Path Integral Implementation of Relational Quantum Mechanics

Jianhao M. Yang (  [email protected] ) Qualcomm (United States)

Research Article

Keywords: Relational Quantum mechanics, Path Integral, Entropy, Inuence Functional

Posted Date: February 18th, 2021

DOI: https://doi.org/10.21203/rs.3.rs-206217/v1

License:   This work is licensed under a Creative Commons Attribution 4.0 International License. Read Full License

Version of Record: A version of this preprint was published at Scientic Reports on April 21st, 2021. See the published version at https://doi.org/10.1038/s41598-021-88045-6. Path Integral Implementation of Relational Quantum Mechanics

Jianhao M. Yang∗ Qualcomm, San Diego, CA 92121, USA (Dated: February 4, 2021) Relational formulation of quantum mechanics is based on the idea that relational properties among quantum systems, instead of the independent properties of a quantum system, are the most fundamental elements to construct quantum mechanics. In the recent works (J. M. Yang, Sci. Rep. 8:13305, 2018), basic relational quantum mechanics framework is formulated to derive quantum probability, Born’s Rule, Schrödinger Equations, and measurement theory. This paper gives a concrete implementation of the relational probability amplitude by extending the path integral formulation. The implementation not only clarifies the physical meaning of the relational probability amplitude, but also gives several important applications. For instance, the double slit experiment can be elegantly explained. A path integral representation of the reduced density matrix of the observed system can be derived. Such representation is shown valuable to describe the interaction history of the measured system and a series of measuring systems. More interestingly, it allows us to develop a method to calculate entanglement entropy based on path integral and influence functional. Criteria of entanglement is proposed based on the properties of influence functional, which may be used to determine entanglement due to interaction between a quantum system and a classical field. Keywords: Relational Quantum mechanics, Path Integral, Entropy, Influence Functional

I. INTRODUCTION to see if a quantum theory constructed based on relational properties can address some of the unanswered fun- Quantum mechanics was originally developed as a damental questions mentioned earlier. Such reconstruc- physical theory to explain the experimental observations tion program was initiated [10] and had some successes, of a quantum system in a measurement. In the early days for example, in deriving the Schrödinger Equation. of quantum mechanics, Bohr had emphasized that the de- Recently, a similar reformulation of quantum mechan- scription of a quantum system depends on the measuring ics was proposed [12, 13]. The reformulation is based apparatus [1–3]. In more recent development of quan- on two basic ideas. 1.)Relational properties between the tum interpretations, the dependency of a quantum state two quantum systems are the most fundamental elements on a reference system was further recognized. The rela- to formulate quantum mechanics. 2.)A physical measure- tive state formulation of quantum mechanics [4–6] asserts ment of a quantum system is a probe-response interaction that a quantum state of a subsystem is only meaningful process. Thus, the framework to calculate the probabil- relative to a given state of the rest of the system. Sim- ity of an outcome when measuring a quantum system ilarly, in developing the theory of decoherence induced should model this bidirectional process. This implies the by environment [7–9], it is concluded that correlation in- probability can be derived from product of two quanti- formation between two quantum systems is more basic ties with each quantity associated with a unidirectional than the properties of the quantum systems themselves. process. Such quantity is defined as relational probability Relational Quantum Mechanics (RQM) further suggests amplitude. Specifically, the probability of a measurement that a quantum system should be described relative to outcome is proportional to the summation of probability another system, there is no absolute state for a quan- amplitude product from all alternative measurement con- tum system [10, 11]. Quantum theory does not describe figurations. The properties of quantum systems, such as the independent properties of a quantum system. In- superposition and entanglement, are manifested through stead, it describes the relation among quantum systems, the rules of counting the alternatives. As results, tradi- and how correlation is established through physical in- tional quantum mechanics formulations can be rediscov- teraction during measurement. The reality of a quantum ered but with new insights on the origin of quantum prob- system is only meaningful in the context of measurement ability. Schrödinger Equation is recovered when there is by another system. no entanglement in the relational probability amplitude The idea that relational properties are more basic than matrix [12]. On the other hand, when there is change in the independent properties of a quantum system is pro- the entanglement measure, the quantum measurement found. It should be considered a starting point for con- theory is obtained [13]. In essence, quantum mechan- structing the formulation of quantum mechanics. How- ics demands a new set of rules to calculate measurement ever, traditional quantum mechanics always starts with probability from an interaction process. an observer-independent quantum state. It is of interest Although the concept of relational probability amplitude is useful to derive the quantum probability, its physical meaning is not obvious to understand. It is desirable to find an explicit calculation of the relational probabil- ∗ [email protected] ity amplitude. It turns out that the path integral method 2 can be used to achieve this goal and is briefly described in another apparatus A′. In an ideal measurement to mea- Ref. [12]. In this paper, the significance of the path inte- sure an observable of S, the apparatus is designed in such gral implementation of relational quantum mechanics is a way that at the end of the measurement, the pointer fully developed. Besides providing the physical meaning state of A has a distinguishable, one to one correlation of relation probability amplitude, the path integral for- with the eigenvalue of the observable of S. mulation also has interesting applications. For instance, The definition of Observer is associated with an appa- it can describe the history of a quantum system that has ratus. An observer, denoted as , is a person who can interacted with a series of measuring systems in sequence. operate and read the pointer variableO of the apparatus. As a result, the double slit experiment can be elegantly Whether or not this observer (a person) is a quantum explained from the formulation developed here. More sig- system is irrelevant in our formulation. An observer is nificantly, the coordinator representation of the reduced defined to be physically local to the apparatus he asso- density matrix derived from this implementation allows ciates with. This prevents the situation that can in- us to develop a method to calculate entanglement entropy stantaneously read the pointer variable of the apparatusO using path integral approach. We propose a criterion on that is space-like separated from . O whether there is entanglement between the system and In the traditional quantum measurement theory pro- external environment based on the influence functional. posed by von Neumann [14], both the quantum system This enables us to calculate entanglement entropy of a and the measuring apparatus follow the same quantum physical system that interacts with classical fields, such mechanics laws. After the interaction during the mea- as an electron in an electromagnetic field. Since entan- surement process, both systems encode information each glement entropy is an important concept in quantum in- other, allowing an observer to infer measurement results formation theory, the method described here may lead of S by reading pointer variable of A. Quantum mea- to new insight on the information aspect of a quantum surement is a question-and-answer bidirectional process. system that interacts with classical fields. The measuring system interacts (or, disturbs) the mea- The paper is organized as following. We first briefly re- sured system. The interaction in turn alters the state of view the relational formulation of quantum mechanics in the measuring system. As a result, a correlation is es- Section II. In Section IIIA the path integral implementa- tablished, allowing the measurement result for S to be tion of the relational probability amplitude is presented. inferred from the pointer variable of A. It is shown to be compatible with the traditional path A Quantum State of S describes the complete informa- integral quantum mechanics, particularly on the definition an observer can know about S. From the exam- tion of the influence functional. Section IIIB generalizes ination on the measurementO process and the interaction the formulation to describe the history of quantum state history of a quantum system, we consider a quantum for the observed system that has interacted with a se- state encodes the information relative to the measuring ries of measuring systems in sequence. The formulation system or the environment that the system previously is applied to explain the double slit experiment in Sec- interacted with. In this sense, the quantum state of S tion IIIC. Section IIID introduces a method to calculate is described relative to A. It is equivalent to say that entanglement entropy between the interacting systems the quantum state is relative to an observer because based on the path integral implementation. A criterion to there is no space-like separation between andO A. determine whether entanglement entropy vanishes based operates A, reads the measurement outcomesO of A, andO on properties of the influence functional is discussed in has the complete control of A. The idea that a quantum Section IV. Section IV also summarizes the conclusions. state encodes information from previous interactions is also proposed in Ref [11]. The information encoded in the quantum state is the complete knowledge an observer II. RELATIONAL FORMULATION OF can say about S, as it determines the possible outcomes QUANTUM MECHANICS of next measurement. When next measurement with another apparatus A′ is completed, the description of quan- A. Terminologies tum state is updated to be relative to A′.

A Quantum System, denoted by symbol S, is an object under study and follows the laws of quantum mechanics. An Apparatus, denoted as A, can refer to the measuring B. Basic Formulation devices, the environment that S is interacting with, or the system from which S is created. All systems are quantum The relational formulation of quantum mechanics [12] systems, including any apparatus. Depending on the se- is based on a detailed analysis of the interaction process lection of observer, the boundary between a system and during measurement of a quantum system. First, from an apparatus can change. For example, in a measurement experimental observations, a measurement of a variable setup, the measuring system is an apparatus A, the mea- on a quantum system yields multiple possible outcomes sured system is S. However, the composite system S + A randomly. Each potential outcome is obtained with a as a whole can be considered a single system, relative to certain probability. We call each measurement with a 3 distinct outcome a quantum measurement event1. De- a particular outcome of variable q is calculated. It turns note these alternatives events with a set of kets s for out such probability is proportional to the sum of weights {| ii} S, where (i = 0,...,N 1), and a set of kets aj for from all applicable measurement configurations, where A, where (j = 0,...,M− 1). A potential measurement{| i} the weight is defined as the product of two relational − outcome is represented by a pair of kets ( si , aj ). Sec- probability amplitudes corresponding to the applicable ond, a physical measurement is a bidirectional| i | process,i measurement configuration. Identifying the applicable the measuring system and the measured system inter- measurement configuration manifests the properties of a act and modify the state of each other. The probability quantum system. For instance, before measurement is of finding a potential measurement outcome represented actually performed, we do not know that which event by a pair of kets ( s , a ), p , should be calculated by will occur to the quantum system since it is completely | ii | j i ij modeling such bidirectional process. This implies pij can probabilistic. It is legitimate to generalize the potential be expressed as product of two numbers, measurement configuration as aj si ak . In other words, the measurement configuration| i → | i → in the| i joint AS SA pij Qji Rij . (1) Hilbert space starts from a , but can end at a , or ∝ | j i | j i AS SA any other event, ak . Indeed, the most general form Qji and Rij are not necessarily real non-negative num- of measurement configuration| i in a bipartite system can ber since each number alone only models a unidirectional be aj sm sn ak . Correspondingly, we process which is not a complete measurement process. generalize| i → Eq.(1) | i →by introducing| i → | i a quantity for such con- On the other hand, pij is a real non-negative number figuration, since it models an actual measurement process. To satisfy such requirement, we further assume ASSA AS SA SA ∗ SA Wjmnk = Qjm Rnk = (Rmj ) Rnk . (4) AS SA ∗ Qji = (Rij ) . (2) The second step utilizes Eq.(2). This quantity is interpreted as a weight associated with the potential measure- Written in a different format, QAS = (RSA)† . This ji ji ment configuration aj sm sn ak . Suppose means QAS = (RSA)†. Eq.(1) then becomes we do not perform actual| i → measurement | i → | i →and | inferencei information is not available, the probability of finding S p = RSA 2/Ω (3) ij | ij | in a future measurement outcome can be calculated by summing W ASSA from all applicable alternatives of mea- where Ω is a real number normalization factor. QAS and jmnk ji surement configurations. SA Rij are called relational probability amplitudes. Given With this framework, the remaining task to calculate the relation in Eq.(2), we will not distinguish the notation the probability is to correctly count the applicable alter- R versus Q, and only use R. SA natives of measurement configuration. This task depends The relational matrix R gives the complete descrip- on the expected measurement outcome. For instance, tion of S. It provides a framework to derive the proba- suppose the expected outcome of an ideal measurement bility of future measurement outcome. Although RSA is ij is event si , i.e., measuring variable q gives eigenvalue a probability amplitude, not a probability real number, | i qi. The probability of event si occurs, pi, is propor- we assume it follows certain rules in the classical prob- tional to the summation of W ASSA| i from all the possible ability theory, such as multiplication rule, and sum of jmnk configurations related to si . Mathematically, we select alternatives in the intermediate steps. ASSA | i all Wjmnk with m = n = i, sum over index j and k, and The set of kets si , representing distinct measure- obtain the probability p . ment events for S{|, cani} be considered as eigenbasis of i Hilbert space with dimension N, and s is an eigen- M HS | ii vector. Since each measurement outcome is distinguish- p (RSA)∗RSA = RSA 2. (5) i ∝ ij ik | ij | able, si sj = δij. Similarly, the set of kets aj j,kX=0 Xj is eigenbasish | i of Hilbert space with dimension{| Ni} HA for the apparatus system A. The bidirectional process This leads to the definition of wave function ϕi = j Rij, ⇋ potential measurement configura- 2 aj si is called a so that pi = ϕi . The quantum state canP be de- tion| i in the| i joint Hilbert space . | | HS ⊕ HA scribed either by the relational matrix R, or by a set To derive the properties of S based on the relational of variables ϕi . The vector state of S relative to A, R, we examine how the probability of measuring S with A { } T is ψ S = (ϕ0, ϕ1, . . . , ϕN ) where superscript T is the transposition| i symbol. More specifically,

A 1 ψ = ϕ s where ϕ = R . (6) In the basic formulation described here, we assume finite number | iS i| ii i ij of quantum measurement events for the convenience of notation. Xi Xj It is possible to extend to infinite number of events. In fact, the path integral implementation, which is the main result of this The justification for the above definition is that the prob- paper, assume infinite number of positions that a system can be ability of finding S in eigenvector si in future measure- located. See Section III A. ment can be calculated from it by| definingi a projection 4 operator Pˆ = s s . Noted that s are orthogonal single relational matrix R. In this case, the entanglement i | iih i| {| ii} eigenbasis, the probability is rewritten as: measure E = H(ρS). H(ρS) enables us to distinguish different quantum dy- ˆ 2 pi = ψ Pi ψ = ϕi (7) namics. Given a quantum system S and its referencing h | | i | | apparatus A, there are two types of the dynamics be- Eqs.(5) and (6) are introduced on the condition that tween them. In the first type of dynamics, there is no there is no entanglement (See Section IIC for the defini- physical interaction and no change in the entanglement tion of entanglement). between quantum system S and measure between S and A. S is not necessarily isolated A. If there is entanglement between them, the summa- in the sense that it can still be entangled with A, but the tion in Eq.(5) over-counts the applicable alternatives of entanglement measure remains unchanged. This type of measurement configurations and should be modified ac- dynamics is defined as time evolution. In the second type cordingly. A more generic approach to describe the quan- of dynamics, there is a physical interaction and correla- tum state of S is the reduced density matrix formulation, tion information exchange between S and A, i.e., the von which is defined as Neumann entropy H(ρS) changes. This type of dynamics is defined as quantum operation. Quantum measurement ρ = RR† (8) S is a special type of quantum operation with a particular outcome. Whether the entanglement measure changes The probability pi is calculated using the projection operator Pˆ = s s distinguishes a dynamic as either a time evolution or a i | iih i| quantum operation [12, 13]. 2 As shown in this introduction section, the relational pi = T rS(PîρˆS) = Rij . (9) X | | probability amplitude Rij provides a complete descrip- j tion of the quantum system relative to a reference sys- The effect of a quantum operation on the relational tem. It is natural to ask what is the physical meaning of probability amplitude matrix can be expressed through this quantity and how to mathematically calculate it. A an operator. Defined an operator Mˆ in Hilbert space concrete implementation of the relational quantum mechanics depends on how R is calculated. This is main S as Mij = si Mˆ sk , The new relational probability ij amplitudeH matrixh | isk obtainedi by question we intend to answer in this paper.

(RSA ) = M (RSA ) , or new ij ik init kj III. RESULTS Xk (10) R = MR . new init A. Path Integral Implementation Consequently, the reduced density becomes, This section shows that the relational probability am- † † ρnew = Rnew(Rnew) = MρinitM . (11) plitude can be explicitly calculated using the Path In- tegral formulation. Without loss of generality, the following discussion just focuses on one dimensional space- C. Entanglement Measure time quantum system. In the Path Integral formulation, the probability to find a quantum system moving The description of S using the reduced density ma- from a point xa at time ta to a point xb at time tb trix ρS is valid regardless there is entanglement between is the absolute square of a probability amplitude, i.e., S and A. To determine whether there is entanglement P (b, a) = K(b, a) 2. The probability amplitude is postu- between S and A, a parameter to characterize the entan- lated as the| sum| of the contribution of phase from each glement measure should be introduced. There are many path [18]: forms of entanglement measure [15, 16], the simplest one b is the von Neumann entropy, which is defined as ~ K(b, a) = e(i/ )Sp(x(t)) x(t) (14) Za D H(ρS) = T r(ρSln(ρS)). (12) − where x(t) denotes integral over all possible paths from D Denote the eigenvalues of the reduced density matrix ρS point xa to point xb. It is the wave function for S moving as λi , i = 0,...,N, the von Neumann entropy is calcu- from xa to xb [18]. The wave function of finding the lated{ as} particle in a region previous to t can be expressed as Rb b ~ H(ρS) = λilnλi. (13) ϕ(x ) = e(i/ )Sp(x(t)) x(t) (15) − X b Z i Rb D

A change in H(ρS) implies there is change of entangle- where xb is the position of particle at time tb. The in- ment between S and A. Unless explicitly pointed out, tegral over region can be interpreted as integral of Rb we only consider the situation that S is described by a all paths ending at position xb, with the condition that 5 each path lies in region b which is previous to time tb. reduced density matrix is The rational of such definitionR can be found in Feynman’s ′ (i/~)∆S original paper [17]. ρ(xb, xb) = dyb e Z ZRS ZRS ZRA ZRA Now let’s consider how the relational matrix element b b′ b b S ′ ′ can be formulated using similar formulation. Define b x(t) x (t) y(t) y (t) R × D D D D is the region of finding system S previous to time tb, and S S ′ A where ∆S = Sp (x(t)) Sp (x (t)) (20) b is the region of finding measuring system A previous − R A A ′ to time tb. We denote the matrix element as R(xb; yb). + Sp (y(t)) Sp (y (t)) Here the coordinates x and y act as indices to the sys- − b b + SSA(x(t), y(t)) tem S and apparatus A, respectively. Borrowing the int ideas described in Eq.(14), we propose that the relational SSA(x′(t), y′(t)). − int matrix element is calculated as ′ ′ Here xb = x(tb) and xb = x (tb). The path integral over ′ (i/~)SSA(x(t),y(t)) y (t) takes the same region (therefore same end point R(xb, yb) = e p x(t) y(t) (16) D ZRS ZRA D D yb) as the path integral over y(t). After the path inte- b b D gral, an integral over yb is performed. Eq.(20) is equiv- SA where the action Sp (x(t), y(t)) consists three terms alent to the J function introduced in Ref [18]. We can rewrite the expression of ρ using the influence functional, SSA(x(t), y(t)) = SS(x(t)) + SA(y(t)) F (x(t), x′(t)), p p p (17) SA 1 ~ S S ′ + Sint (x(t), y(t)). ′ (i/ )[Sp (x(t))−Sp (x (t))] ̺(xb, xb) = e Z ZRS ZRS The last term is the action due to the interaction between b b′ ′ ′ S and A when each system moves along its particular F (x(t), x (t)) x(t) x (t) × D D ′ path. Eq.(16) is considered an extension of Postulate 1. ′ (i/~)∆S ′ We can validate Eq.(16) by deriving formulation that is F (x(t), x (t)) = dyb e y(t) y (t) Z ZZRA D D consistent with traditional path integral. Suppose there b where ∆S′ = SA(y(t)) SA(y′(t)) is no interaction between S and A. The third term in p − p Eq.(17) vanishes. Eq.(16) is decomposed to product of SA + Sint (x(t), y(t)) two independent terms, SSA(x′(t), y′(t)). − int (i/~)SS (x(t)) (21) R(xb, yb) = e p x(t) ZRS D b (18) where Z = T r(ρ) is a normalization factor to ensure ~ A e(i/ )Sp (y(t)) y(t) T r(̺) = 1. × Z A D In summary, we show that the relational probability Rb amplitude can be explicitly calculated through Eq.(16). ~ Noticed that the coordinates y and y are equivalent SA iSp/ a b Rij is defined as the sum of quantity e , where Sp is of the index j in Eq.(6), the wave function of S can be the action of the composite system S + A along a path. obtained by integrating yb over Eq.(18) Physical interaction between S and A may cause change ∞ of Sp, which is the phase of the probability amplitude. iSp/~ ϕ(xb) = R(xb, yb)dyb But e itself is a probabilistic quantity, instead of a Z−∞ quantity associated with a physical property. With this ~ S definition and the results in Section II, we obtain the = e(i/ )Sp (x(t)) x(t) Z S D formulations for wave function in Eq. (19) and prob- Rb ∞ (19) ability in Eq.(29) that are the consistent with those in (i/~)SA(y(t)) e p y(t)dyb traditional path integral formulation. The reduced den- × Z Z A D −∞ Rb sity expression in Eq.(20), although equivalent to the J ~ S function in Ref [18], has richer physical meaning. For in- = c e(i/ )Sp (x(t)) x(t) ZRS D stance, we can calculate the entanglement entropy from b the reduced density matrix. This will be discussed fur- where constant c is the integration result of the second ther in Section IIID. term in step two. The result is the same as Eq.(15) except an unimportant constant. Next, we consider the situation that there is entangle- B. Interaction History of a Quantum System ment between S and A as a result of interaction. The third term in Eq.(17) does not vanish. We can no longer One of the benefits of implementing the relational define a wave function for S. Instead, a reduced density probability amplitude using path integral approach is matrix should be used to describe the state of the par- that it is rather straightforward to describe the inter- ticle, ρ = RR†. Similar to Eq.(16), the element of the action history of a quantum system. Let’s start with a 6 simple use case and later extending the formulation to a Pˆ = ψ(xb) ψ(xb) , the probability is calculated, sim- general use case. ilar to| Eq.(9),ih as | Suppose up to time ta, a quantum system S only interacts with a measuring system A. The detail of interaction p(ψ) = T r(ρPˆ) = ψ(x ) ̺ ψ(x ) h b | | b i is not important in this discussion. S may interact with (29) = ψ∗(x′ )ψ(x )̺(x , x′ )dx dx′ A for a short period of time or may interact with A for ZZ b b b b b b the whole time up to ta. Assume that after ta, there is no further interaction between S and A. Instead S starts to To find the particle moving from a particular positionx ¯b ′ interact with another measuring system A , up to time at time tb, we substitute ψ(xb) = δ(xb x¯b) into Eq.(29), ′ tb. Denote the trajectories of S, A, A as x(t), y(t), z(t), − respectively. Up to time ta, the relational matrix element ′ ′ ′ p(¯xb) = ̺(xb, x )δ(xb x¯b)δ(x x¯b)dxbdx is given by Eq.(16), ZZ b − b − b (30) = ̺(¯x , x¯ ). i SA b b ~ S (x(t),y(t)) R(xa, ya) = e p x(t) y(t). (22) Z S Z A D D ′ Ra Ra Suppose there is no interaction′ between S and A after SA time ta, the action Sp (x(t), z(t)) consists only two inde- ′ ′ Up to time tb, the relational matrix element becomes SA S A pendent terms, Sp (x(t), z(t)) = Sp (x(t)) + Sp (z(t)). This allows us to rewrite Eq.(25) as a product of two R(x , y , z ) = x(t) y(t) z(t) b a b ′ ZRS ZRA ZRA D D D terms: b a b (23) ′ i SA SA i S exp [Sp (x(t), y(t)) + Sp (x(t), z(t))] K(xa, xb, zb) = x(t)exp Sp (x(t)) × {~ } Z S D {~ } Rab S S S i ′ We can split region b into two regions, a and ab, z(t)exp SA (z(t)) S R R R ′ ~ p (31) where is a region between time ta and time tb. × ZRA D { } Rab b This split allows us to express R(xb, ya, zb) in terms of ′ i A R(xa, ya), = K(xa, xb) z(t)exp Sp (z(t)) Z A′ D {~ } Rb R(x , y , z ) = R(x , y )K(x , x , z )dx , (24) ′ ′ b a b Z a a a b b a Consequently, function G(xa, xa; xb, xb) is rewritten as ′ ′ ∗ ′ ′ where G(xa, xa; xb, xb) = K(xa, xb)K (xa, xb) ′ ′ ′ i A A ′ K(xa, xb, zb) dzb z(t) z (t)exp [S (z(t)) S (z (t)) . ′ ~ p p × Z ZZRA D D { − } ′ b i SA (25) = x(t) z(t)exp Sp (x(t), z(t)) (32) Z S Z A′ D D {~ } Rab Rb The integral in the above equation is simply a con- From Eq.(25), one can derive the reduced density matrix ′ ′ stant, denoted as C. Thus, G(xa, xa; xb, xb) = element for S, ∗ ′ ′ CK(xa, xb)K (xa, xb). Substituting this into Eq.(28), one obtains ρ(x , x′ ) = R(x , y , z )R∗(x′ , y , z )dy dz b b ZZ b a b b a b a b 1 (26) ̺(x , x′ ) = K(x , x )ρ(x , x )K∗(x′ , x′ )dx dx′ . b b Z ZZ a b a b a b a a = ρ(x , x′ )G(x , x′ ; x , x′ )dx dx′ ZZ a a a a b b a a (33) The constant C is absorbed into the normalization factor. where Eqs.(33) and (30) together can explain the double slit experiment, as will be shown in next Section. G(x , x ; x′ , x′ ) = K(x , x , z )K∗(x′ , x′ , z )dz . We wish to generalize Eqs.(23) and (28) to describe a b a b Z a b b a b b b a series of interaction history of the quantum system (27) S. Suppose quantum system S has interacted with a Normalizing the reduced density matrix element, we have series of measuring systems A1,A2,...,An along the his- 1 tory but interacts with one measuring system at a time. ̺((x , x′ ) = ρ(x , x′ )G(x , x′ ; x , x′ )dx dx′ b b Z ZZ a a a a b b a a Specifically, S interacts with A1 up to time t1. From (28) t1 to t2, it only interacts with A2. From t2 to t3, it where the normalization factor Z = T r(ρ). only interacts with A3, and so on. Furthermore, we as- The reduced density matrix allows us to calculate the sume there is no interaction among these measuring sys- probability of finding the system in a particular state. tems. They are all independent. Denote the trajectories (1) (2) (n) For instance, the probability of finding the system ini- of these measuring systems as y (t), y (t), . . . , y (t), (1) (1) (2) (2) (n) (n) tially in a state ψ(xb). Defining a project operator and y (t1) = yb , y (t2) = yb , . . . , y (tn) = yb . 7

With this model, we can write down the relational ma- The normalized version of the reduced density matrix trix element for the whole history element is given by (n) (n) (1) (2) (n) R (xb , yb , yb , . . . , yb ) (n) (n) n (n) (n) ρ(xb , xb′ ) ′ (i) ̺(xb , xb ) = (40) = x(t) y (t) T r(ρ(tn)) Z S D {Z A D } Rn iY=1 Ri (34) n (n) i (j) (j) and the probability of ﬁnding S in a positionx ¯b at time exp [Sp (x(t), y (t))] × {~ } tn is Xj=1

S (n) (n) (n) where n is the region of finding the measured system S p(¯x ) = ̺(¯x , x¯ ). (41) previousR to time t , A is the region of finding measuring b b b n Ri system Ai between time ti−1 to ti. Action (n) (n) Recall that x , x ′ are two different positions of finding tj b b S(j)(x(t), y(j)(t)) = L(x(t), x˙(t), y(j)(t), y˙(j)(t))dt S at time tn. The probability of finding S at position p (n) Zt − j 1 x¯b is simply a diagonal element of the reduced density (35) matrix. is the integral of the Lagrangian over a particular path S A p lying in region n j . The reason the total action among S and A ,AR ,...,A∪ R is written as a summation 1 2 n C. The Double Slit Experiment of individual action between S and Ai is due to the key assumption that S only interacts with one measuring system Ai at a time, and the interaction with each of mea- In the double slit experiment, an electron passes suring system Ai is independent from each other. This through a slit configuration and is detected at position assumption further allows us to separate path integral x of the destination screen. Denote the probability that S 2 over region n into two parts, one is integral over region the particle is detected at position x as p1 = ϕ1(x) R S | |2 previous to tn−1, n−1, and the other is the integral over when only slit T1 is opened, and as p2 = ϕ2(x) R S when only slit T is opened. If both slits are| opened,| region between tn−1 and tn, n−1,n. Thus, 2 R the probability that the particle is detected at posi- (n) (n) (1) (2) (n) R (xb , yb , yb , . . . , yb ) tion x after passing through the double slit is given 2 (n−1) (1) (2) (n−1) by p(x) = ϕ1(x) + ϕ2(x) , which is different from (n−1) | 2 2 | = R (xb , yb , yb , . . . , yb ) (36) p +p = ϕ (x) + ϕ (x) . Furthermore, when a detec- Z 1 2 | 1 | | 2 | (n−1) (n) (n) (n−1) tor is introduced to detect exactly which slit the particle 2 2 K(xb , xb , yb )dxb goes though, the probability becomes ϕ (x) + ϕ (x) . × | 1 | | 2 | and This observation was used by Feynman to introduce the concept of probability amplitude and the rule of calculat- (n−1) (n) (n) (n) K(xb , xb , yb ) = x(t) y ing the measurement probability as the absolute square of ZRS D ZRA D n−1,n n (37) the probability amplitude [18]. However, why the proba- i bility is the absolute square of the probability amplitude exp S(n)(x(t), y(n)(t)) × {~ p } is not explained in Ref. [18]. By recognizing the mea- Similar to Eq.(18), the reduced density matrix element surement probability is a directional process, the mea- at t is surement probability is shown to be the absolute square n of probability amplitude [12]. In this section we will show i=n (n) (n) (i) (n) (n) (1) (2) (n) that these observations can be readily explained by ap- ρ(x , x ′ ) = [ dy ]R (x , y , y , . . . , y ) b b Z b b b b b plying Eqs.(33) and (30). iY=1 Denote the location of T1 is x1 and location of T2 is (n) (n) (1) (2) (n) ∗ (R (x , y , y , . . . , y )) x . Suppose both slits are opened. The electron passes × b b b b 2 (n−1) (n−1) through the double slit at ta and detected at the destina- = ρ(x , x ′ ) ZZ b b tion screen at tb. When the electron passes through the (n−1) (n) (n−1) (n) (n−1) (n−1) double slit, there is interaction between the electron and G(x , x , x ′ , x ′ )dx dx ′ × b b b b b b the slit. Denote the relational matrix after interaction as (38) R. Suppose there is no entanglement after the electron where passes through the slits, there is no inference information (n−1) (n) (n−1) (n) on exactly which slit the electron passes through. Fur- G(xb , xb , xb′ , xb′ ) = ther assuming equal probability for the electron passing (n−1) (n) (n) ∗ (n−1) (n) (n) (n) either T1 or T2, the state vector is represented by a super- K(xb , xb , yb )K (xb′ , xb′ , yb )dyb . Z position state ψa = (1/√2)( x1 + x2 ). The reduced (39) density operator| ati t is ̺(t ) =| (1i /2)(| xi + x )( x + a a | 1i | 2i h 1| 8

x ). Its matrix element is Substituted this into Eq.(33), the reduced density matrix h 2| element at tb is ̺(x , x′ ) = x ρˆ x′ a a h a| a| ai ̺(x , x′ ) = 1 b b (42) ∞ = (δ(xa x1) + δ(xa x2)) 1 ′ ∗ ′ ′ ′ 2 − − K(xb, xa)̺(xa, xa)K (xb, xa)dxadxa ′ ′ 2 ZZ−∞ (46) (δ(xa x1) + δ(xa x2)) × − − 1 = [K(x , x )K∗(x′ , x ) + K(x , x )K∗(x′ , x )]. where the property x x = δ(x x ) is used in the last 2 b 1 b 1 b 2 b 2 h a| ii a − i step. Later at time tb, according to Eq.(33), the density From this, the probability of finding the electron at po- matrix element for the electron at a position xb in the sition x is detector screen is b p(x ) = ̺(x , x ) ′ b b b ̺(xb, xb) = 1 ∗ ∗ ∞ = [K(xb, x1)K (xb, x1) + K(xb, x2)K (xb, x2)] 1 ′ ∗ ′ ′ ′ 2 K(xb, xa)̺(xa, xa)K (xb, xa)dxadxa 2 ZZ−∞ 1 2 2 (43) = ( K(xb, x1) + K(xb, x2) ) 1 ∗ ′ ∗ ′ 2 | | | | = [K(xb, x1)K (xb, x1) + K(xb, x2)K (xb, x2) 2 1 2 2 ∗ ′ ∗ ′ = ( ϕ1(xb) + ϕ2(xb) ) + K(xb, x1)K (x , x2) + K(xb, x2)K (x , x1)]. 2 | | | | b b (47) According to Eq. (30), the probability of finding the There is no interference term ϕ1ϕ2 in Eq.(47). This result electron at a position xb is can be further understood as following. The interaction between S and A′ results in the entanglement between p(x ) = ̺(x , x ) b b b these two systems. This is equivalent to say that A′ has 1 ∗ ∗ “measured” S. Therefore, A′ has extract information = [K(xb, x1)K (xb, x1) + K(xb, x2)K (xb, x2) 2 from S. The indeterminacy of which eigenstate S is in ∗ ∗ + K(xb, x1)K (xb, x2) + K(xb, x2)K (xb, x1)] disappears since such information can be inferred if we 1 know which eigenstate A′ is in. Thus, the interference = K(x , x ) + K(x , x ) 2 2| b 1 b 2 | terms should be excluded to avoid over-counting alter- 1 natives that contribute to the measurement probability = ϕ (x ) + ϕ (x ) 2 2| 1 b 2 b | (see more detailed explanation of this probability count- (44) ing rule in Ref. [12]), resulting in Eq.(47). where ϕ1(xb) = K(xb, x1) and ϕ2(xb) = K(xb, x2). Therefore, the probability to find the electron show- D. The Von Neumann Entropy ing up at position xb of the detector D is p(xb, tb) = 2 (1/2) ϕ1(xb, tb) + ϕ2(xb, tb) . The definition of Von Neumann entropy in Eq.(12) Now| suppose there is a weak| measurement such that calls for taking the logarithm of the density matrix. This another apparatus, A′, is added to detect whether the is a daunting computation task when the reduced density ′ electron passes through T1 or T2. A must interact matrix element is defined using path integral formulation with the electron in order to detect whether electron as Eq.(21). Brute force calculation of the eigenvalues of passes through T1 or T2. Denote the pointer states of the reduced density matrix may be possible if one ap- A′ that are corresponding to the electron passing though proximates the continuum of the position with a lattice T1 and T2 as T1 and T2 , respectively. After the in- model with spacing ǫ, and then takes the lattice spacing teraction, there| isi an entanglement| i between the elec- ǫ 0 to find the limit. On the other hand, in quantum → tron and A′. As a weak measurement (or, non-selective field theory, there is a different approach to calculate en- measurement), the quantum state of S is not yet pro- tropy based on the “replica trick” [19–21]. This approach jected to either eigenstate of x1 or x2 . The state allows one to calculate the von Neumann entropy with- vector of the composite system| ofi electron| i and A′ is out taking the logarithm. We will briefly describe this Ψ = 1/√2( x1 T1 + x2 T2 ), thus the reduced density approach and apply it here. In the case of finite degree operator| i for| thei| electroni | afteri| i passing the slit configura- of freedom, the eigenvalues of the reduced density ma- 1 tion isρ â = T rT ( Ψ Ψ ) = ( x1 x1 + x2 x2 ). Its trix λi must lie in the interval [0, 1] such that i λi = 1. | ih | 2 | ih | | ih | n n matrix element is This means the sum T r(ρ ) = i λi is convergent.P This statement is true for any n > 1P even n is not an integer. ̺(x , x′ , t ) = x ρˆ x′ n a a a h a| a| ai Thus, the derivative of T r(ρ ) with respect to n exists. 1 ′ It can be shown that = (δ(xa x1)δ(xa x1) (45) 2 − − ∂ (n) ′ n + δ(x x )δ(x x )). H(ρS) = lim T r(ρS) = lim HS (48) a − 2 a − 2 − n→1 ∂n n→1 9 where the Rényi entropy is defined as Substituting Eq.(21) into (52), we get n (i) (i) (n) 1 n Z(n) = ∆+−dx+− HS = lnT r(ρS). (49) Z { 1 n iY=1 − (i/~)[S(x(i)(t)−S(x(i)(t))] (53) The replica trick calls for calculating ρn for integers n e L R S × ZR(i) ZR(i) first and then analytically continuing to real number n. L R (i) (i) (i) (i) In this way, calculation of the von Neumann entropy is F (xL (t), xR (t)) xL (t) xR (t) . turned into the calculation of T r(ρn). But first, we have × D D } S We have omitted the normalization so far. To remedy to construct ρn from the path integral version of reduced S this, the normalized Z(n) = T r(ρ ) should be rewritten density matrix element in Eq.(21). n as ρn is basically the multiplication of n copies of the S Z(n) same density matrix, i.e., ρn = ρ(1)ρ(2) . . . ρ(n), at time (n) = T r(̺n) = . (54) S S S S Z Z(1)n tb. To simplify the notation, we will drop the subscript S. Since all the calculations are at time tb, we also drop sub- where Z(1) = Z = T r(ρ) as defined in Eq.(21). Once script b in Eq.(21). Denote the element of ith copy of re- (n) is calculated, the von Neumann entropy is obtained (i) (i) (i) (i) Z duced density matrix as ρ(x− , x+ ), where x− = xL (tb) through (i) (i) and x+ = xR (tb) are two different positions at time tb, ∂ n ∂ (i) (i) H(̺) = lim T r(̺S) = lim (n) (55) n→1 n→1 and xL (t) and xR (t) are two different trajectories used − ∂n − ∂nZ to perform the path integral for ith copy of reduced den- Eq.(53) appears very complicated. Let’s validated in sity matrix element as defined in Eq.(21). With these the case that there is no interaction between S and A. n notation, the matrix element of ρS is One would expect there is no entanglement between S and A. Thus, the entropy should be zero. We can check n (1) (n) whether this is indeed the case based on Eq.(53). In ρ (x− , x+ ) this case, the influence functional is simply a constant. = dx(1)dx(2) . . . dx(n−1)dx(n)ρ(x(1), x(1)) ... Eq.(21) is simplified into Z + − + − + − (n) (n) (1) (2) (n−1) (n) ′ 1 ∗ ′ 1 ′ ρ(x , x )δ(x x ) . . . δ(x x ) ̺(xb, x ) = ϕ(xb)ϕ (x ) = ρ(xb, x ) (56) × + − + − − + − − (50) b Z b Z b (n−1) (i) (i+1) (i) (i) where ϕ(x ) has been given in Eq.(19). Taking trace of = [dx dx ρ(x , x ) b Z + − − + the above equation, iY=1 (i) (i+1) (n) (n) 1 ∗ δ(x+ x− )]ρ(x− , x+ ). 1 = T r(̺) = ϕ(x )ϕ (x )dx (57) × − Z Z b b b The delta function is introduced in the calculation above gives to ensure that when multiplying two matrices, the second Z = ϕ(x )ϕ∗(x )dx (58) index of the first matrix element and first index of the Z b b b second matrix element are identical in the integral. From Eq.(50), we find the trace of ρn is Eq.(56) implies that S is in a pure state, since by definition, ϕ(x ) = x ϕ = δ(x x )ϕ(x)dx, so that b h b| i − b (1) (n) (n) (1) (n) (1) R n n ′ 1 ′ T r(ρ ) = ρ (x− , x+ )δ(x+ x− )dx+ dx− ̺(x , x ) = x ϕ ϕ x . (59) Z − b b Z h b| ih | bi n = ρ(x(i), x(i))δ(x(i) x(mod(i,n)+1)) (51) Multiplication of density matrix that represents a pure Z { − + + − − state gives the same density matrix itself. Using the same iY=1 (i) (mod(i,n)+1) notation as in Eq.(50), we obtain dx+ dx− × } 2 (1) (2) ρ (x− , x+ ) where mod(i, n) = i for i < n and mod(i, n) = 0 for i = (1) (1) (2) (2) (1) (2) (1) (2) (i) = ρ(x− , x+ )ρ(x− , x+ )δ(x+ x− )dx+ dx− n. To further simplifying the notation, denote ∆+− = Z − (i) (mod(i,n)+1) (i) (i) (mod(i,n)+1) (1) (1) (1) (2) (1) δ(x+ x− ), dx+− = dx+ dx− and = ρ(x , x )ρ(x , x )dx Z(n) =−T r(ρn), the above equation is rewritten in a more Z − + + + + compact form = ϕ(x(1))ϕ∗(x(1))ϕ(x(1))ϕ∗(x(2))dx(1) Z − + + + + n (i) (i) (i) (i) (1) (2) Z(n) = ρ(x , x )∆ dx (52) = Zρ(x− , x+ ). Z { − + +− +−} iY=1 (60) 10

From this we can deduce that ρn = Zn−1ρ. This gives electromagnetic field between the double slit and the des- Z(n) = T r(ρn) = Zn, and tination screen detector. We will need to apply Eq.(28) instead of (33) to calculate the reduced density matrix Z(n) (n) = = 1. (61) element. In this case, we simply substitute Eq.(42) into Z Z(1)n Eq.(28) and obtain 1 It is independent of n, thus ̺(x , x′ ) = [G(x , x ; x , x′ ) + G(x , x ; x , x′ ) b b 2 1 b 1 b 1 b 2 b (66) ∂ ′ ′ H(̺) = lim (n) = 0, (62) + G(x2, xb; x1, xb) + G(x2, xb; x2, xb)]. − n→1 ∂nZ The probability of finding the electron at position xb is as expected. The von Neumann entropy is only non- zero when there is an interaction between S and A. The p(xb) = ̺(xb, xb) effect of the interaction is captured in the influence func- 1 = [G(x , x ; x , x ) + G(x , x ; x , x ) (67) tional. Concrete form of the influence functional should 2 1 b 1 b 1 b 2 b be constructed in order to find examples where the von + G(x , x ; x , x ) + G(x , x ; x , x )]. Neumann entropy is non-zero. 2 b 1 b 2 b 2 b ′ ′ From the definition of function G(xa, xb; xa, xb) in Eq.(27), it is easy to derive the following property, IV. DISCUSSION AND CONCLUSION ′ ′ ∗ ′ ′ G(xa, xb; xa, xb) = G (xa, xb; xa, xb). (68) A. The G Function ′ ′ When xa = xa and xb = xb, we have G(xa, xb; xa, xb) = ∗ G (xa, xb; xa, xb). Thus, G(xa, xb; xa, xb) must be a real The G function introduced in Eq.(27) can be rewritten function. We denote it as GR(xa, xb). With these prop- in terms of the influence functional. To do this, we first erties, Eq.(67) can be further simplified as rewrite Eq.(25) as 1 p(xb) = [GR(x1, xb) + GR(x2, xb)] i S 2 (69) K(xa, xb, zb) = x(t)exp ~Sp (x(t) ZRS D { } + Re[G(x1, xb; x2, xb)]. ab (63) ′ ′ i A SA z(t)exp [Sp (z(t)) + Sint (x(t), z(t))] This is consistent with the requirement that p(xb) must × Z A′ D {~ } Rb be a real number. The second term Re[G(x1, xb; x2, xb)] is an interference quantum effect due to the fact that the Substituting this into Eq.(27), we have initial state after passing through the double slit is a pure state. This interference term also depends on the inter- ~ S S ′ ′ ′ (i/ )[Sp (x(t))−Sp (x (t))] G(xa, xb; xa, xb) = e action between S and B. If the interaction is adjustable, ZRS ZRS ab a′b′ the probability distribution will be adjusted accordingly. F (x(t), x′(t)) x(t) x′(t) For instance, if there is an electromagnetic field between × D D (64) the double slit and the destination screen that interacts with the electron, tuning the electromagnetic field Consequently, the normalized reduced density matrix el- will cause the probability distribution p(xb) to change ement in Eq.(28) becomes through the interference term in Eq.(69). Presumably, the Aharonov Bohm effect [23] can be explained through 1 ~ S S ′ − ′ (i/ )[Sp (x(t))−Sp (x (t))] Eq.(69) as well. The detailed calculation will be pub- ̺((xb, xb) = e Z ZZRS ZRS lished in a later paper. ab a′b′ ρ(x , x′ )F (x(t), x′(t)) x(t) x′(t)dx dx′ . × a a D D a a (65) B. Influence Functional and Entanglement Entropy This gives the same result as in Ref. [22]. However, there is advantage of using the G function instead of the in- In Section IIID, we show that in the condition that fluence functional F because the complexity of path in- there is no interaction, the influence functional is a con- tegral is all captured inside the G function, making it stant and therefore the entanglement entropy is zero. We mathematically more convenient. This can be shown in can relax this condition to detect whether there is entan- the following modified double split experiment example. glement. Suppose the influence functional can be decom- Suppose after the electron passing the double slit, there posed in the following production form, is no detector next to the slits to detect which split the F (x(t), x′(t)) = f(x(t))f ∗(x′(t)). (70) electron passing through, but there is another system B that continues to interact with S till S reaches the des- Such a form of influence functional satisfies the rule [18] tination screen detector. Alternatively, this can be an F (x(t), x′(t)) = F ∗(x′(t), x(t)). Eq.(70) implies that the 11 entanglement entropy is still zero even there is interaction of Sp, which is the phase of a complex number. This ~ between S and A. The reason for this is that the reduced complex number, eiSp/ , is just a probabilistic quantity, density matrix element can be still written as the form instead of a quantity associated with a physical property. of Eq.(56) but with Second, it gives a natural derivation of the coordinator representation of the reduced density matrix of the ob- ~ ϕ(x ) = e(i/ )S(x(t))f(x(t)) x(t). (71) served system. Based on the coordinator representation b Z Rb D of the reduced density matrix element, it is mathematically convenient to develop formulation for some of inter- Again Eq.(56) shows S is in a pure state therefore the esting physical processes and concepts. For instance, we reasoning process from Eqs.(56) to (62) is applicable here. can describe the interaction history of the measured sys- We now examine the entanglement entropy for tem and a series of measuring systems or environment, some general forms of influence functional discussed in and use this formulation to elegantly explain the double Ref. [18]. The most general exponential functional in slit experiment. A more interesting application of the linear form is coordinator representation of the reduced density matrix is the method to calculate entanglement entropy using F (x(t), x′(t)) = exp i x(t)g(t)dt i x′(t)g(t)dt] path integral approach. This will allow us to potentially { Z − Z } (72) calculate entanglement entropy of a physical system that where g(t) is a real function. Clearly this form interacts with classical fields, such as an electron in an satisfies the condition specified in Eq.(70) since we electromagnetic field. Section IVB gives a criterion on can take f(x(t)) = exp[i x(t)g(t)dt] and f(x′(t)) = whether there is entanglement between the system and exp[i x′(t)g(t)dt]. The entanglementR entropy is zero external environment based on the influence functional. with thisR form of influence functional. Since entanglement entropy is an important concept in On the other hand, the most general Gaussian influ- quantum information theory, the method described here ence functional [18] may lead to new insight on the information aspect of a physical system that interacts with classical fields. This is a topic for further research. F (x(t), x′(t)) = exp [α(t, t′)x(t′) {ZZ (73) α∗(t, t′)x′(t′)][x(t) x′(t)]dtdt′ Ref. [12, 13] and this paper together show that quan- − − } tum mechanics can be constructed by shifting the start- where α(t, t′) is an arbitrary complex function, defined ing point from the independent properties of a quantum only for t > t′. This form of influence functional can- system to the relational properties between a measured not be decomposed to satisfy the condition specified in quantum system and a measuring quantum system. In Eq.(70). The entanglement entropy may not be zero in essence, quantum mechanics demands a new set of rules this case. It will be of interest to further study the influ- to calculate probability of a potential outcome from the ence functional of some actual physical setup and calcu- physical interaction in quantum measurement. The re- late the entanglement entropy explicitly. lational formulation of quantum mechanics is not only significant at the conceptual level, but also valuable at the application level. At the conceptual level, the differ- C. Conclusions ence between the relational formulation and traditional formulation results in fundamental consequence in some We show that the relational probability amplitude can special scenario. This is demonstrated in resolving the be implemented using path integral approach. The for- EPR paradox [13]. At the application level, the path mulation is consistent with the results from the tradi- integral implementation of relational probability ampli- tional path integral quantum mechanics. The signifi- tude leads to new tool to calculate entanglement entropy cance of such implementation is two-folds. First, it gives based on the coordinator representation of the reduced a clearer meaning of the relational probability amplitude. density matrix. Ultimately, we hope the relational for- SA iSp/~ Rij is defined as the sum of quantity e , where Sp is mulation presented in Ref. [12, 13] and this paper can the action of the composite system S + A along a path. be one step towards a better understanding of quantum Physical interaction between S and A may cause change mechanics.

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