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Quantum Information Physics I
TR2021-
Davi Geiger and Zvi M. Kedem Courant Institute of Mathematical Sciences New York University, New York, New York 10012
Abstract
Quantum physics, despite its intrinsic probabilistic nature, is formulated as time- reversible. We propose an entropy for quantum physics, which may conduce to the emer- gence of a time arrow. That entropy is a measure of randomness over the degrees of freedom of a quantum state and is quantified in phase spaces. Its minimum is positive due to the uncertainty principle. To study the relation of the entropy to physical phenomena, we classify the behaviors of quantum states according to their entropy evolution. We revisit transition probabilities and Fermi’s golden rule to show their close relation to states with oscillating entropy. We study collisions of two particles in coherent states, and show that as they come closer to each other, their entanglement causes the total system’s entropy to oscillate. We conjecture an entropy law whereby the entropy never decreases, and speculate that entropy oscillations trigger the annihilations and the creations of particles.
CONTENTS
- Introduction and Summary
- Quantum Entropy in Phase Spaces
- Coordinate-Entropy
- Spin-Entropy
- The Minimum Entropy Value
- QCurves and Entropy-Partition
- The Coordinate-Entropy of Coherent States Increases With Time
- Time Reflection as a Mechanism to Convert QCurves in I to D and Vice-Versa
- Entropy Oscillations
- Physical Scenarios with Particle Creation
- A Two-Particle Collision
- The Hydrogen Atom and Photon Emission
- An Entropy Law and a Time Arrow
- Conclusions
- References
QM and Quantum Field Theory (QFT). To analyze particles’ evolution, we introduce a QCurve structure imposed on the evolutions of a quantum state. We partition the set of all the QCurves according to their entropies’ behavior during an evolution. The study of the QCurves in the blocks of the partition leads us to conjecture that there is an entropy law, whereby entropy never decreases, applicable to all particle physics.
QUANTUM ENTROPY IN PHASE SPACES
The quantum entropy must account for both the coordinate and the internal (spin) DOFs, and we define the entropy in light of this requirement.
Coordinate-Entropy
We associate with a state |휓〉푡 its projection onto the QM eigenstates of the operators r ˆ and p ˆ , i.e., | r 〉 and | p 〉. Either one, | r 〉 or | p 〉, is sufficient to recover the other one via a Fourier transform. As illustrated by the uncertainty principle, the randomness of the coordinates of a particle is described in the coordinate phase space (휓( r , 푡) = 〈 r |휓푡 〉 , 휙( p , 푡) = 〈 p |휓푡 〉). By Born’s rule, 휌r ( r , 푡) = |휓( r , 푡)|^2 and 휌푝 ( p , 푡) = | 휙˜( p , 푡)|^2 are the probability densities of the phase-space representation of the state. Motivated by Gibbs [13] and Jaynes [15], we will define S, the coordinate- entropy of a particle. Let k = (^1) \ p be the spatial frequency, 휌푘 ( k , 푡) = (^) ^13 휌푝 ( p , 푡) the associated probability density, Sr = −
휌r ( r , 푡) ln 휌r ( r , 푡) d^3 r ; and analogously for Sk. Then we define
S = −
æ 휌r ( r , 푡)휌푘 ( k , 푡) ln (휌r ( r , 푡) 휌푘 ( k , 푡) ) d^3 r d^3 k = Sr + Sk. (1)
The entropy is dimensionless and invariant under changes of the units of measure- ments. For an extension to 푁-particle systems, see [11]. Fields in QFT are described by the operators 훹 ( r , 푡), where ( r , 푡) is the space-time, and 훷( k , 푡) is the spatial Fourier transform of 훹 ( r , 푡). A represen- tation for a system of particles is based on Fock states with occupation number 푛푞 1 , 푛푞 2 , ,... , 푛푞푖 ,.. .〉, where 푛푞푖 is the number of particles in a QM state |푞푖〉. The number of particles in a Fock state is then 푁 = ∑퐾푖= 1 푛푞푖 , and a QFT state is described in a Fock space as |state〉 = ∑푚 훼푚 푛푞 1 , 푛푞 2 , ,... , 푛푞푖 ,.. .〉, where 푚 is an index over configurations of a Fock state, 훼푚 ∈ ℂ, and 1 = ∑푚 |훼푚 |^2. The QFT operators act on a state producing a phase space state (훹 ( r , 푡) |state〉 , 훷( k , 푡) |state〉). We then define the probability density function for the spatial coordinates as
휌QFT r ( r , 푡) = |훹 ( r , 푡) |state〉 |^2 = 〈state|훹 †^ ( r , 푡)훹 ( r , 푡) |state〉.
Analogously, 휌QFT k ( k , 푡) = |훷( k , 푡) |state〉 |^2 = 〈state| 훷†^ ( k , 푡)훷( k , 푡) |state〉. One may call the coefficients “the wave function” and interpret them as distributions of the information about the position and the space frequency of the state of the field. The QFT coordinate-entropy is then described by (1), where we dropped the superscript QFT, as it will be clear which framework is used, QM or QFT. In [11], we proved that the coordinate-entropy is invariant under continuous 3D coordinate transformations, continuous Lorentz transformations, and discrete CPT transformations.
Spin-Entropy
The DOFs associated with the spin are captured by a vector or a bispinor repre- sentation of the states in both frameworks. It is not possible to know simultaneously the spin of a particle in all three dimensional directions, and this uncertainty, or
The coordinate-entropy (1) is Sr + Sk. Due to the entropic uncertainty principle Sr + Sk ≥ 3 ln eπ as shown in [1, 2, 14], with Sk = Sp − 3 ln . To complete the proof, in [11] we showed that the minimum spin-entropy is 2 θ(푠) ln 2.
Higgs bosons in coherent states have the lowest possible entropy 3 ( 1 + ln π). The dimensionless element of volume of integration to define the entropy will not contain a particle unless d^3 r d^3 k ≥ 1 , due to the uncertainty principle, and this may be interpreted as a necessity of discretizing the phase space. We note that the minimum entropy of the discretization of (1) is also 3 ( 1 + ln π), as shown in [7].
QCURVES AND ENTROPY-PARTITION
We introduce the concept of a QCurve to specify a curve (or path) in a Hilbert space parametrized by time. In QM a QCurve is represented by a triple ( (^) |휓 0 〉^ , 푈^ (푡),^ δ푡
) (^) where |휓 0 〉^ is the initial state,^ 푈^ (푡)^ =^ e−i퐻푡^ is the evolution operator, and [ 0 , δ푡] is the time interval of the evolution. Alternatively, we can represent the initial state by (〈 r |휓 0 〉 , 〈 k |휓 0 〉) and in QFT by (훹 ( r , 0 ) |state〉 , 훷( k , 0 ) |state〉).
Definition 1 (Partition of E). Let E to be the set of all the QCurves. We define a partition of E based on the entropy evolution into four blocks:
C : Set of the QCurves for which the entropy is a constant. I : Set of the QCurves for which the entropy is increasing, but it is not a constant. D : Set of the QCurves for which the entropy is decreasing, but it is not a constant. O : Set of the oscillating QCurves, with the entropy strictly increasing in some subinterval of [ 0 , δ푡] and strictly decreasing in another subinterval of [ 0 , δ푡].
It is straightforward to show that all stationary states are in C (see [11]).
The Coordinate-Entropy of Coherent States Increases With Time
Coherent states, represented by the state |훼〉, the eigenstates of the annihilator operator, yield in position and momentum space representations
휓k 0 ( r − r 0 ) = 〈 r |훼 = 0 〉 = 1 23 π^32 (det 횺) 12
N ( r | r 0 , 횺) ei k^0 · r^ ,
훷r 0 ( k − k 0 ) = 〈 k |훼 = 0 〉 = 1 23 π^32 (det 횺−^1 ) 12
N
k | k 0 , 횺−^1
ei( k − k^0 )· r^0 , (2)
where 횺 is the spatial covariance matrix. The foundational material follows from most common textbooks, e.g., [6, 10, 17, 19]. In [11] we proved that for a QCurve with an initial coherent state (2) and evolving
according to the energy \휔(k) = \
k^2 푐^2 +
\
, the entropy evolves as 3 ( 1 + ln π) + 12 ln det ( I + 푡^2 (횺−^1 H )^2 ), where
H 푖 푗 = H 푖 푗 ( k 0 ) = 푚\
( \푘 0
) 2 )−^32 [
δ푖, 푗
( \푘 0
( \푘푖
) ( \푘 푗
)]
and H is positive definite. Thus, the QCurve is in I. This suggests that quantum physics has an inherent dispersion mechanism to increase entropy for free fermion particles.
Time Reflection as a Mechanism to Convert QCurves in I to D and Vice-Versa
We consider a time-independent Hamiltonian and investigate the discrete sym- metries C and P, and Time Reflection, the augmentation of Time Reversal with Time Translation, i.e., the classical mapping 푡 7 → 푡′^ = −푡 + δ푡. We define the Time Reflection quantum field, Tδ, as 훹 Tδ^ ( r , −푡 + δ푡) = T훹 ( r , 푡) = 푇훹 ∗^ ( r , 푡). Note that in contrast to the case of Time Reversal, 훹 T^ ( r , −푡) = T훹 ( r , 푡), the
Entropy Oscillations
Consider a Hamiltonian 퐻′^ = 퐻 +퐻I, where |퐻I^ | |퐻|, and the initial eigenstate 휓퐸푖^ 〉^ of 퐻 associated with the eigenvalue 퐸푖 = \휔푖. The time evolution of 휓퐸푖^ 〉^ is
|휓푡 〉 = e−i^ (퐻^ +퐻^ \ I)푡^ 휓퐸푖^ 〉^ =
∑^ 푛
푘= 1
훼푘 (푡) 휓퐸푘^ 〉^ ,
where 푛 is the number of the eigenvectors of 퐻. Fermi’s golden rule [8, 9] approxi- mates the coefficients of transition for 푘 ≠ 푖 and short time intervals by
퐻I 푖,푘
(휔푖 − 휔푘 )
−2 sin^2
Theorem 2 (Entropy Oscillations). Consider the QCurve (^ 휓퐸푖^ 〉^ , 푈 (푡) = e −i^ (퐻^ +퐻^ \ I)푡^ , 푇 )^ with \휔 1 the ground state value of 퐻 and 푇 = (^) |휔푖^2 −π휔 1 |. Assume that |훼 1 (푡)|^2 , |훼푖 (푡)|^2 |훼푘 (푡)|^2 for 푘 ≠ 1 , 푖 and 푡 ∈ [ 0 , 푇]. Then the QCurve is in O_._
Proof. With the theorem’s assumptions, we can approximate the position and the momentum probability densities associated with |휓푡 〉 by
휌r ( r , 푡) ≈
1 − |훼 1 (푡)|^2 〈 r 휓퐸푖^ 〉^ + 훼 1 (푡) 〈 r 휓퐸 1 〉^2 , 휌k ( k , 푡) ≈
1 − |훼 1 (푡)|^2 〈 k 휓퐸푖^ 〉^ + 훼 1 (푡) 〈 k 휓퐸 1 〉^2.
The time coefficients are the same for 휌r ( r , 푡), and 휌k ( k , 푡) and they all return to the same values simultaneously after a period of 푇, and so the entropy will return to its previous value too. As the entropy is not a constant, it must be oscillating.
PHYSICAL SCENARIOS WITH PARTICLE CREATION
A Two-Particle Collision
Consider a two-fermions or a two-bosons system
|휓푡 〉 = (^) √^1 퐶 푡
휓^1 푡^ 〉^ 휓^2 푡^ 〉^ ∓ 휓^2 푡^ 〉^ 휓^1 푡^ 〉
where 퐶푡 is the normalization constant that may evolve over time and the signs “∓” represent fermions (“−”) and bosons (“+”). When 휓^1 푡^ 〉^ and 휓 푡^2 〉^ are orthogonal to each other, 퐶푡 = 2. Projecting on 〈 r 1 | 〈 r 2 | and on 〈 k 1 | 〈 k 2 |,
휓( r 1 , r 2 , 푡) = √^1 퐶 푡
(휓 1 ( r 1 , 푡)휓 2 ( r 2 , 푡) ∓ 휓 1 ( r 2 , 푡)휓 2 ( r 1 , 푡)) ,
휓( k 1 , k 2 , 푡) = √^1 퐶 푡
(휙 1 ( k 1 , 푡)휙 2 ( k 2 , 푡) ∓ 휙 1 ( k 2 , 푡)휙 2 ( k 1 , 푡)).
From [11], the entropy of the two-particle system, discarding the spin-entropy which is constant throughout the collision, is then
푆
휓 푡^1 〉^ , 휓^2 푡^ 〉
æ d^3 r 1
æ d^3 r 2 휌r ( r 1 , r 2 , 푡) ln 휌r ( r 1 , r 2 , 푡) −
æ d^3 k 1
æ d^3 k 2 휌k ( k 1 , k 2 , 푡) ln 휌k ( k 1 , k 2 , 푡).
Consider a collision of two particles, each one in an initial coherent state with position variance 휎^2 centered at 푐 1 and 푐 2 , and moving towards each other along the 푥-axis with center momenta 푝 0 = \푘 0 and −푝 0. They can be represented in position
(a) (^) 푚\ = 1 : Entropy vs. time; 휌푥 (푥 1 , 푥 2 , 푡) overlaid over time
(b) (^) 푚\ = 0. 5 : Entropy vs. time; 휌푥 (푥 1 , 푥 2 , 푡) overlaid over time
Figure 2. Collision of two fermions with individual amplitudes (3), parameters 푘 0 = 1 , 푐 2 = −푐 1 = 300 , speed of light 푐 = 1 , a grid of 1 000 points for 푥 1 , 푥 2 , 푘 1 , 푘 2. The left column shows entropy vs. time. The right column shows snapshots of the density at initial time, final time, and intervals of 100 time units, overlaid on single plots. The 푧-axis represents the density and the 푥-푦 axes represent 푥 1 - 푥 2 values. As the particles approach each other, their individual densities disperse, the maximum values are reduced, and the entropy increases. Only when the particles are close to each other, the interference reduces the total entropy.
and in the Coulomb Gauge (∇·퐴 = 0 ), for 푞 = |푞|(sin 휃푞 cos 휙푞, sin 휃푞 sin 휙푞, cos 휃푞), the polarizations satisfy 휖 1 (푞) = (cos 휃푞 cos 휙푞, cos 휃푞 sin 휙푞, sin 휃푞) and 휖 2 (푞) = (− sin 휙푞, cos 휙푞, 0 ).
The state of the atom can be described by |푛, 푙, 푚〉e− |푞, 휆〉γ, where 푛, 푙, 푚 are the quantum numbers of the electron e−, and 푞 and 휆 are the momentum and the helicity of the photon γ. We next consider the Lyman-alpha transition, |푛 = 2 , 푙 = 1 , 푚 = 0 〉 | 0 〉 → |푛 = 1 , 푙 − 0 , 푚 = 0 〉 |푞, 휆〉 with the emission of a pho-
ton with wavelength 휆 ≈ 121. 567 × 10 −^9 m. We first evaluate the electron’s entropy at both states |푛 = 2 , 푙 = 1 , 푚 = 0 〉 and |푛 = 1 , 푙 − 0 , 푚 = 0 〉. For simplicity, we consider the Schrödinger approximation to describe the electron state with the energy change in this transition of ∆퐸푛= 2 →푛= 1 ≈ −
22 −^1
× 13 .6 eV = 10 .2 eV. We now compute the difference between the final and initial state entropy following three steps.
(i) The position probability amplitudes described in [3] and the associated en- tropies are
휓 2 , 1 , 0 (휌, 휃, 휙) = √^1
32 π
휌e−^ 휌^2 cos(휃) → Sr (휓 2 , 1 , 0 ) ≈ 6. 120 + ln π + 3 ln 푎 0 ,
휓 1 , 0 , 0 (휌, 휃, 휙) = (^) √^1 π
e−휌^ → Sr (휓 1 , 0 , 0 ) ≈ 3. 000 + ln π + 3 ln 푎 0 ,
where 푎 0 ≈ 5. 292 × 10 −^11 m is the Bohr radius, and 휌 = 푟/푎 0. (ii) The momentum probability amplitudes described in [3] and the associated entropies are
2 π푝^30
cos(휃푝^ )^ ,
→ S푝 (훷 2 , 1 , 0 ) ≈ 0. 042 + 3 ln 푝 0 ,
훷 1 , 0 , 0 ( 푝, 휃푝, 휙푝) =
π 푝^30
) 2 )−^2
→ S푝 (훷 1 , 0 , 0 ) ≈ 2. 422 + 3 ln 푝 0 ,
where 푝 0 = /푎 0. (iii) Therefore, ∆S 2 , 1 , 0 → 1 , 0 , 0 = Sr (휓 1 , 0 , 0 ) + S푝 (훷 1 , 0 , 0 ) − Sr (휓 2 , 1 , 0 ) − S푝 (훷 2 , 1 , 0 ) ≈
AN ENTROPY LAW AND A TIME ARROW
In classical statistical mechanics, the entropy provides a time arrow through the second law of thermodynamics [5]. We have shown that due to the dispersion property of the fermionic Hamiltonian, some states, such as coherent states, evolve with an increasing entropy. However, current quantum physics is time reversible and we have just studied in the previous section several scenarios where the entropy oscillates. This study lead us to think that entropy oscillations do not occur in nature, instead and inspired by the second law of thermodynamics, we conjecture
Law (The Entropy Law). The entropy of a quantum system is an increasing function of time.
Let us review some of the physical scenarios where oscillations may not take place:
- A high-speed collision e+^ + e−^ → 2 γ may produce new particles instead of allowing the entropy to decrease (see Figure 2).
- According to QED, and due to photon fluctuations of the vacuum, the state of an electron in an excited state of the hydrogen atom is in a superposition with the ground state, and by Theorem 2 the entropy would decrease within a time interval 2 π/|휔푛= 2 ,푙= 1 ,푚= 0 − 휔푛= 1 ,푙= 0 ,푚= 0 |. Instead, the electron jumps to the ground state and a photon is created/emitted, increasing the entropy.
- We speculate that the QCurve of a neutral K meson (kaon K^0 ) [4], 푒 0 = (휓 0 ( r ), 푈 (푡), (^) ∆^2 π푤 ), is in O. Then, a K^0 particle in state 휓 0 ( r ) evolves with increasing entropy until, say at time 푇, it enters the remaining segment of QCurve 푒푇 = (휓푇 ( r ), 푈 (푡), [푇, (^) ∆^2 π푤 ]) in D. To block such a decrease (forbid- den by the entropy law), a transformation takes place, with quarks exchanging bosons to transform K^0 7 → K¯^0 to create an antiparticle’s QCurve 푒 1 in I.
We conjecture that the entropy law is the trigger for those particles’ creation. Finally, in [11] we studied the spin-entropy in more depth, and this law would impact which spin state evolutions would be physically allowed.
CONCLUSIONS
Capturing all the information of a quantum state requires the specifying of the parameters associated to the DOFs of a quantum state as well as as the intrinsic randomness of the quantum state. We proposed an entropy defined in the phase spaces of position and spatial frequency as well as in the spin phase space addressing the space and the spin DOFs. This definition of the entropy possesses desirable properties, including invariance in special relativity, and invariance under CPT transformations. We characterized the behaviors of all quantum states according to their entropy evolution. To this end, we introduced a QCurve structure, a triple representing the initial state, the unitary evolution operator, and a time interval. We partitioned the set of all the QCurves into four blocks, characterized by the entropy during an evolution. A QCurve is in C if the entropy is a constant, in I if it is increasing, in D if it is decreasing, and in O if it is oscillating. We showed that due to the dispersion property of a fermionic Hamiltonian, QCurves of initially coherent states are in I. We extended the CPT transformation to allow for Time Reflection, consequently mapping C, I, O, D, to C, D, O, I, respectively. Then we revisited Fermi’s golden rule, discussing its relation to QCurves in O. We showed that the entropy increases when an electron in excited state of the hydrogen atom falls to the ground state emitting a photon. We studied the collision of two particles, each in a coherent state. The entropy of each particle alone is increasing, but as they approach each other, an entropy oscillation can occur in the two-particle system due to their entanglement.
[10] R. P. Feynman and S. Weinberg, editors. Elementary particles and the laws of physics: The 1986 Dirac memorial lectures. Cambridge University Press, 1999. [11] D. Geiger and Z. M. Kedem. Quantum information physics II. New York University, Dept. of Computer Science,TR 2021-997, 2021. [12] W. Gerlach and O. Stern. Der experimentelle Nachweis des magnetischen Moments des Silberatoms. Z. Phys. , 8(1):110–111, 1922. [13] J. W. Gibbs. Elementary principles in statistical mechanics. Courier Corporation,
[14] I. I. Hirschman. A note on entropy. Am. J. Math. , 79(1):152–156, 1957. [15] E. T. Jaynes. Gibbs vs Boltzmann entropies. Am. J. Phys. , 33(5):391–398, 1965. [16] H. P. Robertson. The uncertainty principle. Physical Review , 34:163–164, Jul 1929. [17] J. J. Sakurai. Modern Quantum Mechanics. Cambridge University Press, 2 edition,
[18] J. von Neumann. Mathematical foundations of quantum mechanics: New edition. Princeton University Press, 2018. [19] S. Weinberg. The quantum theory of fields: Volume 1, (Foundations). Cambridge University Press, 1995.
