Quantum Production and Cost Functions
Introduction
This paper explores quantum information processing of an organization’s
production. Quantum information processing is inherently uncertain and enriches the
probability structure relative to classical information processing via the possibility of
interference.1 Quantum production is represented as a unitary transformation applied to
an economic agent’s initial perception of the production possibilities.2 Production
success is uncertain but becomes almost sure with sufficient inputs. Efficient production
attempts balances likelihood of success with cost of the inputs. The cost function is the
expected cost of efficiently combining inputs and measurement timing of production
success.
The cost function has two prime attributes: constant returns to inputs, for
simplicity, and uncertainty in the mapping of inputs into outputs. That is, the quantity of
inputs required to successfully produce outputs is stochastically determined by the
quantum production function.
The ideas are presented in sequential fashion from simplest to richest setting.
Initially, production employs homogeneous factor inputs in an input-observable setting.
The first result compares fully aggregate with individual measures of production. Fully
aggregate measures are taken after sufficient inputs are applied to ensure production
success. On the other hand, individual measures occur after each input of fixed amount.
We demonstrate that there exists a range of inputs in which fully aggregate measures are
preferred to individual measures. The second result identifies an optimal level of
aggregation in the same homogeneous input-observable setting. This involves trading-off
1
Quantum interference gives rise to a variety of information riches including quantum
teleportation, super-dense coding, and substantial enhancements in computationally
challenging tasks such as factoring. For example, Nielsen, 2003, reports that to factor a
300-digit number (the key to secret, RSA-type codes employed in electronic commerce)
the best classical factoring algorithm employs 5 x 1024 steps while the best quantumfactoring algorithm employs 5 x 1010 steps. This translates into approximately 150,000
years versus a second in real time.
2
See Fuchs, 2002, for a discussion of the nature of quantum states. Similar to Savage’s,
1954, classical statistical arguments, Fuchs argues that quantum states are perceived
possibilities.
1
benefits of aggregation (increase the likelihood of successfully producing) against the
chance that success occurs early (no additional costly inputs are needed).
The third result identifies an optimal level of aggregation for heterogeneous factor
inputs. While the analysis confronts inherent uncertainty, the results are otherwise
analogous to classical Cobb-Douglas or CES production and cost functions (Shephard,
1970). The penultimate result identifies a quantum cost function and economies of scope
associated with multiple products and/or tasks some of which are complementary. When
complementarity of products/tasks exists, entangled quantum information processing
yields substantial scope economies.
Finally, given the optimal level of aggregation for heterogeneous factor inputs,
the efficiency loss associated with unobservable agent inputs is analyzed in a moral
hazard setting. The optimal contract is a “piece-rate” payment scheme that pays a
constant for success regardless of the number of measurements taken before production is
successfully completed. Numerical examples illustrate the results throughout. The final
set of examples compares the cost of supplying incentives in a setting characterized by
complementary products/tasks with the cost in an independent product/task setting. As in
the input observable setting, complementarity yields substantial economies.
To summarize, interfering with production too frequently is harmful. In an input
observable setting, interfering to measure is harmful as it destroys synergy among the
inputs. In an asymmetric information setting, interfering too frequently may mute
incentives (see also Arya, 2002, for a classical information processing setting, and
Fellingham and Schroeder, 2006, for a quantum information processing setting). It is
also inefficient to measure too late as costly inputs are over-utilized.
Homogeneous factor input
Suppose a firm is committed to producing q units in an inherently uncertain
(quantum) production environment from a factor input with uniform factor price p per θ0.
Quantum information processing of production identifies whether or not it is successfully
completed with each measurement. As the firm is committed to producing q units, the
firm produces until q successes are achieved. The question is how frequently should
production be measured. In other words, a standard accounting choice arises – what level
2
of aggregation is desirable? The trade-offs are measurement interferes with production
but if measurement indicates success then no additional (costly) inputs are applied.
Initially two production regimes involving a single product are evaluated. One
involves little interference as production is measured after sufficient inputs have been
supplied to assure successful production; this is dubbed fully aggregate measurement.
The other involves frequent interference as production is measured after each input. If
success is achieved, the current unit of output is complete; otherwise another input is
supplied and measurement is repeated. This process repeats until success is achieved.
This alternative is dubbed individual measurement.
Proposition 1:
For a given input level 0 < θ0 < π/2, fully aggregate measurement yields lower
expected production cost than does individual measurement.
E[costfully aggregate] = π/θ0 p q since the probability of success given fully
n
"0
= 1 if n = π/θ0 inputs (fractional inputs can be
j=1 2
aggregate measurement is sin2 #
employed).3 E[costindividual] =
!
1
p q since the probability of success given a
sin 2 "20
"
single input is sin 0 and the expected number of trials (inputs and measurements) for a
2
!
1
success follows a geometric probability distribution with expectation
. Hence,
sin 2 "20
!
E[costfully aggregate] < E[costindividual] if 0 < θ0 < π/2.
2
!
3
Derivation of quantum probabilities from the quantum production function and initial
perceptions is discussed in the appendix. By the axioms of quantum information
processing, it is equivalent to think of this as one production cycle with nθ0 inputs
followed by measurement, or n successive cycles each employing θ0 inputs followed by a
single measurement at the end of the nth cycle.
3
Examples
E[costfully aggregate]
E[costindividual]
θ0 = π/4
40
68.28
θ0 = π/2
20
20
θ0 = 3/5π
16.67
15.28
Table 1. Homogeneous, observable inputs comparing fully aggregate with individual
measurement of production. p = 1 and q = 10 units of output.
The above suggests that there is an intermediate level of aggregation where θ
inputs with unit factor price p are supplied before measurement. This is stated as
proposition 2.
Proposition 2:
The optimal level of aggregation is min
" >0
The solution to min
" >0
condition: 0 < θ = tan
"
.
sin 2 "2
"
!, involves iterative evaluation of the first order
sin 2 "2
"
or θ ≈ 2.33112. E[cost] = 27.601 for p = 1 and q = 10 units of
2
output. Next,!we extend the analysis to accommodate heterogeneous inputs in an input
observable setting.
!
Heterogeneous factor inputs
Consider employing θ1 through θn units, respectively, of n heterogeneous factor
inputs (analogous to K, L, etc. in a classical production setting4), with unit factor prices p1
4
In a classical production setting, the CES (constant elasticity of substitution) production
function is q = A[α K-β + (1-α) L-β]-1/β. If the elasticity of factor substitution
fK fL
=
f KL q
1
is one, CES reduces to a Cobb-Douglas production function q = AKαL(1-α) where fj
1+ "
!
is the partial derivative with respect to factor j.
!
4
through pn to produce q units (successes) of a product. What is the expected cost
minimizing aggregation level?
Quantum production involving n heterogeneous factor inputs are combined
n
#j
according to $ " j
j=1
n
where θj, αj > 0, # " j = 1 rather than θ (the homogeneous factor
j=1
input).5 As in classical information processing of production, heterogeneity arises
through factor prices pj and !
factor productivity αj. The cost function associated with
!
heterogeneous factor inputs is presented in proposition 3.
Proposition 3:
The expected cost minimizing aggregation level for a product employing n
n
#" p
j
j
j=1
heterogeneous factor inputs is min
n
" 1 ,L," n >0
$j
q where θj, αj > 0 " j and
%" j
sin 2
j=1
2
!
n
# " j = 1.
j=1
!
Examples illustrating heterogeneous factor input cost functions are presented
!
below.
5
Notice if the original quantum production function is employed with additive,
heterogeneous inputs, the program is
"1 p1 + " 2 p2
q, where θ1 and θ2 ≥ 0.
" 1 ," 2 >0 sin 2 " 1 +" 2
2
min
Since optimal aggregation produces expected costs that are linear in inputs, if p1 < p2 then
efficient production utilizes only θ1 (θ2 = 0). Therefore, similar to classical production
! functions richer interactions amongst inputs are needed to produce a heterogeneous factor
input-based cost function.
5
Examples
pθ(1)
pθ(2)
θ1
θ2
E[cost]
1
1
2.33112
2.33112
55.202
1
2
3.2967
1.64835
78.067
1
5
5.21255
1.04251
123.435
Panel A: α1 = 1/2
pθ(1)
pθ(2)
θ1
θ2
E[cost]
1
1
2.9370
1.4685
52.163
1
2
3.7004
0.925
65.721
1
5
5.0222
0.5022
89.197
Panel B: α1 = 2/3
pθ(1)
pθ(2)
θ1
θ2
E[cost]
1
1
1.4685
2.9370
52.163
1
2
2.33112
2.33112
82.803
1
5
4.29397
1.71759
152.524
Panel C: α1 = 1/3
Table 3. Two heterogeneous, observable inputs with optimal aggregation. q = 10 units
of output.
As in the classical case, the expected cost minimizing factor mix yields expected factor
costs proportional to α:
" j #j pj
=
for all j, k. Next, we analyze economies of scope
" k #k pk
arising from quantum information processing in a multi-product setting.
!
Economies of scope
Consider a setting with heterogeneous factor inputs applied to multiple products.
Economies of scope arise from complementarity amongst products and/or tasks. Suppose
there are k products/tasks each with ni heterogeneous factor inputs (denoted by the set
{hi}) but a subset of w products/tasks are complementary. Complementary
products/tasks are intertwined to accommodate entangled information processing of their
joint production and measurement. On the other hand, non-complementary
6
products/tasks involve independent processing and can be measured independently. The
probability of success associated with independent processing of each non# ij
% "ij
complementary product/task i is sin 2 j${h i }
, and the probability of success associated
2
with entangled information processing of complementary products/tasks is
$"
sin 2
% xj
xj
+L+
j#{ h x }
$!"
j#{h y }
2
% yj
yj
, where θij is input for product/task i from factor j, θij, αij >
0, and $ " ij = 1.6 To ensure strictly positive inputs for all factors employed in all
j#{h i }
!
! complementary products/tasks,
$"
j
c # xj
xj
$"
%
nc# xj
xj
j
w
where " xjc is the quantity of factor j
input employed for complementary product/task x, " xjnc is the quantity of factor j input
!
that would be employed
! if product/task x were non-complementary, and w is the number
of complementary products/tasks processed. This restriction ensures that each
!
product/task contributes to the likelihood of successful production.7 Complementary and
non-complementary multi-product/task cost functions are identified in the proposition
below.
Proposition 4:
The expected cost minimizing aggregation level for k products employing ni
heterogeneous factor inputs on product i for
(a) independent processing of non-complementary products/tasks is
6
See the appendix for derivation of these quantum probabilities.
Without the constraint the only meaningful cost report regarding complementary
products/tasks is total cost for all complementary products/tasks. Ambiguity of this sort
in the assignment of costs to individual complementary products is analogous to the
nonexistence (in general) of average cost in a classical, multi-product setting (Demski,
1994).
7
7
n
n
#"
1j
#"
pj
j=1
min
%"1j
" 11 ,L," kn k >0
sin 2
&1j
q1 + … +
j${h 1 }
pj
%"kj
sin 2
2
kj
j=1
& kj
qk, and
j${h k }
2
(b) entangled processing of w complementary products/tasks x through y (and
!
non-complementary or independent processing of all other products/tasks) is
!
n
n
#"
#"
1j p j
j=1
min
&1j
%"
" 11 ,L," kn k >0
j=1
q1 + … +
%"
1j
sin 2
j${h 1 }
sin 2
2
n
xj p j + L + #" yj p j
j=1
& xj
xj
+L+
j${h x }
%"
& yj
qx + …
yj
j${h y }
2
n
#"
+
!
pj
%"
sin
such that
!
kj
j=1
!qk
& kj
kj
2 j${h k }
2
%"
c # xj
xj
%"
&
nc# xj
xj
j${h x }
w
j${h x }
M
!
!
%"
c # yj
yj
%"
&
nc# yj
yj
j${h y }
w
j${h y }
,
where " xjc ( " yjc ) is the quantity of factor j employed for complementary
nc
nc
!product/task x (y), " xj ( " yj ) is the quantity of factor j employed if product/task x
! (y)!were non-complementary, w is the number of complementary products/tasks
ni
processed,
! θij,!αij > 0, # " ij = 1, and qx = … = qy.
j=1
!
8
Examples
Scope economies produced by entangled processing of complementary
products/tasks are compared with expected costs for independent processing of noncomplementary products/tasks in the examples below.
Information
θ11
θ12
θ21
θ22
E[cost]
independent
2.33112
2.33112
2.33112
2.33112
11.040
entangled
1.16556
1.16556
1.16556
1.16556
5.520
processing
Panel A. k = 2, n1 = n2 = 2, q1 = q2 = 1, p1 = p2 = 1, and α11 = α21 = 1/2.
Information
θ11
θ12
θ21
θ22
E[cost]
independent
3.2967
1.64835
3.2967
1.64835
15.613
entangled
1.64835
0.82418
1.64835
0.82418
7.807
processing
Panel B. k = 2, n1 = n2 = 2, q1 = q2 = 1, p1 = 1, p2 = 2, and α11 = α21 = 1/2.
Information
θ11
θ12
θ21
θ22
E[cost]
independent
2.9370
1.4685
2.9370
1.4685
10.433
entangled
1.4685
0.73425
1.4685
0.73425
5.216
processing
Panel C. k = 2, n1 = n2 = 2, q1 = q2 = 1, p1 = 1, p2 = 1, and α11 = α21 = 2/3.
Table 4. Comparison of entangled and independent information processing of two
heterogeneous inputs employed to produce two complementary and two noncomplementary products, respectively.
Quantum entanglement produces substantial scope economies.8 Next, we analyze
a setting in which the agent supplies input that is unobservable by the principal (moral
hazard).
8
It is likely that complementary products/tasks are relatively rare, say, two products/tasks
in twenty are complementary. The benefits then are on the order of 104.88 versus 110.40
(or 5% rather than the apparent 50% savings).
9
Moral hazard setting
Suppose production involves m agents supplying unobservable factor inputs. Let
an input be denoted θutij for factor input j supplied by agent u for (possibly,
complementary) product/task i during period t and θuij is { " uij , " uij} where u = 1, …, m, t
= 1, …, T, i = 1, …, k, and j = 1, …, ni. In other words, factor input heterogeneity may
apply to an agent’s tasks as well as across agents. The agents are strictly risk averse and
! !
bear personal cost c(θuij) for supplying input θuij. Each agent has reservation utility Uu0.
Production involves T measurements of k products/tasks by m agents to achieve success.
In other words, T is a random variable with a geometric probability distribution and
probability of success for each input depends on each of the agents’ inputs involved in
the product/task.
The principal acquires inputs from agent u for product/task i at price sui for
production success achieved at time T by contracting with the agents according to the
following program.
T
Program P:
s.t.
m
k
min E[ " " " sutiq i  " u111,K," u1kn k , …, " uT11,K," uTkn k ]
s1 ,s 2 ,K,s T
t=1 u =1 i=1
E1[U(s11, …, sTk, "111,K,"1kn k , …, " T11,K," Tkn k )] ≥ U0
(IR1)
!
!
M!
ET[U(sT1, …, sTk, "111,K,"1kn k )]
≥ U0
!
!
! E1[U(s11, …, sTk, "111,K,"1kn k , …, " T11,K," Tkn k )] ≥
(IRT)
!
!
E1[U(s11, …, sTk, "111,K,"1kn k ,…, " T11,K," Tkn k )]
!M
!
E1[U(s11, …, sTk, "111,K,"1kn k , …, " T11,K," Tkn k )] ≥
!
!
E1[U(s11, …, sTk, "111,K,"1kn k ,…, " T11,K," Tkn k )]
!
!M
!
ET[U(sT1, …, sTk, " T11,K," Tkn k )] ≥ ET[U(sT1, …, sTk, " T11,K," Tkn k )]
!
!
!
M
!
ET[U(sT1, …, sTk, " T11,K," Tkn k )] ≥ ET[U(sT1, …, sTk, " T11,K," Tkn k )]
!
!
(IC11)
( IC1n )
1
(ICT1)
( IC Tn )
k
!
!
!
!
10
where the constraints apply to each agent u = 1, …, m.9
An agent is provided incentives for each input to be supplied during the current
period conditional on incentives being supplied for all future periods and for all other
agents. Further, individual rationality must be satisfied each period as the contract is
settled whenever success is achieved (the agent can leave the firm at will).
Proposition 5
In a moral hazard setting, the optimal arrangement is a piece-rate contract that
pays a fixed amount for success sT > 0, s1 = s2 = … = sT-1 = 0 where measurement
T indicates successful production.
Sketch of proof: A piece-rate contract is optimal since conditional on previous failure
(otherwise production is complete and the contract is settled) the likelihood ratios
across periods are identical. Hence, payment for success is a constant sT. Ex ante,
the payment for success must compensate the agent for his current input and the
risk associated with anticipated future input.
The optimal arrangement is a piece-rate payment to supply incentives and there is
efficiency loss due to the imposition of risk on risk-averse agents.10 Risk bearing is a
result of the uncertainty regarding the timing of payment and the amount of personal cost
(input) the agent invests before success.
9
The subscript u (for the agent) has been suppressed in the constraints to simplify
notation.
10
Lazear, 1986, and Gibbons, 1987, discuss piece-rate arrangements in classical
information processing settings with emphasis on adverse selection issues.
11
Examples
The examples illustrate the benefit of entangled information processing applied to
complementary products/tasks relative to independent processing of non-complementary
products/tasks by comparing expected costs of supplying incentives to agents in a moral
hazard setting. Each agent’s preferences are represented by U(s, a) = -exp(-r(s – c(θ)),
where c(θ) is the agent’s personal cost of supplying the input. Suppose U0 = 0, r = 0.1,
α11 = α12 = ½, "ui equals the optimal input in the input observable setting, hence "ui =
1.16556 for complementary products/tasks and " ui = 2.33112 for non-complementary
products/tasks, " ui = 0, and c(θu) = θu for the two agents (u = 1, 2) each supplying two
!
!
# " " + " " &
12 22
! is sin 2 % 11 21
factor inputs. The probability of success
( for complementary
2
$
'
!
products/tasks where the first subscript refers to the agent and the second subscript is the
product/task. The optimal contract for each agent is a piece-rate contract where s1 = … =
!
sT-1 = 0, sT = 1.413 per task, and the expected cost of the two agents’ inputs under moral
hazard is 5.653. This exceeds the input-observable (first best) expected cost (5.520) by
0.096 (efficiency loss is approximately 2.4 %).
# " " &
# " " &
The probability of success is sin 2 % 11 21 ( and sin 2 % 12 22 ( for the first and
$ 2 '
$ 2 '
second independent products/tasks, respectively. Hence, the probabilities of success are
equal for the complementary and non-complementary products/tasks settings (≈ 0.8446);
!
!
the difference between the two settings is the expected quantity of inputs needed to
achieve success. The optimal contract for non-complementary products/tasks is a piecerate contract where s1 = … = sT-1 = 0, sT = 2.826 per task. The expected cost of the two
agents’ inputs facing non-complementary tasks is 11.305 and exceeds the expected cost
for complementary tasks (5.6525) by 5.6525.11 This result reaffirms that the potential
efficiency gains due to quantum entangled information processing in an observable input
setting are repeated in a moral hazard setting.12
11
The difference in expected costs for non-complementary products/tasks in the moral
hazard setting compared with the input observable (first best) setting is 0.265 (11.305 –
11.040; approximately 2.4 % efficiency loss).
12
See footnote 8.
12
Conclusions
Production and cost functions are central to an organization’s planning and
control activities. This paper explores cost functions when quantum information
processing is applied to production. In this setting, production success is inherently
uncertain. Consequently, the quantity of factor inputs needed to achieve success is
random. The cost function reflects the expected cost minimizing combination of possibly
heterogeneous factor inputs and aggregation in production measurement. A key
determinant of the cost function involves identification of an optimal level of
aggregation. Optimal measurement timing (or level of aggregation) is identified for both
the input observable and input unobservable (moral hazard) settings. Interfering with
production to measure too frequently destroys synergy and mutes incentives while
waiting until production is almost surely successful over-utilizes costly factor inputs.
An important extension of this paper is discovery of Bell inequality-like (Bell,
1964) empirically testable implications for quantum information processing of
production. Such a test could supply evidence on whether quantum information
processing arises naturally in economic settings. Also, a potentially important limitation
associated with quantum information processing of production is decoherence (Zuric,
1991). If the quantum bits interact with the environment then the apparent gains in
information processing due to quantum interference are lost. Hence great care in the
implementation of quantum information processing such as discussed in this paper is
required. Finally, Penrose, 2005, argues that the measurement axiom in quantum
mechanics is a stopgap measure. Penrose believes that unification of quantum mechanics
with relativity theory requires substantive rethinking of the measurement principle. As
measurement is at the center of the accounting question perhaps accounting scholarship
can contribute to this important matter. These issues are topics of future research.
13
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14
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15
Appendix
This appendix contains an axiomatic development of the quantum probabilities
used in the paper. There are two fundamental probabilities to derive: independent
processing of non-complementary products/tasks and entangled processing of
complementary products/tasks. For independent processing the probability of success is
# ij
$ "ij
sin 2
j
2
, where " ij , αij > 0, # " ij = 1 and " ij are the j = 1, …, ni (possibly,
j
heterogeneous) inputs to the non-complementary i = 1, …, k products/tasks. The
!
probability
! of success!associated with
! entangled processing of w complementary
$"
products/tasks x through y is sin 2
# xj
xj
# yj
+ L + $" yj
j
j
. These probabilities are derived
2
axiomatically (see Nielsen and Chuang, 2002).
!
Axiom 1 — Superposition
$" '
A quantum bit (qubit) is represented by a two element vector & ) where " and "
%# (
2
2
are (possibly complex) numbers, known as amplitudes, such that " + # = 1. Dirac
!
!
"1%
"0%
$" '
!
notation will ease the discussion. 0 = $ ' and 1 = $ ', so ψ〉 = & ) = " 0 + # 1 , and
#0&
#1&
%# (
!
2
2
〈ψ= [α β], the conjugate transpose of ψ〉, so that 〈ψψ〉 = " + # = 1. The
superposition axiom captures
the inherent
!
! uncertainty of!quantum objects.
Axiom 2 — Transformation
!
Evolution of the system or qubit transformation is accomplished by a linear
operator – a unitary matrix. An important transformation in quantum production is the
Hadamard operation H =
1 #1 1 &
%
( . H has the unitary property; that is, after
2 $1 "1'
multiplication by H, the resulting qubit has amplitudes which, when squared, add to one.
!
16
For example, H 0 =
0 +1
0 "1
and H 1 =
. The other unitary transformation
2
2
& i $ # ij%ij
central to quantum production is " = (e
(' 0
!
!
> 0, # " ij = 1. " 0 = e
j
$ij
i % # ij
)
0+– a phase shifter, where i =
1+*
"1 , " ij , αij
0 and " 1 = 1 . The two unitary transformation matrices
!
!
!
are combined to form the quantum production function; this is called an interferometer
% %!
(
!
$ (
#e i"
'exp'i#" j j * 0*
H ' '& j
*) *H or for the homogeneous factor input case (n = 1) H %
$0
'&
0
1*)
!
0&
(H (see
1'
Bouwmeester and Zeilinger, 2000).
!
!
Axiom 3 — Measurement
The probability that qubit ψ〉 = " 0 + # 1 is measured in basis 0 is
2
2
〈ψ0〉〈0ψ〉 = " , or measured in basis 1 is 〈ψ1〉〈1ψ〉 = " .
!
!
Superposition collapses when an observation takes place. The probability of the
!
!
!
measurement yielding a particular state is the square (of the modulus) of the amplitude of
that state. See, for example, Feynman (1963).
Axiom 4 — Combination
Two (or more) qubits are combined into one system according to tensor
multiplication of vectors. Tensor multiplication is defined as follows.
$"1" 2 '
$"1' $" 2 ' &"1# 2 )
)
& )*& )=&
%#1 ( %# 2 ( &#1" 2 )
&
)
%#1# 2 (
!
17
Dirac (1958) notation simplifies tensor multiplication. For example, 0 " 0 = 00 ,
"1%
$'
0
where 00 = $ '. Also, 0 " 1 = 01 , 1 " 0 = 10 , and !
so forth.
$0'
$'
#0&
"1
$
$0
! The most important two qubit operation is CNOT = $0
$
#0
!
!
0
1
0
0
0
0
0
1
0%
'
0'
. The first qubit is the
1'
'
0&
control and the second qubit is the target such that CNOT 00 = 00 , CNOT 01 = 01 ,
!
CNOT 10 = 11 , and CNOT 11 = 10 .
!
An important two qubit state is called an “entangled” pair of qubits, so important
!
!
it has its own conventional notation, denoted " 00 . Start with 00 and perform, in order,
!
an H transformation on the first qubit (denote the operation H1), and then a CNOT
transformation on the pair.
!
" 00 = CNOT H1 00 = CNOT
!
00 + 10
2
=
00 + 11
2
The resulting two-qubit system is referred to as an entangled state; note that it cannot be
created by the tensor combination of any two individual qubits (Aczel, 2001, Nielsen and
!
Chuang, 2002, and Zeilinger, 2000). " 00 is a Bell or EPR state and its orthogonal
01 + 10
00 # 11
01 # 10
, "10 =
, and "11 =
. Together,
2
2
2
!
the four are said to form the Bell basis (Bell, 1964).
complements are " 01 =
n-qubit entangled states are identified in similar fashion. Entangled qubits and
!
!
!
measurements are formed by successive transformations CNOT1,2 H1 … CNOTn-2,n-1 Hn-2
CNOTn-1,n Hn-1 applied to 00…0〉 ≡ β00…0〉 or applied to 00…1〉 ≡ β00…1〉, where
CNOTij is the controlled not operator in which qubit i is the control qubit and qubit j is the
target qubit such that if the control qubit is 1〉 then the target is bit flipped and Hk is a
Hadamard operator applied to qubit k (see Nielsen and Chuang, 2000, for more details).
Examples include
β00〉 = CNOT12H100〉 =1/√2 (00〉 + 11〉),
18
β01〉 = CNOT12H101〉 =1/√2 (01〉 + 10〉),
β000〉 = 1/√2 (0〉β00〉 + 1〉β01〉),
β001〉 = 1/√2 (0〉β01〉 + 1〉β00〉),
β0000〉 = 1/√2 (β00〉β00〉 + β01〉β01〉),
β0001〉 = 1/√2 (β00〉β01〉 + β01〉β00〉),
β00000〉 = 1/√2 (0〉β0000〉 + 1〉β0001〉), and
β00001〉 = 1/√2 (0〉β0001〉 + 1〉β0000〉).
Probability Derivations
Independent processing of non-complementary products/tasks begins with initial
perceptions 00…0〉. Quantum production evolves via H1"1H1⊗…⊗ H k " k H k where the
subscript refers to the qubit on which the transformation operates. Equivalently, this can
be represented by individual qubit transformation and measurement as independent
!
!
measurement follows from tensor operations on the qubits. Hence, H i" i H i applied to
initial perception 0〉 describes the transformation of product/task i. Measurement in
basis 1〉 signals “success” while measurement in basis 0〉 signals “failure”. That is,
!
#
$ "ij ij
the probability of success is 〈0 H j" hj H j1〉〈1 H j" jH j0〉 = sin 2 j
, while the
2
# ij
$ "ij
probability of failure is 〈0 H j" hj H j0〉〈0 H j" jH j0〉 = cos 2
!
!
!
j
2
where superscript h
refers to conjugate, transpose operation.
Entangled processing of complementary
products/tasks proceeds similarly.
!
!
!
Measurement in basis β00…1〉 signals “success” while measurement in basis β00…0〉
signals “failure”. The probability of success is
〈β00…0 H x " hx H x LH y " hy H y β00…1〉〈β00…1 H x " x H x LH y " y H y β00…0〉 =
$"
!
sin 2
# xj
xj
# yj
+ L + $" yj
j
j
2
, while the probability of failure is
!
!
19
〈β00…0 H x " hx H x LH y " hy H y β00…0〉〈β00…0 H x " x H x LH y " y H y β00…0〉 =
$"
!
cos2
# xj
xj
# yj
+ L + $" yj
j
j
2
.
!
!
20