Quantum Information and Accounting Information: Their Salient Features and Applications1
by Joel S. Demski,* Stephen A. FitzGerald,** Yuji Ijiri,*** Yumi Ijiri,** and Haijin Lin* *University of Florida, Fisher School of Accounting
**Oberlin College, Department of Physics and Astronomy ***Carnegie Mellon University, Tepper School of Business
Introduction
In this paper, we explore the nature of quantum information in order to search for
promising conceptual applications to accounting. After first introducing salient features, such as quantum superposition, randomness, entanglement, and unbreakable cryptography, we examine the research methods in quantum information and evaluate useful possibilities in accounting. In particular, we conclude that important lessons can be derived from quantum information’s attention to the fundamental laws of the discipline, consistency with past principles, causality of events, and ways to cope with a paradigm shift.
We then draw parallels to double-entry information through a discussion of the work of nineteenth century mathematician Arthur Cayley. He developed important mathematical concepts, such as matrix algebra, that later became indispensable for quantum mechanics. In addition, Cayley wrote a small booklet on double-entry bookkeeping and praised the system embedded in double-entry bookkeeping as an “absolutely perfect one.” We describe the parallels made by Cayley between Euclid’s ratio theory and double-entry theory and conclude that Cayley was able to give such high praise to accounting because of the isomorphism he saw between the
1 To be presented at Carnegie Mellon University, Tepper School of Business, Accounting mini-Conference on
August 26 and 27, 2005. CMU Tepper School Working Paper No. 2005-E42. A preliminary version of the paper is to be presented at Goizueta School of Business, Emory University, on August 15, 2005. Its Japanese version was presented at Department of Management, Komazawa University in Tokyo on June 26, 2005.
“ratio matrix” and the “double-entry matrix.” The feasibility of a hybrid, “quantum double-entry information,” is briefly explored.
Accounting information, in our point of view, is a broader concept than double-entry information. We emphasize its endogeneity issues including recognition and aggregation. The measurement school and the information content school are contrasted and critically analyzed from both physical and social perspectives. Some conceptual applications of quantum
information to accounting information are identified. By exploring quantum information, we hope to find a way to integrate the measurement process and its interactions with the
environment when the double-entry feature of the information is both preserved and emphasized.
Part I: Understanding Quantum Information 1. Salient Features of Quantum Information
In a cross-disciplinary effort, we wish to examine developments in quantum information with an eye toward their possible conceptual application to accounting.2 In recent years, there has been much attention in physics to the application of quantum mechanics to information
processing as opposed to using it to explain the behavior of matter evidenced in certain circumstances. We wish to take this a step further and seek a connection between so-called “quantum information” and accounting.3
To begin, we review some salient features of quantum information.
2 Here, we are using the term “information” broadly to include “computation,” except where the latter term fits in
more specifically.
3 Interest in quantum information has been almost non-existent in accounting, with the notable exception of
1A) Quantum superposition and quantum parallel processing: The most important feature of quantum information as compared with traditional information centers on the nature of the fundamental state of the data. The basic unit of classical physical information is a “bit” which is always either 0 or 1. The basic unit of quantum information is a “qubit” which is also either 0 or 1 if not “suspended.” If suspended in a so-called “superposition” state, it exists as a linear combination of states. However, a qubit can take on such a superposition state only as long as no measurements of the state are taken. Once a measurement is taken, the state of the qubit goes back to being either 0 or 1 (see, for example, Nielsen (2002)).
The power of using qubits as opposed to a series of bits lies in these more complex states that allow for intricate parallel processing, as considered in the following example. Suppose a quantum computer X operates with 2 qubits, A1 and A2. X can represent 4 states since A1 can be 0 or 1 and A2 can be 0 or 1; hence 00, 01, 10, and 11 are the 4 states. Thus a quantum computer X with 2 qubits is actually probing 4 possible conditions or states at the same time. A classical computer Y can also do the same parallel processing but with 4 classical computers acting together in parallel, B1, B2, B3, and B4, in order to investigate the 4 different states.
If a quantum computer X operates with 3 qubits, A1, A2, and A3, it can then represent and operate 8 states simultaneously, namely 000, 001, 010, 011, 100, 101, 110, and 111. To do the same with a classical computer Y, we must have 8 computers and operate them in parallel. Thus the following comparison shows the power of the quantum computer, reflecting the difference between arithmetic and geometric progressions (Gribbin 2002).
Number of qubits in a quantum computer: 2 3 4 5 6 7 8 9 10 Number of classical computers needed: 4 8 16 32 64 128 256 512 1024
This illustrates the enormous difference in computational power between quantum and classical computers.
1B) Irreducible randomness, inexhaustible uncertainty, and inexplicable probability amplitude: The second most important feature of quantum information is a randomness that is a fundamental, intrinsic, and physical reality. Milburn (1998) states, “The quantum principle is this: physical reality is irreducibly random, but random in a way we could never have expected (p. 1).” For example, consider a situation in which you place red and white particles into a bag, draw from it, and guess the color. You can attribute the degree of statistical randomness to a good or bad shuffling before drawing a particle. Classically there is a statistical randomness in what color particle you draw, but the particles themselves are intrinsically either red or white, and they do not switch colors. But from a quantum mechanics perspective, they do. In this example, a particle may be observed to be red at one time and white at another. Quantum randomness means the randomness is in the particle itself, and so there is an “inexhaustible uncertainty.” In quantum information theory, that fundamental unit of uncertainty is associated with the qubit.
The uncertainty attributed to a particle or a qubit’s state also translates into uncertainties in ascertaining other physical features. As Werner Heisenberg discovered, it is not possible to determine with infinite precision a particle’s position and velocity (or momentum) at the same time.4 A rough interpretation of this statement is that the measurement of one feature interacts and disturbs the particle’s other features.
Note that the existence of quantum randomness and uncertainty means that the notion of probability appears quite naturally. However, one peculiar aspect of probabilities in quantum mechanics is that probabilities themselves are not fundamental but are determined by a so-called
“probability amplitude” which is multiplied by itself to obtain a probability.5 Probability amplitude is a notion introduced to capture how the environment affects the probability for an event. This modified definition of probability means that the additive nature of probability in classical probability theory has to be treated differently in quantum probability theory. Bayes’ rule in classical theory states that “the probability for an event which can happen in two indistinguishable ways is the sum of the probability for each considered separately,” while the quantum version, called Feynman’s rule, states that “the probability amplitude of an event that can occur in two or more indistinguishable ways is the sum of the probability amplitude for each way considered separately (Milburn 1998, pp. 195, 198).” Again, in quantum mechanics, the resulting sum of the probability amplitudes is then “squared” to obtain the probability. The reason for this difference comes from the possibility of interference or intermixing which may be neglected in classical theory but can be quite significant in quantum theory.
This intrinsic uncertainty and associated probabilistic character to quantum information mean that quantum computers must be prepared for the occurrence of computation ambiguities or errors. Furthermore, the quantum information state (a superposition state) is often very fragile and can “decohere” easily. Therefore, the data must be refreshed before they become corrupted by outside interferences. A practical implementation will focus on computations where if and when the computer generates a correct answer, it is immediately recognized. Such computations are observed in some fields, for example, data mining, factorization, etc. Also, any computations where approximately correct answers are acceptable and the error range can be bound are among the candidates for quantum computation.
5 Technically, the probability amplitude or “wavefunction” is multiplied by the complex conjugate of itself. In other
1C) Quantum entanglement and quantum teleportation: The third salient and perhaps most mysterious feature of quantum information centers on quantum entanglement and quantum teleportation. Here entanglement refers to the intermixing of states of more than one particle or entity. Essentially, after entanglement, individual particles that became a part of the group lose their own quantum identity. Instead, the group is now given a state of its own, and all member particles assume that state.
Once entangled, the particles can be separated far apart from each other, and yet still maintain the characteristics of the state of the group. Hence, for a two-particle system, if a property of one member is measured, instantaneously, a particular property for the other member becomes known with certainty. The interaction of the particles in the past helps prediction in the future. Entanglement can be useful as a resource for solving information-processing problems in new ways. Examples include using entanglement to assist sending classical information from one to another location (a.k.a., superdense coding), and using entanglement to teleport a quantum state from one to another location (a.k.a., quantum teleportation). Quantum teleportation has been verified in a number of experiments including one involving a fiber optic cable over a 10 kilometer distance between the two entangled particles.6, 7
6 Gribbin (2002) states “In these experiments, an atom is induced to emit two photons simultaneously in opposite
directions. The common origin of these photons means that they are correlated with one another and, according to the equation of quantum physics, they remain “entangled” even when they are far apart--as if they formed a single particle. The experiments proved that measuring the properties of one of the photons on one side of the lab affected the other photon on the other side of the lab instantaneously. [However, it should be noted that this procedure did not violate the principle of special relativity, in that no information was instantaneously transmitted (the
authors).]….By the middle of 1990s, researchers in Geneva had extended these experiments and sent photons along fiber-optic cable 6.2 miles (10 km) in length...The two photons acted like one particle even when they were more than six miles apart.” See also Aczel (2001, p. 237).
7 The following excerpt reported in a national newspaper, Asahi Shimbun, in Japan on June 12, 2005 might also be
an indication of the direction where this type of research is headed. “While quantum entanglement has been thought to be difficult to create beyond a single pair of particles at a time, researchers at University of Tokyo reported in the
Entanglement exhibits an analogous behavior to energy in that different forms of entanglement are found to be qualitatively equivalent just as light, heat and electricity can be viewed as different forms of energy (see, for example, Aczel 2001.)
1D) Quantum cryptography and the discovery of unbreakable keys: In information theory, protecting the privacy of information has become increasingly important as hackers and identity thieves proliferate the Internet and other communication channels. Hence, cryptography is, along with speed and accuracy, one of the fundamental subjects of the field of quantum information.
The most commonly used coding system is called, “RSA coding.” It was developed by three professors at MIT, Ronald Rivest, Adi Shamir, and Leonard Adleman (1978) and became the most popular coding method in 1980s as the use of the Internet exploded. It takes advantage of the fact that if two large prime numbers, say each a few hundred digits long, are multiplied, the resulting product is extremely difficult to factor into the original two prime numbers even using the fastest supercomputers currently available. The text to be transmitted is converted into a series of digits and multiplied by the product of the two prime numbers and sent to the receiver, who decodes it using one of the two prime numbers as the key.
The popularity of RSA coding was threatened in 1994 when Peter Shor of AT&T Bell Labs reported a quantum algorithm that can make the code-breaking far easier than previously thought. The computational steps for factoring a 500-digit number using a classical computer was said to take 100 million times more steps than for a 250-digit number. Shor says the use of his algorithm will reduce it to merely eight times because the cost of his algorithm goes up only polynomially not exponentially as in the case of classical computers (see Shor 1997).
The threat of losing RSA coding without a competent replacement hung over the Internet for sometime, but an idea developed in 1984 by Charles Bennett and Gills Brassard (1984) has now been implemented and become commercially available. This method called “BB84” makes use of Heisenberg’s uncertainty principle to detect an eavesdropper who tries to intercept a transmission. Although it is impossible to say the system is foolproof, as bribery of insiders and other loopholes may exist, it is said to be foolproof quantum mechanically, i.e. as long as quantum mechanical laws are relied upon.8
2. Research Methods and Research Emphases in Quantum Information
After examining the salient features, we now change our focus to the research methods used in quantum information.
2A) Making quantum phenomena visible with a Bose-Einstein Condensate: One important component to research in quantum information focuses on experimental efforts, particularly in the creation of a so-called “Bose-Einstein Condensate” (BEC) and related “atomic chips,” putting atoms at only one millionth of a degree above absolute zero. A recent article in Scientific American states “Over the past few years, BEC studies have given fresh experimental life to many quantum effects that formerly were considered very remote and inaccessible in practice. In this sense, BEC research has made quantum phenomena become more real, like a rock you can directly see and kick instead of something talked about in the abstract (Reichel 2005).”
2B) Quantum algorithms by Shor, Grover, and others: Beyond experimental manifestations, quantum algorithms have played especially important roles in quantum
information. The tradition goes back to the 1930s when Alan Turing developed the “Turing machine,” a very early computational machine. It was astonishing at the time Turing did so without any models or blue prints of computers because they were to be developed some two decades later. Many of the algorithms that have been developed for quantum information have been of the same kind, as quantum computers are still in a very early experimental stage. It is still remarkable that such algorithms can be constructed with actual computers “sight unseen.”9
Lov Grover’s search algorithm (Grover 1997) also acted as a wake up call to get people prepared for the forthcoming changes in computational tools. To briefly explain the algorithm, consider a search in a telephone directory that has a million entries. The search goes normally from the names to the phone numbers; hence the telephone directory is in alphabetical order by the owner’s name. Suppose a search in reverse is necessary; namely we know the phone number and want to search the name of the owner of that phone number. Clearly, the search requires on average 500,000 entries to be checked. Grover’s algorithm requires on average only 1,000 accesses to the database, namely only the square-root, √N, of the total number of entries N, taking advantage of quantum parallel processing.
2C) An insightful way to think about the fundamental laws of physics: Andrew Steane’s (1998) article provides an excellent in-depth review of the state of quantum information at that time. He expresses a pessimistic view of quantum computers but strong optimism about the field itself. He states “As things stand, no QC [quantum computer] has been built, nor looks likely to be built in the author’s lifetime, if we measure it in terms of Shor’s algorithm, and ask for factoring of large numbers. However, if we ask instead for a device in which quantum-information ideas can be explored, then only a few quantum bits are required and this will
9
It reminds us of Euclidean geometry, in which after the five axioms are accepted, the whole field can be developed purely deductively from the axiom. Thus, for example, the existence of Shor’s factoring algorithm, mentioned earlier, indicated issues with quantum computers well before their physical manifestation.
certainly be achieved in the near future. Simple 2-bit operations have been carried out in many physics experiments, notably magnetic resonance, and work with three to ten qubits now seem feasible (p. 128).”
Steane concludes the article with insightful comments, reproduced below. “The idea of ‘quantum computing’ has fired many imaginations simply because the words themselves suggest something strange but powerful, as if the physicists have come up with a second revolution in information processing to herald the next millennium. This is a false impression. Quantum computing will not replace classical computing for similar reasons that quantum physics does not replace classical physics: no one ever consulted Heisenberg in order to design a house and no one takes their car to be mended by a quantum mechanic. If large QCs are ever made, they will be used to address just those special tasks which benefit from quantum information processing... A more lasting reason to be exited about quantum computing is that it is a new and insightful way to think about the fundamental laws of physics (p. 166).”
2D) Stretching the domain without damaging the classical interpretation: Steane’s last comment above seems to indicate a method that has worked over many centuries, namely extension takes place by expanding the coverage or domain of interpretation without altering the original, more limited, region. A simple illustration using a function space might best clarify the concept of domain stretching.
For example, consider a function S(x) = 1 + x + x2 + x3 + ... which has a value defined only for -1 < x < 1, since outside the range, the sum does not converge. On the other hand, R(x) = 1/(1 - x) = 1 + x + x2 + x3 + ... takes the same form as S(x) if expanded, but in the form of R(x) = 1/(1 - x), it has values defined for all x, except for x = 1. Hence the domain of R(x) is larger than S(x). If S(x) is analogous to a classical method within a limited domain, R(x) is analogous
to a quantum method. Notice the important issue is S(x) = R(x) everywhere the classical domain is defined, namely, -1 < x < 1. If this is not true even for a single x in -1 < x < 1, then the
classical method is threatened by the domain stretching of the quantum approach (see Derbyshire 2003).
Why should we be so careful not to damage an old more limited theory while replacing it with a more expanded one? Here we see a tremendous tradition in physics to respect what worked in the old domain. Of course, if the old theory is found to be definitely wrong by new observations, it will be discarded immediately. But as long as doubt exists, the old principle will be maintained even under skepticism. Why?
Making analogy to chess, Nielsen (2002) states “The goal of quantum information science is to understand the general high-level principles that govern complex quantum systems such as quantum computers. These principles relate to the laws of quantum mechanics in the way that heuristics for skillful play at chess relate to the game’s basic rules (p. 68)...By applying what we have learned from quantum information science, we may greatly enhance our skills in the on-going chess match with the complex quantum universe (p. 75).”
2E) Emphasis on causality of events: Quantum information theory pays attention to the issue of causality of events. “Most of the laws in physics are descriptions of cause-and-effect relationships (Omori 2004 [the quotation translated from the Japanese original by the authors of this article]).” This is exactly the same as in double-entry bookkeeping. We may in fact restate the above quotation as: “Most journal entries in accounting are descriptions of cause-and-effect relationships.”
Why is causality so important in physics and likewise in accounting? It is because causality allows one to predict the future much more reliably. “If A then B” expresses repeated
observations in which “if A occurs, then B follows.” This may be used for prediction
(“observational causality”) or for action (“manipulative causality”, i.e. “make A happen, then B will occur”). Note that in both quantum and accounting information, there are limits to the degree to which causal relations may be stated, limits which may be expressed by the probabilities associated with different states or events. Hence, the issues of irreducible
randomness and uncertainty in quantum information translate into key constraints on predictive power.
Another important aspect of considering the causal nature of information is that it allows the recording of cause and effect in tandem, which is not possible if the information is “single-entry.” This leads to another advantage of causal information. Namely, the control of information is much easier in double-entry framework than in single-entry framework, because the data form a network of causal linkages.
2F) Coping with a paradigm shift such as quantum entanglement: As noted earlier, entanglement is a unique feature of quantum information, which requires a shift in paradigm from classical information theory. Nielsen and Chuang (2000) state, “Entanglement is a uniquely quantum mechanical resource [emphasis added] that plays a key role in many of the most
interesting applications of quantum computation and quantum information; entanglement is iron to the classical world’s bronze age. In recent years there has been a tremendous effort to better understand the properties of entanglement considered as a fundamental resource of Nature, of comparable importance to energy, information, entropy, or any other fundamental resource. Although there is as yet no complete theory of entanglement, some progress has been made in understanding this strange property of quantum mechanics. It is hoped by many researchers that
further study of the properties of entanglement will yield insights that facilitates the development of new applications in quantum computation and quantum information (pp. 11~12).”
The way in which entanglement is assimilated into quantum information theory will also be interesting from the standpoint of evolving research methods in particular, how classical principles need to be modified. In any event, quantum information is a field that is full of excitement. It can easily influence its neighboring disciplines positively. We hope this paper makes a small step toward exploring this benefit suggesting renewed emphasis on fundamentals, new approaches, and the sheer excitement of enhanced understanding.
Part II: Quantum Information and Mathematical Structure of Double-Entry Systems 3. Salient Features of Classical Double-Entry Information
3A) Double-entry information vs. single-entry information: Having examined quantum information in some depth, let us now turn to the double-entry information channel, which has been at the core of accounting information. Generally speaking, the difference
between “double-entry” information and “single-entry” information lies in the “connectivity” of information. For example, a firm finds an extra $10,000 of cash in the safe. For anyone else this is a good news, but not for accountants. Accountants cannot record the event until they find a reason why cash increases. If it is because the customer paid in advance on the previous day for goods to be delivered on the following day, then this information must be matched or connected to another piece of information, namely an increase in the firm’s obligation to deliver the goods. If an investor paid for the firm’s newly issued stock, cash increase must be matched with an increase in stockholders’ equity. If it is an outright donation with no strings attached, cash
increase must be matched with income increase. No such connectivity is required in single-entry information.
Figure 1: Double-Entry Information (Left) and Single-Entry Information (Right)
Connectivity is based on the accountants’ judgment on causality. As shown in the left graph of Figure 1, if the circle in the center represents “cash” and an arrow coming into the circle means an increase in cash and an arrow going out from the circle means a decrease in cash, the cash circle at the center is connected from one of the four circles with arrows pointing to cash. While connectivity is based on causality, the causal relations are often difficult to determine, in which case accountants may have to be satisfied by mere correlation or by “nominal accounts” as against “real” accounts.
Nevertheless, what is remarkable is that for over five centuries, double-entry information has been used diligently throughout the world. More importantly, this causal connection is not an option that they can use in some cases and not in others. It is to be practiced 100% of the time. For non-accountants, the habit of always finding another event causally related to the event on hand is not a natural talent. It is the hardest part of accounting classes to foster this double-entry perspective. New students are all single-entry accountants. They are happy to learn that $10,000
increase in cash balance and adjust the cash balance accordingly in the center circle on the right of Figure 1 but they are under no pressure to connect the change with anything else.
Usefulness of double-entry information for planning is obvious. Having accumulated causal judgments thousands of times in one period, double-entry information offers a highly reliable data source for planning purposes. A successful firm can explain their success with various strategies they undertook. Single-entry information provides none of this helpful data, as they merely record what happened but not why. (Of course there are limits to this projection exercise; for example, absent separability and constant returns to scale, accounting product costs cannot well approximate the firm’s vector of marginal costs.)
Double-entry information is also useful for control purposes.10 Double-entry information can offer a solid base for looking back and evaluating the past performance. Again this is
because the information is supported by the numerous causal judgments, some successful some failures. At the same time, double entry accounting should not be confused with the mythological hydra, as information useful for planning or valuation is not necessarily useful for control.
What is even more interesting is the fact that double-entry information is rarely observed outside accounting, nowhere near the100% practice level as observed in accounting throughout the world. This is strange in that there is nothing in double-entry information that should be limited to accounting. In fact, the recording of “What” and “Why” in tandem is so natural that a full system of double entry information can come out of science or engineering laboratories. For example, recording the change in the temperature of equipment is single entry. It can be made double-entry information if every recorded temperature change is causally connected with various events that occurred in the laboratory during the period.
10 The Sarbanes-Oxley Act of 2002, Section 404, Management Assessment of Internal Controls, has elevated the
importance of double-entry records to a much higher level than before. This section is only 169 words long but has forced enormous changes in business.
There may be a good reason for the non-use of double-entry information in science--Causality itself seems to have been replaced by predictability ever since Kepler’s three laws were used to predict planetary motion, prior to a complete understanding of gravity. They predicted planetary motion extremely accurately, but no one was able to provide the reason why the laws work. Especially in more recent years in quantum mechanics, there have been so many
predictable and reproducible phenomena that can never be causally determined. If this is the main reason of non-use, the basis of connectivity in double-entry information may be moved away form causality and be satisfied with recording correlation or contemporaneously salient phenomena.
3B) Mathematical Structure: Comparing ratio matrix and double-entry matrix: Double-entry bookkeeping was originally invented and published in 1494 by an Italian
mathematician, Luca Pacioli (1494), as a part of a series of mathematical volumes. Since then, it has spread widely throughout the world, even in communist countries, as the merchants’ best way of organizing and recording business transactions. Over five centuries later, it is still an indispensable tool in business. As German economic historian Werner Sombart (1902) stated, “Double-entry bookkeeping is borne of the same spirit as the system of Galileo and Newton… With the same means as these, it orders the phenomenon into an elegant system, and it may be called the first cosmos built upon the basis of a mechanistic thought. Double-entry bookkeeping discloses to us the cosmos of the economic world by the same method as, later, the cosmos of the stellar universe was unveiled by the great investigation of natural philosophy.”
While trying to relate quantum information with accounting may sound far-fetched, the nineteenth century British mathematician, Arthur Cayley, is perhaps the best person to provide a bridge between quantum information and accounting. Cayley developed matrix algebra, an
essential basis for quantum mechanics, and published a twenty-page booklet on double-entry bookkeeping from Cambridge University Press (Cayley 1894).11 Cayley himself commented on the parallels between double-entry bookkeeping and mathematical theory. He gave the highest praise imaginable for double-entry bookkeeping in the preface of his booklet: “The Principles of Book-keeping by Double Entry constitute a theory which is mathematically by no means
uninteresting: it is in fact like Euclid’s theory of ratios an absolutely perfect one, and it is only its extreme simplicity which prevents it from being as interesting as it would otherwise be.1213”
In order to further explore the mathematical roots of double-entry information, we are anxious to explore what Cayley saw in common between double-entry bookkeeping and Euclid’s theory of ratios as well as between double-entry bookkeeping and the matrix algebra he
developed.
Euclid’s theory of ratios referred to by Cayley appears in Book V and Book VII of Euclid’s Elements. Searching for a clue as to what Cayley saw in these Books is not easy. But note the fact that Cayley was the inventor of matrix algebra. What did he see in Euclid’s theory of ratios that may have something to do with matrix algebra? We think it must have been what may be called the “ratio matrix.”
The ratio matrix is a square matrix whose rows and columns are identified by the series of natural numbers, 1, 2, 3, ... The value of a cell in the i-th row and j-th column are expressed as
11 We are all indebted directly or indirectly to Professor William W. Cooper for his alerting one of us as early as
1961 about the existence and importance of Cayley’s publication. As an authority on linear programming, Cooper has contributed many models of matrix accounting and their applications as shown in Charnes and Cooper (1961), Charnes, Cooper, and Ijiri (1963), etc.
12 Here Cayley refers to the theory of proportions discussed in Books V and VII of Euclid’s
Elements. See Heath
(1956).
13 Another version of praise appears in the biographical notice of Arthur Cayley in Forsyth (1895). “Financial
matters and accounts also interested him; and only a few months before his death he published a brief pamphlet on book-keeping by double entry, which he has been known to declare one of the two perfect sciences. He could not resist some reference to the subject in his Presidential Address, making the remark that the notion of a negative magnitude ‘is used in a very refined manner in book-keeping by double-entry.”
the ratio, “i/j,” where both i and j are natural numbers. Thus the first row is 1/1, 1/2, 1/3..., the second row is 2/1, 2/2, 2/3... Reading vertically, the first column is 1/1, 2/1, 3/1..., (which is in fact a series of natural numbers), the second column is 1/2, 2/2, 3/2... and so on. Number theory indicates that any positive rational number can be expressed by a ratio of two natural numbers. Hence it can easily be seen that the matrix covers all positive rational numbers as shown in Table 1 below. Evidently then, since any rational number k can be expressed as a ratio of two natural numbers k = i/j, it will appear in the cell at the intersection of row number i and column number j. Since k can also be expressed k = (ni)/(nj) for any natural number n, k will also appear in every cell where the numerator and the denominator are commonly divisible by n, but this redundancy does not affect the fact all rational numbers appear in the ratio matrix.
Table 1: Ratio matrix containing all positive rational numbers ith\ jth 1 2 3 .. row \col. _____________________ 1 1/1 1/2 1/3 .. 2 2/1 2/2 2/3 .. 3 3/1 3/2 3/3 .. .. .. .. .. ..
Cayley’s statement (1894) now can be reiterated as 1) Cayley regards Euclid’s theory of ratios as an absolutely perfect one; and 2) Cayley also regards the theory constituted by the principles of Double-Entry Bookkeeping (Double-Entry theory) as an absolutely perfect one. Clearly, Cayley is finding commonality between the theory of ratios and the theory of double entry in some abstract form. We think the ratio matrix must be the one to represent the theory of
ratios since all rational numbers are neatly organized in rows and columns. If so, is there a comparably neat expression to represent the theory of double-entry?
We think there is and it must be the matrix in Table 2, but with different interpretations.
Table 2: Double-entry matrix with all transaction amounts between two accounts ith\ jth 1 2 3 .. credits row \col. _____________________ 1 1|1 1|2 1|3 .. debits 2 2|1 2|2 2|3 .. 3 3|1 3|2 3|3 .. .. .. .. .. ..
Notice that the two matrices are structurally isomorphic, only the interpretations are different. In a ratio matrix, we interpret the row and the column numbers as natural numbers. In a double-entry matrix, we interpret them as account numbers. The value of the cell is the rational number in the ratio matrix. In the double-entry matrix, the transaction amount is entered in the cell at the intersection of account i on the debit and account j on the credit. The “law of double entry accounting” requires that the amount debited and the amount credited must always be equal in every transaction. Hence, instead of the value of the rational number in the ratio matrix, the value of the cell in the double-entry matrix indicates the same amount of transaction entered in account i and account j simultaneously. This meaning is indicated by the vertical line rather than the division sign in this table.
While our use of a matrix specification of the double entry system follows a long list of authors, as discussed later, we also note that the claimed isomorphism to Cayley’s ratio matrix is
somewhat deceiving. The ratio matrix is countably infinite, while the double entry matrix is finite. Moreover, the accounts themselves must be defined, and this leads to the fact the account numbers in double-entry matrix are “nominal” numbers. It serves only for the identification
purpose. However, by means of double-entry matrix, a “network of transactions” is created in the form shown in Table 2 above.14
We now can see why Cayley was able to be so confident in praising double-entry bookkeeping. It is his discovery of beauty in Euclid’s theory of ratios coupled with his
translation and attribution based on the isomorphism of the beauty to double-entry bookkeeping. It is true that the above interpretation by means of the isomorphism between the ratio matrix and the double-entry matrix may not be totally satisfactory as to whether double-entry bookkeeping deserves such highest praise as “absolutely perfect.15” We should keep in mind, however, that we need to give Cayley credit for what he said until we can come up with a definite answer that he was wrong, as many physics principles have been given such benefit of doubt. Hence anyone who finds the above interpretation not satisfactory should try to see whether there exists a “better reason” for Cayley’s statement.16
14 Here, we are simplifying journal entries by assuming all journal entries are “simple,” consisting of exactly one
debit entry and one credit entry. All “compound” transactions are broken into simple ones by applying a
“proportionality” assumption. For example, [Land $30 and Building $20] purchased in exchange for [Cash $25 and Loan $25] will be recorded by means of four simple entries, i) (debit) Land $15| (credit) Cash $15, ii) (debit) Land $15| (credit) Loan $15, iii) (debit) Building $10| (credit) Cash $10, and iv) Building $ 10| (credit) Loan $10.
15 At the beginning of Book V., Heath makes a comment that might somewhat mitigate the meaning of
“perfectness.” He states “Thus we are told that the Pythagoreans distinguished three sorts of means, the arithmetic,
the geometric and the harmonic mean, the geometric mean being called proportion par excellence; and further
Iamblichus speaks of the ‘most perfect proportion consisting of four terms and specially called harmonic,’ in other
words, the proportion a : (a+b)/2 = 2ab/(a+b) : b, which was said to be a discovery of the Babylonians and to have been first introduced into Greece by Pythagoras (Heath 1956, p. 112).” Such possibility of many degrees of “perfectness” makes us think that “well balanced” or “harmonious” might be more appropriate. Keep in mind, however, the word Cayley actually used in the booklet is “absolutely perfect,” leaving no room for ambiguity. See also Ellerman (1995) for the analyses of “the Pacioli Group,” one of a rare and notable attempt to establish double-entry bookkeeping on a firm mathematical ground (Ellerman has a few series of work on this subject, alerted to us by Yoshitaka Fukui).
3C) Equal Exchange--Additive and multiplicative inferences: One of the important principles impounded in double-entry bookkeeping is the historical cost principle—a rule that sets the initial book value of an item obtained equal to the book value of the item foregone in exchange. (Thus, the rule may also be stated as a rule of “equal exchange.”) There are growing exceptions to this rule but a large majority of transactions have followed the rule since the double-entry bookkeeping system was introduced in the fifteenth century. We think Cayley’s praise for double-entry bookkeeping largely stems from the simplification and elegance that the principle provides and would like to attribute the principle to him because of his great
enthusiasm for the system that is built on the notion of equal exchange.
Natural science has comparable approaches. Consider Laplace’s rule of insufficient reason: “If we know no better, the probability for obtaining a particular outcome in a game of chance or other random process is the same for every outcome. All results are equally likely (Milburn 1998, p. 199).” Similarly, we may consider “Cayley’s rule of insufficient reason: If we know no better, a journal entry is an exchange of two items with equal amounts.” In Table 2, every cell of the double-entry matrix consists of a transaction amount that applies commonly to the corresponding row and column accounts. As long as the majority of transactions are equal exchanges, the double-entry matrix such as the one in Table 2 works well since only one amount is needed for each transaction. When two unequal amounts are needed in one transaction, such as a $10 sale whose cost of sale is only $6, this compound transaction has to be split between the cost recovery of $6 as a transaction and the profit recognition of $4. In any event, the fact that the double-entry matrix consists of one transaction amount linking two accounts, a debit and a credit account, is fundamental to the double-entry matrix structure such as the one in Table 2.
Any sound mathematical structure, such as the ratio matrix or the double-entry matrix, should be capable of generating, by means of inferences, new information heretofore unavailable. We wish to explore the two “inference rules” under double-entry bookkeeping which take the form of the following equations:
“Rule 1: The Intra-Account Inference Rule, which states, for each account, that: (1) Beginning Balance + Increases = Decreases + Ending Balance
by which a missing number can be found if three of the four numbers are known.”
“Rule 2: The Inter-Account Inference Rule, which states that a change in an account cannot occur without causing a corresponding change in another account, thereby allowing the inference to jump from one account to another in the network of accounts (Ijiri 1993).”
Note that the inference need not stop at the first account. Having found the 4th number may make it possible for the inference to continue in a new account, and a domino like effect might happen as a result of combined intra- and inter-account inference. For example, an
inventory account has only two known numbers, a beginning balance and an ending balance. As soon as an entry for cost of goods (decrease) is recorded, the fourth number, purchases (increase) can be deduced using Equation (1). When the purchase amount is entered in the debit side of the inventory account, the same amount must be entered in the credit side of accounts payable, thereby creating a chain reaction. These inference rules are just as useful as any other rules or laws in narrowing or eliminating options, thereby tightening the slack that might otherwise remain in the system.
Equation (1) allows an additive inference; let us consider a multiplicative inference rule in order to bring the double-entry matrix closer to the ratio matrix. We first consider splitting the transaction amount into two components, most simply expressed as the product of a price P and
quantity Q. Hence, under the rule of “equal exchange” between an item foregone (P’Q’) and an item obtained (PQ) after an equal exchange, we obtain:
(2) P’Q’ = PQ.
A multiplicative inference rule is created from (2) as: (3) P’ = (PQ)/Q’
which means that P’ can be deduced when P, Q, and Q’ are known.
Similarly, for the exchange just prior to the exchange in (2), we have; (4) P”Q” = P’Q’
A multiplicative inference rule is created from (4) as: (5) P” = (P’Q’)/Q”
If we know the values of only two variables in (2) or in (4), we cannot make any compact inference. But once we know the values of three variables out of four in the equation, the
inference engine becomes highly explicit, and a chain reaction can be created. Namely, we convert (2) into (3), solving for P’. Then we can put this newly found value of P’ into (5) to solve for P”. We thus obtain:
(6) P” = (P’Q’)/Q” = ([(PQ)/Q’]Q’)/Q” = (PQ)/Q”
If we want to measure the price of, say, the wheat we obtained, we can do so by measuring the price of the rice foregone in exchange, or the price of anything foregone in the chain of exchanges.17 Every journal entry is potentially a source for finding a price or quantity we wish to know. We can replace every cell entry in the double-entry matrix with PQ with proper subscripts and start taking all kinds of ratios, discovering many pieces of economic and market data. This cannot be said if bookkeeping were single-entry, without the redundancy
17 This may be related to entanglements mentioned in Section 1C in which a property of a particle has to be imputed
which offers hidden data. Indeed, it is the use of these inferential possibilities that forms the basis for Arya, Fellingham, Glover, Schroeder, and Strang (2000)’s inversion process.18
3D) The power of the equal-exchange rule: If no information is available (we shall call this situation “Level 0 knowledge”), the P and Q in any given transaction can be at any place on the positive quadrant of a plane. If it is known that the transaction satisfies the equal-exchange rule (“Level 1 knowledge”), again any combination of P and Q can occur and the information seems to be equally useless.
However, contingently, Level 1 knowledge can be very useful since if and when the amount A=PQ (for example A = 180, 120, or 60) is given, we can determine the PQ curve for the given amount (which we shall call “Level 2 knowledge”). Then eventually when relevant
information becomes available, we can get to the final stage of determining the value of both P and Q (“Level 3 knowledge”). Here, as mentioned earlier, the relevant information may come from a seemingly irrelevant source such as the measurement of P” or Q” in a transaction that occurred a few steps earlier in a chain of transactions.
The equal-exchange rule may not be useful currently if we are at Level 1 knowledge, its existence changes the usefulness of the “future” information since we can move to Level 2 knowledge with readiness to deal with it before the information arrives. It can also alter the direction of search in the future.
Indeed, the equal-exchange rule may be compared with Euclid’s fifth “axiom of
parallels” and none of the hyperbolas can cross each other under the equal-exchange rule. (In fact it appears in hyperbolic geometry, also called “Lobačevskii’s non-Euclidean geometry,”) Just as
18 Note that the equal-exchange rule, the historical cost principle, and accounting conservatism are all related in that
in a normal economy, costs are incurred first in anticipation for benefits that are expected to be higher than costs. The historical cost principle is thus more conservative than the market value accounting. But the equal-exchange
geometry should be built on a set of simple axioms, double-entry accounting should be built on a set of simple axioms so that numerous complex objects or phenomena can be sorted out
uniformly and related to each other systematically.19
Finally, relating back to Cayley’s statement, we believe the praise must have come from the observations that both matrices are comprehensive, symmetric, organized, and tightened to allow inferences to propagate.2021 While past mathematicians and economists seem to have great respect for entry bookkeeping, most contemporary accountants seem to regard double-entry bookkeeping as only a practice of writing the same number twice, once in each of the two columns set aside for a binary classification. It reminds us of a person patiently waiting for the arrival of the 6 o’clock train but in desperation turns the hands of the station clock to the vertical position. Recording of the same two numbers is the culmination of a long endeavor in searching for causality of events just like scientists do. That is at the heart of double-entry information.
4. Symbiotic Relationship between Quantum Information and Double-Entry Information 4A) Information “threads”: To continue, we would like to explore a more symbiotic relationship between quantum and double entry information. This might be thought of in terms
19 See Ijiri (1967) for an axiomatic structure of double-entry accounting based on the historical cost principle. The
structure consists of three axioms, axiom of control, axiom of quantities, and axiom of exchanges. See also, an axiomatic system of quantum operations in Nielsen and Chuang (2000, Ch. 8))
20 No wonder Cayley had to use the highest praise possible, “absolutely perfect,” as “the most perfect” was already
assigned to harmonic mean. See footnote 14.
21 By the way, use of matrix algebra in accounting has a long history. Even before Cayley invented matrix algebra,
De Morgan (1846) first introduced the notion of a matrix to accounting. Rossi (1889) offered examples in which accounting matrices played a decisive role. Matrix algebra was first used in general accounting theory by Mattessich (1957). Ijiri (1966) introduced a rectangular matrix in an accounting context. Butterworth (1972) used a matrix representation of double-entry accounting and extended this to value of information assessment using the economics of information. Shank (1972), in turn, stressed a more computational use of matrix algebra in accounting. More recently, Ayra, Fellingham and Schroeder (2000) examined the issue of aggregation (from transactions to accounts) and determined the cost of aggregation—information loss in a decision-making setting. Arya, Fellingham, Glover, Schroeder and Strang (2000) employed the tool of an incidence matrix and the inferential apparatus to represent accounting systems and examined how financial statement readers make their “best guess” of the underlying transactions.
of 1) take quantum information and “make it” double-entry information, or 2) take double-entry information and “make it” quantum information. Obviously, the latter is far more difficult than the former since the latter calls for “quantizing” existing information, which we do not know how to do. But the former is a possibility because we know how accounting evolved from single-entry information to double-single-entry information.
Italian merchants in the medieval centuries seem to have kept diaries in a narrative form to track debtors and creditors as well as ins and outs of their merchandise.22 They were written in a narrative form. Gradually, certain words such as the names of debtors and creditors started appearing repeatedly and special columns or ledgers for them began to emerge. They eventually formed “accounts” whose names were all listed in a column next to “amounts.” They were still single-entry, yet they needed connectivity among the sizeable number of data daily in order to minimize search time. Then they discovered adding minute notes or our computer term, “threads,” improved the speed of search. For sometime, transactions were recorded with “accounts”, “amounts”, and “notes or threads.” Then threads began to grow in frequency and some of them were turned into “accounts.” When more than one account started appearing in a transaction, some order among accounts in a transaction became necessary. That is when a mathematician Luca Pacioli, mentioned earlier, wrote a volume on double-entry bookkeeping as one of the series of mathematical volumes in 1494.
4B) Thick parallel threads in double-entry information for better connectivity: Now the same evolutionary process might be observed in quantum information. In some areas, partial double-entry information may be practiced; in some other areas a 100% practice may be
22 See, for example, Littleton and Yamey (1956) on the historical events that led to the emergence of double-entry
observed as in the current accounting practice. The thick parallel threads observed in double-entry information, which are the strong building block of the accounting’s connectivity and network, will be very attractive for any networks where connectivity is a crucial requirement. In any case, there seems to be no question that the potential new areas opened by quantum
information will bring about new challenges to accounting as well.
Part III: Quantum Information and Accounting Information 5. Endogeneity Issues in Accounting Information
We now turn to the design side of double entry information. By analogy, the quantum experiment rests on careful design of the apparatus and measurements that are to be recorded; and double entry information rests on careful design of what measurements are to be recorded, and when they are to be recorded. Likewise, the historian is selective about his observations, and seeks to paint broad themes in man’s various activities. Too little detail is uninformative, and too much detail is, well, equally uninformative.
5A) Recognition and aggregation: This important feature of double entry information comes to the fore when we examine the actual implementation of such a system. General Motors, for example, sells automobiles in all 50 states in the United States, let alone in countries around the globe. Though individual records are essential for transaction controls, it does not publish its sales of each vehicle type on a state by state basis. It also does not report its sales on a minute by minute basis. Rather, sales data are aggregated for reporting purposes, and how this aggregation is effected is an important design consideration. Parallel comments apply to virtually any release of financial data. And to move toward the absurd, oil traders do not keep records on a barrel by
barrel basis. Rather broad category groups are used, as is evident from the structure of organized commodity markets.
Inter-temporal aggregation is also at work. For example, the time at which a sale is
recorded is also an important design consideration. General Motors, for example, considers a sale finalized and recorded well before any warranty obligations for which it is responsible have been determined, let alone satisfied. Similarly, a new product innovation, regardless of estimated profitability, is not recorded before sales are consummated. Indeed, special techniques, such as qualifying special purpose entities, have been developed to accelerate recognition of gains in the double-entry framework.
Importantly, then, even when we are committed to double entry recording, we face important design questions of the level of detail to preserve and the very time at which
measurements are to be taken. Moreover, extant regulations call for specific disclosures as well, in fact mandating production of information that is outside the formal double entry system. Operating lease obligations and the LIFO reserve are cases in point.23 For this reason we refer, from this point on, to the more inclusive term “accounting information” as opposed to simply double-entry information.
5B) Measurement school vs. information content school: This leads to the question of how to deal with, to conceptualize, these important design issues, including the distinction between formal double entry inclusion and mere disclosure. The “measurement school” in accounting has historically focused on the importance of well reasoned economic measurement, but has kept the design issues somewhat removed from the fundamentals. Here we find the broad notions that the measures produced by the double entry system should be “relevant” and
“reliable” but the precise meaning of these terms, or social consequences of one vs. another answer to the aggregation and recognition questions, are kept at bay. Rather, the functioning of classical markets, in an economic sense, provides the conceptual underpinnings (Ijiri and Jaedicke (1966), Christensen and Demski (2002), and Demski, Fellingham, Ijiri and Sunder (2002)).
The imprecise nature of the central terms of relevance and reliability leaves accounting researchers and practitioners with little guidance on how to measure or understand accounting information, let alone connect to the underlying resource allocation issues as in, say, a value of information analysis. Moreover, when the notion of perfect and complete market collapses, not only “perfect measurement” is impossible, but what we try to measure (i.e., the underlying economic values) becomes ill-defined.
More recently, exploiting developments in the economics of uncertainty, the “information content” approach has emerged, e.g., Feltham (1972), Demski and Feltham (1976), and
Christensen and Demski (2002), where the users of the double entry product are explicitly identified, such as equity valuation markets or labor contracts, and the consequences of one versus another answer to the aggregation and recognition questions are sorted out. In this way, economic forces are central to the analysis, and we avoid the imprecise guidance of relevance, reliability and even perfect markets. Concepts such as recognition, aggregation and conservatism are explored in terms of carefully identified economic trade-offs. The costs of emphasizing these forces are complexity and being forced to abstract away from the very structure of double-entry.
To be sure, the measurement school identifies these forces, but keeps them at somewhat of a distance by replacing them with the characteristics of relevance and reliability. This allows the measurement school to remain tightly connected to the double entry framework, just as
dealing explicitly with these forces forces us to abandon the tight connection to the double entry framework. Arguably, accounting information cannot be well understood if either the double-entry structure or the economic forces are missing from the analysis. Unfortunately, the classical debate between the measurement school and the information content school leads us to
emphasize only one, but not the other.
5C) Physical vs. social sciences perspective: Broadly, the measurement school takes more of a physical than a social science perspective in its approach, while the information content school takes more of a social science perspective. The difference is the role played by endogeneity, in particular whether the design choices and attendant behaviors are open to self directed choice themselves. In the physical sciences the focus is on scientists or measurers and objects, while in the social sciences the focus is on scientists or measurers, objects, and
“humans.” This does not mean the physical sciences are free of human dimensions, as the experiments must be designed and executed. See, for example, Blackwell’s (1951) famous comparison of experiments paper and its linkage to processing what is learned from the
experiments. (We should also interpret the phrase “humans” in the broad sense of embracing all types of endogeneity. For example, shrimp boats in Florida are under regulatory mandate to tie their nets in such fashion that they protect (endangered) manatees; but dolphins have learned to exploit the ties to their own culinary advantage.)
Another layer of social dimension in accounting information is regulations and standards. Producing accounting information is a highly regulated, political process. Regulations and
standards not only guide how accounting information is measured and disclosed (and audited), as noted, but also help information sharing and information extraction. Accounting rules and
and the political class. This renders the rules and standards highly dependent on the manner in which conflicts of interests are resolved. (Parenthetically, we witness a vastly different approach to the regulation and production of government statistics.) In sharp contrast, physicists appear to struggle with information extraction issues, the very essence of quantum information, but are relatively free of regulatory constraint.
The two approaches are not, in the end reconcilable. Auditors, for example, are first order, endogenous players in the reporting game, and the measurement school treats them as essentially second order importance. Value of information is another stumbling point. The qualitative characteristics of relevance and reliability do not, in general, well substitute for the underlying economic forces, just as entropy in classical information science does not well substitute for the underlying economic forces. This follows from Blackwell (1951), the fact one information source or one experiment is better than a second, absent detailed specification of the setting, if and only if the possible outcomes of the second can be modeled as if they are statistically equal to the possible outcomes of the first plus noise. Otherwise, the two sources are non-comparable, and a measure such as value or entropy simply does not exist.
6. Conceptual Applications of Quantum Information to Accounting Information Now we are ready to speculate about a few applications of quantum information to accounting, from an information content school perspective where we combine the information science and economics perspective by viewing accounting as a developed, disciplined and managed information channel.
6A) Uncertainty and probability assessment: The measurement school implicitly and the information content school explicitly focuses on what can be learned about an entity’s
prospects or a manager’s behavior by observing the output of the accounting system, in conjunction with that of other information sources. The central feature is partial resolution of uncertainty. By analogy with quantum information, it is surely true that all uncertainty cannot be resolved, and that the mere presence of accounting measurements affects the entity and its manager. Moreover, the modeling of the accounting system, with rare exception, is abstracted away from the double-entry structure and stresses, in its place, a likelihood structure. (The channel or structural side of the coin is thus relatively unexploited.) In this sense, the struggles with measurement that are being experienced in quantum information theory have a parallel flavor in accounting. As such, any solution in one arena may shed light in the other. For example, what can we learn from entanglement as a way to cope with irresolvable uncertainty?
A closely related concept is probability assessment that is central to the behavior modeling in the information content approach, but is also important structurally in such accounting areas as depreciation, R&D, liability recognition, pension accounting, and
uncollectibility of accounts receivables. As Ijiri (1975, 175) states, probability measurement has limited implications “partly because there is no proper measurement system to generate a reasonably hard probability measure.” Savage (1954), recall, treats probability assessment as inherently personalistic. Probability assessment in accounting is subjective, affected by transaction designs, and limited by human cognition (e.g., Libby, 1981).
As mentioned earlier, in the world of quantum mechanics and information, there is a much richer definition of probability that includes the possibility of mixing between different conditions. As noted earlier, when several different changes in states can occur, the possibility for interference between different states is included through the summing of probability amplitudes, as opposed to probabilities. But if the uncertainty as to which transition may have
occurred disappears, interference also disappears, and the summing of probability amplitudes leads to the same result as summing up probabilities as in a classical sense. The very nature of probability in quantum information theory and its use of “probability amplitude” provide an avenue for rethinking our own use of this concept, in a setting where the classical notion of probability is fundamentally inoperative, albeit for human as opposed to physical reasons.
6B) Endogenous expectation: When combining probability amplitudes of different alternatives in accordance with Feynman’s rule, the design of the apparatus may affect the interference between the probability amplitudes, which further affect the probability of an event. This widely acceptable concept in quantum mechanics has a parallel in accounting in the form of endogenous expectations. Accounting estimates reflect the managers’ expectations, which presumably stem from managers’ actual and anticipated transactions, transactions that may well be affected by the accounting itself.
The current state of accounting research can be described as excessively taking an exogenous view of how raw data are produced. For example, the managers’ opportunistic behavior in the accrual process should be considered when examining the impact of accounting information on the market, and digging deeper we should recognize that opportunistic behavior is itself endogenous. The standard setters should, presumably, consider how the reporting units react to the actual and anticipated behavior of the standard setters. Demski (2004) surveys the use of expectations in current accounting research and emphasizes how endogenizing
expectations may help to create a broader and more integrated view of accounting. It is hopeful that bringing the accounting process, or standard, into the “equation” will shed new light on behavior and the associated regulations, just as bringing the experimental apparatus into the “equation” has shed new light on behavior in a quantum mechanical setting.
Closely related is the distinction between valuation and evaluation, with the former emphasizing the entity’s conjectured equilibrium path and the latter the distinction between behaviors on vs. off that path. Here, we know usefulness of any particular accounting measure is not coextensive across these two venues. We also know, from Fellingham and Schroeder (2005) that if synergy implies quantum-like interference, that the entity’s control problem is best dealt with in terms of aggregated performance measures. Aggregation, that is, follows naturally. 6C) Error-correction mechanism: After a particle or entity is prepared in a particular quantum state, it may lose this state as a result of interactions with the environment. All quantum systems can suffer from this problem known as decoherence, which leads to information loss and errors. The remedy is error-correcting codes that reverse and undo the errors (DiVincenzo and Terhal 1998).
Accounting systems also have unavoidable measurement errors, but some are actually intentional. The double entry model itself is at work here; auditors play a role to minimize the errors in the process of producing accounting information (including application of the required measurement methods). The current research on auditors focuses on how to improve audit judgment and decision and how auditors interact with clients (Antle and Nalebuff 1991 and Kinney, 2000). A noticeable difference between auditors and error-correcting schemes in quantum information lies in the response of managers to the auditors. This interaction is presumably less neutral than a correction code that restores the quantum mechanical state of a particle. Thus, the nature of that human interaction adds another level of uncertainty to the accounting system.
Here we encounter an important distinction between the measurement or principles school in accounting and the information content school. The former treats the effects of the
measurement and the roles played by the affected actors as second order in importance, while the latter treats them as central to the fundamentals.
6D) Environment: An important lesson we learn from quantum information is the impact of the environment and its interaction with the system are central to characterizing the data. In particular, quantum mechanical states can be viewed as extremely delicate. Any
interaction of the system with the environment may destroy the information or change the system completely. An accounting system has its own environment, which consists of reporting units and their governance mechanisms, regulators, and standard setters. It may be possible to design an experiment so that the interaction between the quantum system and its environment is controlled. But it is never possible to control or isolate an accounting system from its
institutional settings. For that matter, fair value measures, in contrast with their historical cost predecessors, open the door to vastly more extensive interaction between the accounting measure and the environment in which it takes place.
7. Closing Remarks: In Search for More Concrete Fundamentals in Accounting 7A) Necessity of fundamentals: As physicists are excited about quantum mechanics and quantum computing (Steane 1998), we believe exploring the theory of quantum information indeed provides us “a new and exciting way to think about the fundamental laws of accounting.” In general, every discipline must have fundamentals that people agree to adopt in order to avoid a chaotic collection of knowledge. A discipline cannot survive without its fundamentals.
Such thinking will undoubtedly enrich our approach to accounting in dealing with accounting information. It will offer a bird’s-eye-view of the science and economics of
enormously rich disciplines of economics and “information science.” Furthermore, we may now consider a variety of perspectives in dealing with information, namely the quantum perspective, the classical (non-quantum) perspective, the economic, and the structural accounting (the double-entry, in particular) perspective.
7B) Integration with the past: The emphasis on interpreting new phenomena in a
manner that is consistent with past principles as much as possible seems to be widely observed in physics. As Steane (1998) notes, doing so in the case of quantum information leads to a deeper understanding of fundamentals.
The preservation of old principles is not just respect for those principles that worked well in the past, but may come from economic reasons--heuristics that work are not abundantly available, so it makes sense to preserve them as much as possible. With this interpretation, we can now understand the tremendous joy Steane expressed when “the ideas of the classical information theory seem to fit into quantum mechanics like a hand into a glove, giving us the feeling that we are uncovering something profound about nature.”
We see the same kind of emphasis on consistency with past principles in accounting. Sanders, Hatfield, and Moore (1938), Paton and Littleton (1940), Moonitz (1961), Sprouse and Moonitz (1962) are all examples of attempts to establish principles in accounting. The
Conceptual Framework project by the Financial Accounting Standards Board in the 1970s is an example of principles established under a comprehensive framework. Now all accounting
standards established by the FASB must contain statements of how the new standard might relate to the Conceptual Framework, despite the Framework’s inherent flaws (e.g., Christensen and Demski 2005).
Note that it is not just a question of how many principles have been developed in a given discipline but how many people agree to be bound by them. This in turn depends upon how objective the principles are. In this sense, principles in physics are much more objective than those in a
