantum Mechanics

References

[1]Lieb EH. Convex trace functions and the Wigner-Yanase-Dyson conjecture.

Advances in Mathematics. 1973;11:

267-288

[2]Luo S, Zhang Q. On skew information. IEEE Transactions of Information Theory. 2004;50:1778-1782, and Correction to “On skew

information”, IEEE Transactions of Information Theory. 2005;51:4432 [3]Kosaki H. Matrix trace inequality related to uncertainty principle.

International Journal of Mathematics.

2005;16:629-646

[4]Yanagi K, Furuichi S, Kuriyama K. A generalized skew information and uncertainty relations. IEEE Transactions of Information Theory. 2005;IT-51(12):

4401-4404

[5]Hansen F. Metric adjusted skew information. Proceedings of the National Academy of Sciences of the United States of America. 2008;105:

9909-9916

[6]Yanagi K. Metric adjusted skew information and uncertainty relation.

Journal of Mathematical Analysis and Applications. 2011;380(2):888-892 [7]Yanagi K, Furuichi S, Kuriyama K.

Uncertainty relations for generalized metric adjusted skew information and generalized metric adjusted correlation measure. Journal of Uncertainty Analysis and Applications. 2013;1(12):

1-14

[8]Luo S. Heisenberg uncertainty relation for mixed states. Physical Review A. 2005;72:042110

[9]Yanagi K. Uncertainty relation on Wigner-Yanase-Dyson skew

information. Journal of Mathematical Analysis and Applications. 2010;365:

12-18

[10]Gibilisco P, Hansen F, Isola T. On a correspondence between regular and non-regular operator monotone functions. Linear Algebra and its Applications. 2009;430:2225-2232 [11]Kubo F, Ando T. Means of positive linear operators. Mathematische Annalen. 1980;246:205-224

[12]Yanagi K. On the trace inequalities related to left-right multiplication operators and their applications. Linear and Nonlinear Analysis. 2018;4(3): 361-370

[13]Audenaert KMR, Calsamiglia J, Masancs LI, Munnoz-Tapia R, Acin A, Bagan E, et al. The quantum Chernoff bound. Review Letters. 2007;98: 160501-1-160501-4

[14]Yanagi K. Generalized trace inequalities related to fidelity and trace distance. Linear and Nonlinear Analysis. 2016;2(2):263-270

[15]Maassen H, Uffink JBM. Generalized entropic uncertainty relations. Physical Review Letters. 1988; 60(12):1103-1106

[16]Nielsen MA, Chuang IL. Quantum Computation and Quantum

Information. Cambridge University Press; 2000

[17]Hirschman II Jr. A note on entropy. American Journal of Mathematics. 1957; 79:152-156

[18]Yanagi K. Sum types of uncertainty relations for generalized quasi-metric adjusted skew informations.

International Journal of Mathematical Analysis and Applications. 2018;4(4): 85-94

Uncertainty Relations

DOI: http://dx.doi.org/10.5772/intechopen.92137

Since I^ð_A,B^g,f^ÞðX, XÞis the second power of the Hilbert-Schmidt norm,∥X∥¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

I^ð_A,B^g,f^Þð ÞX

q . We then obtain the result by substituting (13),

2. By the triangle inequality of a general norm, we apply the triangle inequality for∥X∥¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I^ð_A,B^g,f^Þð ÞX q

3. We prove the following norm inequality:

∥X∥þ∥Y∥ ≤ ∥XþY∥þ∥X�Y∥: (14) Since

∥X∥¼∥1

2ðXþYÞ þ1

2ðX�YÞ∥ ≤1

2∥XþY∥þ1

2∥X�Y∥ and

∥Y∥¼∥1

2ðYþXÞ þ1

2ðY�XÞ∥ ≤ 1

2∥YþX∥þ1

2∥Y�X∥,

we add two inequalities and obtain (14). □

Author details

Kenjiro Yanagi^1,2

1 Faculty of Science, Department of Mathematics, Josai University, Japan 2 Yamaguchi University, Japan

*Address all correspondence to: [email protected]

by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Quantum Mechanics

References

[1]Lieb EH. Convex trace functions and the Wigner-Yanase-Dyson conjecture.

Advances in Mathematics. 1973;11:

267-288

[2]Luo S, Zhang Q. On skew information. IEEE Transactions of Information Theory. 2004;50:1778-1782, and Correction to “On skew

information”, IEEE Transactions of Information Theory. 2005;51:4432 [3]Kosaki H. Matrix trace inequality related to uncertainty principle.

International Journal of Mathematics.

2005;16:629-646

[4]Yanagi K, Furuichi S, Kuriyama K. A generalized skew information and uncertainty relations. IEEE Transactions of Information Theory. 2005;IT-51(12):

4401-4404

[5]Hansen F. Metric adjusted skew information. Proceedings of the National Academy of Sciences of the United States of America. 2008;105:

9909-9916

[6]Yanagi K. Metric adjusted skew information and uncertainty relation.

Journal of Mathematical Analysis and Applications. 2011;380(2):888-892 [7]Yanagi K, Furuichi S, Kuriyama K.

Uncertainty relations for generalized metric adjusted skew information and generalized metric adjusted correlation measure. Journal of Uncertainty Analysis and Applications. 2013;1(12):

1-14

[8]Luo S. Heisenberg uncertainty relation for mixed states. Physical Review A. 2005;72:042110

[9]Yanagi K. Uncertainty relation on Wigner-Yanase-Dyson skew

information. Journal of Mathematical Analysis and Applications. 2010;365:

12-18

[12]Yanagi K. On the trace inequalities related to left-right multiplication operators and their applications. Linear and Nonlinear Analysis. 2018;4(3):

361-370

[13]Audenaert KMR, Calsamiglia J, Masancs LI, Munnoz-Tapia R, Acin A, Bagan E, et al. The quantum Chernoff bound. Review Letters. 2007;98:

160501-1-160501-4

[14]Yanagi K. Generalized trace inequalities related to fidelity and trace distance. Linear and Nonlinear Analysis.

2016;2(2):263-270

[15]Maassen H, Uffink JBM.

Generalized entropic uncertainty relations. Physical Review Letters. 1988;

60(12):1103-1106

[16]Nielsen MA, Chuang IL. Quantum Computation and Quantum

Information. Cambridge University Press; 2000

[17]Hirschman II Jr. A note on entropy.

American Journal of Mathematics. 1957;

79:152-156

[18]Yanagi K. Sum types of uncertainty relations for generalized quasi-metric adjusted skew informations.

International Journal of Mathematical Analysis and Applications. 2018;4(4):

85-94 Uncertainty Relations

DOI: http://dx.doi.org/10.5772/intechopen.92137

Chapter 5

Complex Space Nature of the

Quantum World: Return Causality to Quantum Mechanics

Ciann-Dong Yang and Shiang-Yi Han

Abstract

As one chapter, we about to begin a journey with exploring the limitation of the causality that rules the whole universe. Quantum mechanics is established on the basis of the phenomenology and the lack of ontology builds the wall which blocks the causality. It is very difficult to reconcile the probability and the causality in such a platform. A higher dimension consideration may leverage this dilemma by expanding the vision. Information may seem to be discontinuous or even so weird if only be viewed from a part of the degree of freedoms. Based on this premise, we reexamined the microscopic world within a complex space. Significantly, some knowledge beyond the empirical findings is revealed and paves the way for a more detailed exploration of the quantum world. The random quantum motion is essen- tial for atomic particle and exhibits a wave-related property with a bulk of trajectories. It seems we can break down the wall which forbids the causality entering the quantum kingdom and connect quantum mechanics with classical mechanics.

The causality returns to the quantum world without any assumption in terms of the quantum random motion under the optimal guidance law in complex space.

Thereby hangs a tale, we briefly introduce this new formulation from the fundamental theoretical description to the practical technology applications.

Keywords: random quantum trajectory, optimal guidance law, complex space

1. Introduction

It took scientists nearly two centuries from first observation of flower powder’s Brownian motion to propose a mathematical qualitative description [1]. Time is an arrow launched from the past to the future, every event happens for a reason. “The world is woven from billions of lives, every strand crossing every other. What we call premonition is just movement of the web. If you could attenuate to every strand of quivering data, the future would be entirely calculable. As inevitable as mathematics [2].” All physical phenomena are connected to the same web. As long as we can see through the quivering data and cut into the very core, we can glimpse the most elegant beauty of nature. As precise as physics.

It took nearly 30 years for physicists to establish quantum mechanics but nearly 100 years to seek for its essence. Quantum mechanics is the most precise theory to describe the microscopic world but also is the most obscure one among all theories.

It collects lots data but not all. Just like what we can observed is the shadow on the

ground not the actual object in the air. It is impossible to see the whole appearance of the object by observing its shadow. The development of the quantum era seems started in such circumstances and missed something we call the essence of nature.

In this chapter, we hope to recover the missing part by considering a higher dimension to capture the actual appearance of nature. At the end, we will find out that nature dominates the web where we live as well as the theories we develop. Every- thing should follow the law of the nature, and there is no exception.

Trajectory is a typical classical feature of the macroscopic object solved by the equation of motion. The trajectory of the microscopic particle is supposed to be observed if the law of nature remains consistent all the way down to the atomic scale. However, such an observation cannot be made till 2011. Kocsis and his coworkers propose an observation of the average trajectories of single photons in a two-slit interferometer on the basis of weak measurement [3]. Since then quantum trajectories are observed for many quantum systems, such as superconducting quantum bit, mechanical resonator, and so on [4–6]. Weak measurement provides the weak value which is a measurable quantity definable to any quantum observable under the weak coupling between the system and the measurement apparatus [7].

The significant characteristic of the weak value does not lie within the range of eigenvalues and is complex. It is pointed out that the real part of the complex weak value represents the average quantum value [8], and the imaginary part is related to the rate of variation in the interference observation [9].

The trajectory interpretation of quantum mechanics is developed on the basis of de Broglie’s matter wave and Bohm’s guidance law. In recent years, the importance of the quantum trajectory in theoretical treatment and experimental test has been discussed in complex space [10–21]. All these research indirectly or directly show that the complex space extension is more than a mathematical tool, it implies a causal essence of the quantum world.

On the other hand, it is found out that the real part of momentum’s weak value is the Bohmian momentum representing the average momentum conditioned on a position detection; while its imaginary part is proportional to the osmotic velocity that describes the logarithmic derivative of the probability density for measuring the particular position directed along the flow generated by the momentum [22]. This not only implies the existence of randomness in a quantum system, but also discloses that the random motion occurs in complex space. Numerous studies with the complex initial condition and the random property have been discussed [23–25]. A stochastic interpretation of quantum mechanics is proposed which regards the random motion as a nature property of the quantum world not the interference made by the measurement devices [26, 27]. These investigations suggest that a complex space and the random motion are two important features of the quantum world.

Based on the complex space structure, we propose a new perspective of quantum mechanics that allows one to reexamine quantum phenomena in a classical way. We will see in this chapter how the quantum motion can provide the classical description for the quantum kingdom and is in line with the probability distribution. One thing particular needed to be emphasized is that the stochastic Hamilton Jacobi Bellman equation can reduce to the Schrödinger equation under some spe- cific conditions. In other words, the Schrödinger equation is one special case of all kinds of random motions in complex space. A further discussion of the relationship between the trajectory interpretation and probability interpretation is presented in Section 2. In particular, the solvable nodal issue is put into discussion, and the continuity equation for the complex probability density function is proposed. In Section 3, we demonstrate how the quantum force could play the crucial role in the force balanced condition within the hydrogen atom and how the quantum potential forms the shell structure where the orbits are quantized. A practical application to Quantum Mechanics

the Nano-scale is demonstrated in Section 4. We consider the quantum potential relation to the electronic channel in a 2D Nano-structure. In addition, the conduc- tance quantization is realized in terms of the quantum potential which shows that the lower potential region is where the most electrons pass through the channel.

And then, concluding remarks are presented in Section 5.

Dalam dokumen Quantum Mechanics (Halaman 67-72)