PHYSICAL REVIEW B 67, 165324 共2003兲

Quantum-limited measurement and information in mesoscopic detectors
A. A. Clerk, S. M. Girvin, and A. D. Stone
Departments of Applied Physics and Physics, Yale University, New Haven, Connecticut 06511
共Received 4 November 2002; published 28 April 2003兲
We formulate general conditions necessary for a linear-response detector to reach the quantum limit ofmeasurement efficiency, where the measurement-induced dephasing rate takes its minimum possible value.
These conditions are applicable to both noninteracting and interacting systems. We assess the status of theserequirements in an arbitrary noninteracting scattering-based detector, identifying the symmetries of the scattering matrix needed to reach the quantum limit. We show that these conditions are necessary to prevent theexistence of information in the detector that is not extracted in the measurement process.
DOI: 10.1103/PhysRevB.67.165324

PACS number共s兲: 73.23.⫺b, 03.65.Ta

I. INTRODUCTION

ally considered in mesoscopic systems. For example, we findthat time-reversal symmetry is not necessary for reaching thequantum limit. We also find that, surprisingly, an adiabatic
point-contact7 system remains a quantum-limited detector
even for voltages large enough that several channels contribute to transport and that the energy dependence of scattering
is important; previous studies8 –10 have only shown that thequantum limit is achieved in the small voltage regime. Our
results for the mesoscopic scattering detector are complementary to those obtained in Ref. 6.

Issues of quantum measurement in mesoscopic systems
have recently garnered considerable interest, both because of
their relevance to attempts at quantum computation1 andquantum-limited amplifiers.2 A general consequence of any
quantum measurement is that it must induce decoherence inthe system variable conjugate to that being measured. This
basic fact naturally leads to the issue of measurement efficiency: what conditions must a particular detector satisfy sothat it induces the absolute minimum amount of dephasing
required by quantum mechanics? This minimum dephasingrate is identical to the measurement rate ⌫ meas , the rate at
which information is extracted during the measurement process; thus, the measurement efficiency ratio ␹ ⭐1 is definedby ␹ ⫽⌫ meas /⌫ ␸ , where ⌫ ␸ is the measurement-induceddephasing rate. Besides being of great conceptual interest,near-ideal measurement schemes are necessary to detect signatures of coherent qubit oscillations in the output noise of adetector,3,4 and are essential if one wishes to construct aquantum-limited amplifier 共i.e., an amplifier whose noise energy is the minimum allowed by quantum mechanics兲.2
While the question of measurement efficiency has received
attention in the context of general measurement theory,5 it is
only recently that it has been considered in the context ofsolid-state detectors. Averin3 has considered the status of thequantum limit in a number of solid-state detectors, while
recently Pilgram and Büttiker6 considered the quantum limit
for a system in which a mesoscopic conductor acts as a detector.
In this paper, we formulate general conditions that areneeded for an arbitrary detector in the linear-response regime
to reach the quantum limit of detection, where ␹ ⫽1. These
general conditions are valid for both interacting and noninteracting systems, and can be given a direct physical interpretation. We also discuss the quantum limit in terms of asimple concept from quantum information theory, the accessible information. To make these considerations more concrete, we apply them to a mesoscopic scattering detector
similar to that considered in Ref. 6, identifying precise conditions and symmetries needed to reach the quantum limit.
We find that the required symmetries are most easily understood if one considers the scattering detector in terms of
information; these symmetries are not the same as those usu0163-1829/2003/67共16兲/165324共12兲/$20.00

II. GENERAL CONDITIONS
A. Model and derivation of the quantum limit

We start by considering a generic system consisting of aqubit 共i.e. a two-level system described as a spin 21 ) coupled
to an arbitrary detector. The system Hamiltonian is H
⫽H qubit⫹H detector⫹H int , where H qubit⫽⫺ 21 ⍀ ␴ z , H int⫽A ␴ z Q, and we leave H detector unspecified. Q is the detector‘‘input’’ operator that couples to the qubit, while A characterizes the strength of the qubit-detector coupling. Mixing
effects, where the detector causes transitions in the qubit, areneglected by taking 关 H int ,H qubit兴 ⫽0; such effects always
cause a deviation from the quantum limit. We work in the
weak-coupling regime (A→0), and can thus use the linearresponse theory to describe the output of detector. Taking I to
be the detector observable that is measured 共i.e., the ‘‘output’’ operator兲, one has to lowest order in A,

具 I 共 t 兲 典 ⫽ 具 I 共 t 兲 典 ␳ 0 ⫹A␭ 具 ␴ˆ z 共 t 兲 典 ␳ Q ,共1兲

where the zero-frequency linear-response coefficient 共or‘‘forward gain’’兲 ␭ is given by
␭⬅

⫺i
ប

冕 ␶具 ␶

共2兲

冕

共3兲

⬁

d

0

关 I 共 兲 ,Q 共 0 兲兴 典 ␳ 0⬁
2⫽ Im d ␶ 具 I 共 ␶ 兲 Q 共 0 兲 典 ␳ 0 .
ប
0Here, ␳ 0 is the initial density matrix of the detector and ␳ Q is
the initial density matrix of the qubit. We have assumed thatthe qubit splitting frequency ⍀ is much smaller than the ratethat characterizes the detector, which allows us to approxi-

67 165324-1©2003 The American Physical Society

PHYSICAL REVIEW B 67, 165324 共2003兲

A. A. CLERK, S. M. GIRVIN, AND A. D. STONE

mate the detector’s response to the qubit as instantaneous.
Alternatively, one can restrict attention to the case where thequbit is in a ␴ z eigenstate, and thus 具 ␴ z (t) 典 is time independent. The operators on the right-hand side 共RHS兲 in theabove equation evolve in the Heisenberg picture generatedby H 0 ⫽H qubit⫹H detector .
Next, we connect the detector noise in the output operator
I and input operator Q to, respectively, the measurement rate
⌫ meas and the dephasing rate ⌫ ␸ . Defining the fluctuating
part of an operator A as Ã⫽A⫺ 具 A 典 ␳ 0 , the required zerofrequency noise correlators are given by,
S I ⫽2

S Q ⫽2

冕

⫹⬁

⫺⬁

冕

⫹⬁

⫺⬁

dt 具 Ĩ 共 t 兲 Ĩ 共 0 兲 典 ␳ 0 ⫽4 ␲ ប

P i ␦ 共 E i ⫺E f 兲 兩 Ĩ i f 兩 ,兺
i, f
P i ␦ 共 E i ⫺E f 兲 兩 Q̃ i f 兩 2 ,
兺
i, f
共4b兲

S IQ ⫽2冕

⫹⬁⫺⬁⫽4 ␲ ប

dt 具 Ĩ 共 t 兲 Q̃ 共 0 兲 典 ␳ 0

P i ␦ 共 E i ⫺E f 兲共 Ĩ i f 兲共 Q̃ f i 兲 .
兺
i, f

共4c兲

A2
ប

S ,
2 Q

⌫ meas⫽A 2␭ 2.
SI

共5兲

We briefly review the origin of Eqs. 共5兲. The dephasingrate describes the measurement-induced decay of the offdiagonal elements of the qubit density matrix. It can be derived by looking at the decay at long times of the phasecorrelator V(t)⫽ 具 ␴ ⫹ (t) ␴ ⫺ (0) 典 , where ␴ ⫹ ( ␴ ⫺ ) is the spinraising 共lowering兲 operator:冓 冋 冕V 共 t 兲 ⫽ exp ⫺i⯝e ⫺i⍀̃t exp

冉

t

0

dt ⬘ 共 ⍀⫹2AQ 共 t ⬘ 兲 /ប 兲

⫺2A 2
ប2

冕 冕 具
t

0

t

dt 1

0

→e ⫺i⍀̃t e ⫺⌫ ␸ t .

册冔

dt 2 Q̃ 共 t 1 兲 Q̃ 共 t 2 兲 典

冊

共6兲
共7兲

Here, ⍀̃⫽⍀⫹2A 具 Q 典 ␳ 0 /ប.
The measurement rate describes how long the measurement must be on before the signal associated with the two
qubit states can be distinguished from the noise in I. The
quantity of interest is the time integral of the detector output,m(t)⫽ 兰 t0 dt ⬘ I(t ⬘ ). One needs that the distributions of m(t)corresponding to the two different qubit states 关i.e.,

共8兲

where ␴ denotes the variance of the distribution and the 冑2factor is included in order to make the final upper bound on␹ unity. Using Eq. 共1兲 for 具 I(t) 典 , and letting ␶ meas⫽1/⌫ meas , the condition becomes2A␭ ␶ meas⭓2 冑2冑冉 冊 ␶
1
S
2 II

meas,

共9兲

which directly yields the expression in Eq. 共5兲 for ⌫ meas .
Note that we have taken ␴ ↑ ⫽ ␴ ↓ in the last step; this issufficient to obtain the leading-order-in-A expression for⌫ meas .
To relate ⌫ ␸ and ⌫ meas , we first note that the right-handsides of Eqs. 共4a兲–共4c兲 implicitly define an inner product
共i.e., interpret the matrix elements 兵 Ĩ i f 其 and 兵 Q̃ i f 其 as defining
vectors兲. The Schwartz inequality then immediately yields
S I S Q ⭓ 兩 S IQ 兩 2 ⫽ប 2 共 ␭⫺␭ ⬘ 兲 2 ⫹ 共 Re S IQ 兲 2 ,Here, we use the short hand O i f ⫽ 具 i 兩 O 兩 f 典 , where 兩 i 典 , 兩 f 典 are
eigenstates of H detector with energies E i ,E f . The probability
P i is defined as 具 i 兩 ␳ 0 兩 i 典 ; we assume that ␳ 0 is diagonal in thebasis of eigenstates. Taking the detector noise to be Gaussian, the standard expressions for the dephasing rate ⌫ ␸ andmeasurement rate ⌫ meas are given by1⌫ ␸⫽

具 m 共 t 兲 典 ↑ ⫺ 具 m 共 t 兲 典 ↓ ⭓ 冑2 关 ␴ ↑ 共 t 兲 ⫹ ␴ ↓ 共 t 兲兴 ,

2

共4a兲

dt 具 Q̃ 共 t 兲 Q̃ 共 0 兲 典 ␳ 0 ⫽4 ␲ បp„m(t) 兩 ↑… and p„m(t) 兩 ↓…] be statistically distinguishable.
Assuming Gaussian distributions, distinguishability is defined as

共10兲

where we have introduced the reciprocal response coefficient
共or ‘‘backwards gain’’兲 ␭ ⬘ :冕⬁2␭ ⬘ ⬅ Im d ␶ 具 Q̂ 共 ␶ 兲 Î 共 0 兲 典 ␳ 0 .
ប
0

共11兲

␭ ⬘ would describe the response of 具 Q(t) 典 to a perturbationthat couples to the operator I. Note that as ␭ and ␭ ⬘ aredefined in terms of commutators, we may substitute I→ Ĩ
and Q→Q̃ in their definitions. General stability considerations lead to the condition ␭␭ ⬘ ⭐0. Using Eqs. 共5兲, we thushave
ប 2␭ 2⌫ meas ប 2 ␭ 2
⫽
⭐ 2
⭐1.
⌫␸
S Q S I ប 共 ␭⫺␭ ⬘ 兲 2 ⫹ 共 ReS IQ 兲 2

共12兲

The best one can do is measure the qubit as quickly as one
dephases it.11 Note that this derivation only requires the validity of linear-response and the weak-coupling approximations which give rise to Eqs. 共5兲; very little is specified of thedetector. Similar derivations of the quantum limit are presented in Refs. 3 and 5.
The inequality in Eq. 共12兲 is in many ways intuitively
reasonable. Both dephasing and measurement involve entangling the state of the qubit with states in the detector. In
principle, there may be degrees of freedom in the detector
which become entangled with the qubit without providing
any detectable information in a measurement of 具 I 典 ; anysuch entanglement would lead to ⌫ ␸ ⬎⌫ meas . More precisely,
imagine that when the measurement is initially turned on, thesystem is in a product state:

165324-2

兩 ␺ 共 t⫽0 兲 典 ⫽1

冑2

共兩↑典⫹兩↓典) 丢 兩D典,

共13兲

PHYSICAL REVIEW B 67, 165324 共2003兲

QUANTUM-LIMITED MEASUREMENT AND INFORMATION . . .

where 兩 D 典 is the initial state of the detector, and 兩 ↑ 典 , 兩 ↓ 典
denote qubit ␴ z eigenstates. At some later time t, the state of
the system may be written as兩␺共 t 兲典⫽1

冑2

共 兩 ↑ 典 丢 兩 D ↑ 共 t 兲 典 ⫹ 兩 ↓ 典 丢 兩 D ↓ 共 t 兲 典 ).

共14兲

To say that we have measured the state of the system implies
that the states 兩 D ↑ (t) 典 and 兩 D ↓ (t) 典 are distinguishable; to saythat the qubit has been dephased only implies that the detector states 兩 D ↑ (t) 典 and 兩 D ↓ (t) 典 are orthogonal. While distinguishability implies orthogonality, the opposite is not true;
thus, in general, ⌫ ␸ ⬎⌫ meas . Note that, in this formulation,
the dephasing rate will be related to the overlap between thetwo detector states:␭ ⬘ implies that this additional coupling will not contribute to
具 Q(t) 典 , and thus cannot further dephase the qubit 关cf. Eq.
共6兲兴.
On a technical level, Eq. 共16兲 follows from the optimization of the Schwartz inequality and the requirement that
Re S IQ ⫽0 关i.e. conditions 共i兲 and 共ii兲 above兴. The vanishingof ␭ ⬘ 关Eq. 共17兲兴 can be interpreted in terms of causality. To
see this, we first introduce the frequency-dependent crosscorrelator S IQ (E):S IQ 共 E 兲 ⫽2

冕

⬁

⫺⬁

⫽4 ␲ ប

dt 具 Ĩ 共 t 兲 Q̃ 共 0 兲 典 ␳ 0 e iEt/ប兺 P i ␦ 共 E⫹E i ⫺E f 兲 Ĩ i f Q̃ f i .

共18兲

i, f ⫽i

We may use this to write兩 具 D ↑ 共 t 兲 兩 D ↓ 共 t 兲 典 兩 ⯝e ⫺⌫ ␸ t .

共15兲
␭共 ␭⬘兲⫽B. Necessary conditions for reaching the quantum limit

We have thus seen that on a heuristic level, reaching thequantum limit requires that the detector have no ‘‘extraneous’’ degrees of freedom which couple to the qubit. Equivalently, all information on the state of the qubit residing in thedetector should be accessible in the measurement of 具 I 典 . The
virtue of the derivation presented in the preceding section isthat these statements can be given a precise meaning. One
sees that three conditions are necessary to reach the quantum
limit: 共i兲 the Schwartz inequality in Eq. 共10兲 must be optimized, 共ii兲 the cross-correlator Re S IQ must vanish, and 共iii兲
the backwards gain ␭ ⬘ must vanish. Conditions 共i兲 and 共ii兲can be succinctly reexpressed as a single condition, leading
to the following necessary and sufficient requirements:

兵 ᭙ i, f 兩 P i ⫽0,E f ⫽E i 其 ,冕

具 f 兩 Ĩ 兩 i 典 ⫽iC具 f 兩 Q̃ 兩 i 典 ,⬁2␭ ⬘ ⬅ Im d ␶ 具 Q̂ 共 ␶ 兲 Î 共 0 兲 典 ␳ 0 ⫽0.
ប
0

共16兲

⫺

1
P
␲

冕

⬁

⫺⬁

dE

冊

Re 关 S IQ 共 E 兲兴
.
E

共19兲

If ␭ ⬘ ⫽0, it follows from the above that at E⫽0, the imaginary part of S IQ (E) coincides with the Hilbert transform of
the real part of S IQ (E):Im 关 S IQ 共 E 兲兴

冏

冉

1
⫽ ⫺ P
␲
E⫽0

冕

⬁

⫺⬁

dE ⬘

Re 关 S IQ 共 E ⬘ 兲兴E ⬘ ⫺E

冊冏

.
E⫽0

共20兲

If this held for all E, it would follow from the Titchmarsh
theorem13 that S IQ (t)⫽ 具 Ĩ (t)Q̃(0) 典 ␳ 0 is causal: it would vanish for t⬍0. This would clearly be sufficient to satisfy Eq.
共17兲. More generally, the vanishing of ␭ ⬘ only requires theweaker zero-frequency causality condition in Eq. 共20兲.

共17兲

Here, C is a real number that is independent of the detector
eigenstates 兩 i 典 and 兩 f 典 . 12 Equations 共16兲 and 共17兲 are central
results of this paper. The first of these equations expresses
the fact that to reach the quantum limit, there must be a close
similarity between the detector’s input and output
operators—as far as the zero-frequency noise correlators areconcerned, the operators I and Q must be proportional toone another. This required similarity between the detector
input and output is a formal expression of the intuitive ideathat a quantum-limited detector has no ‘‘extraneous’’ internal
degrees of freedom. The second condition, Eq. 共17兲, expresses the fact that a quantum-limited detector must have astrong intrinsic directionality that discriminates between the
input and output. The output operator is influenced by behavior at the input, but not vice versa. This requirement is consistent with our tacit assumption that the quantity 具 I 典 can bemeasured without problems. To measure I, one needs to introduce a coupling in the Hamiltonian to I; the vanishing of冉

1⫹ 共 ⫺ 兲 Im 关 S IQ 共 0 兲兴
2ប

C. The quantum limit and information theory

We close this section by formalizing the connection between the quantum limit and information. A deviation fromthe quantum limit 共i.e., ␹ ⬍1) implies the existence in thedetector of ‘‘missing information’’ regarding the state of thequbit, information that is not revealed in the measurement of
具 I 典 . The dephasing rate thus corresponds to what the measurement rate would be if we could make use of all the available information. This notion can be quantified by borrowing
a concept from quantum information theory, the accessible
information.14 –17 To define this, note first that if we choose a
specific detector quantity 共or set of quantities兲 Y to measure共described by, e.g., a set of commuting observables兲, we canthink of our system as a noisy classical communication channel. The two possible inputs to the channel are the qubit
states 兩 ↑ 典 and 兩 ↓ 典 ; interaction with the detector for a time tthen leads to two corresponding detector states 兩 D ↑ (t) 典 and兩 D ↓ (t) 典 关c.f. Eq. 共14兲兴.18 Finally, the outputs from the channel are the outcomes of the measurement of Y. The ‘‘noise’’here is a result of the intrinsic uncertainties of Y in the states

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PHYSICAL REVIEW B 67, 165324 共2003兲

A. A. CLERK, S. M. GIRVIN, AND A. D. STONE兩 D ↑ (t) 典 and 兩 D ↓ (t) 典 ; the output will thus be described by theconditional probability distributions p(y 兩 ↑) and p(y 兩 ↓),which are determined by these states, where y represents
possible outcomes of the measurement. Letting p̄(y)⫽ 关 p(y 兩 ↑)⫹ p(y 兩 ↓) 兴 /2, the mutual information R of thischannel is19

1
R 关 Y 兴 ⫽H 关 p̄ 共 y 兲兴 ⫺ „H 关 p 共 y 兩 ↑ 兲兴 ⫹H 关 p 共 y 兩 ↓ 兲兴 …, 共21兲2
where H 关 p(y) 兴 is the Shannon information entropy associated with the distribution p:
H 关 p 共 y 兲兴 ⫽⫺兺y p 共 y i 兲 ln关 p 共 y i 兲兴 .

共22兲

i

Note that we have chosen to equally weight our two inputs tothe channel. Assuming that this choice is optimal, Shannon’s
noisy channel coding theorem implies that R 关 Y 兴 is the maximum rate at which messages can be reliably transmitted
down the channel by modulating the state of the qubit andmaking measurements of Y.19 Alternatively, R 关 Y 兴 may beconsidered as being related to a generalized measurement
rate describing the chosen measurement Y. For example, if
the distributions p„y(t) 兩 ↑… and p„y(t) 兩 ↓… are Gaussian, one
finds that at small times 共i.e., before the two distributions arewell separated兲
R 关 Y 兴 Gaussian⫽

1 共 具 y 共 t 兲典 ↑⫺ 具 y 共 t 兲典 ↓ 兲2.
8
␴ ↑共 t 兲 ␴ ↓共 t 兲

共23兲This corresponds to our definition of the measurement rate,
cf. Eqs. 共8兲 and 共9兲. We thus have a new way to interpret themeasurement rate ⌫ meas : given that one is monitoring 具 I 典 ,⌫ meas represents the maximum rate at which information canbe sent to the detector by modulating the qubit.
The quantum-mechanical accessible information I is now
defined by maximizing the mutual information R 关 Y 兴 over all
possible measurement schemes Y. Remarkably, for the caseconsidered here 共where the detector is described by a purestate兲, it can be calculated exactly;14 a simplified proof ispresented in Appendix A, where we also demonstrate that
there are several possible optimal measurement schemes.
Letting 兩 具 D ↑ (t) 兩 D ↓ (t) 典 兩 2 ⫽cos2„␣ (t)…, we have1I⫽maxR⫽ 兵 关 1⫹sin ␣ 共 t 兲兴 ln关 1⫹sin ␣ 共 t 兲兴2
兵Y 其⫹ 关 1⫺sin ␣ 共 t 兲兴 ln关 1⫺sin ␣ 共 t 兲兴 其 .

共24兲

This expression corresponds to having equally weighted our
two input states, as we did in Eq. 共21兲; one can check that
this choice maximizes I. At small times (⌫ ␸ tⰆ1), comparison against Eq. 共15兲 yields ␣ (t)→0, and we haveI⯝ ␣ 共 t 兲 2 ⫽⌫ ␸ t.

共25兲

As expected, the growth of the accessible information is determined by the dephasing rate. Achieving ␹ ⫽1 thus implies
that the rate that we actually obtain information, ⌫ meas , coincides with the growth of the total accessible information.

FIG. 1. Schematic diagram of the mesoscopic scattering detector, in which the current through a phase coherent scattering regionis used to detect the qubit. Q denotes the charge in the scattering
region, while I R (I L ) is the current in the right 共left兲 contact.

Thus, there is no ‘‘missing’’ information in the detector. We
can also think of Eqs. 共24兲 and 共25兲 as providing an alternate
route for deriving the quantum-limit inequality ⌫ ␸ ⭓⌫ meas ,i.e.,
R 关 Y 兴 ⯝⌫ meast⭐I⯝⌫ ␸ t.

共26兲The utility of thinking about back-action effects and thequantum limit in terms of information will become clear in
the following section, where we discuss the mesoscopic scattering detector. Note also that the relation between information and state disturbance has been studied in a slightly different context by Fuchs et al.15
III. MESOSCOPIC SCATTERING DETECTOR

To make the preceding discussion more concrete, we now
consider the status of the quantum limit in a slightly lessgeneral detector setup, the mesoscopic scattering detector
considered in Ref. 6. We determine the conditions needed toreach the quantum limit of detection by directly applying thegeneral conditions derived in the preceding section, namely
the proportionality condition in Eq. 共16兲 and the causality
condition in Eq. 共17兲. This is in contrast to Ref. 6, whichdeveloped conditions needed for the quantum limit by directly calculating ⌫ ␸ and ⌫ meas . We explicitly show that theviolation of Eq. 共16兲 implies the existence of unused information in the detector, information that is not extracted in themeasurement process.
The detector here is a two terminal scattering region 共seeFig. 1兲 characterized by a scattering matrix s. Taking thecontact to both the right and left reservoirs to have N propagating transverse modes, s will have dimension 2N. The
output operator of the detector I is simply the current throughthe region; the state of the qubit alters 具 I 典 by modulating thepotential in the scattering region. Note that while we focus
on the limit of a weak coupling between the qubit and detector, so that the linear-response approach of the preceding
section is valid, we do not assume that the voltage is smallenough that 具 I 典 ⬀V. 20 The mesoscopic scattering detector describes the setup used in two recent ‘‘which path’’experiments.21,22 These experiments used a quantum point
contact to detect the presence of an extra electron in a nearby

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QUANTUM-LIMITED MEASUREMENT AND INFORMATION . . .

quantum dot. As the dot was imbedded in an AharanovBohm ring, the dephasing induced by the measurement could
be studied directly.
We start by considering the simplest situation, also considered in Ref. 6, where the state of the qubit provides a
uniform potential change in the scattering region. In thiscase, the input operator Q is the total charge in the scattering
region. Unlike Ref. 6, we do not explicitly consider theeffects of screening here. Within the random-phase approximation, consideration of such effects allows an explicit calculation of the qubit-detector coupling strength A, but does
not result in any other changes over a noninteracting approach. In the weak-coupling regime, the particular value of
A does not affect the approach to the quantum limit.
Letting a ␣† n (E) represent the creation operator for an incident wave in contact ␣ ⫽L,R, transverse mode n, and atenergy E, the detector current operator for contact ␣ takes
the form;23
I ␣⫽

e
h

冕 冕 ⬘

⫻

dE

兺

冉

A positive current corresponds to a current incident on the
scattering region; note that throughout this section, we neglect electron spin for simplicity. The total charge Q in the
scattering region may be defined in terms of the total current
incident on the scattering region—in the Heisenberg picture,⳵ t Q(t)⫽I L (t)⫹I R (t) . One obtainsdEdE

␤ , ␥ ⫽L,R

N共 E,E⫹ប ␻ 兲 ⫽

册

冋

册

d
1
s共 E 兲 .
s †共 E 兲
2␲i
dE

共31兲

Finally, the assumption that the qubit couples to the total
charge in the scattering region is equivalent to assuming thatthe potential it creates is smooth in the WKB sense. We canuse the fact that the sensitivity of the scattering matrix s to a
global change of potential in the scattering region is the sameas its sensitivity to energy. Thus, the linear-response coefficient ␭ has the form

␮R

d␧

兺j

dT j 共 ␧ 兲
,
d␧

共32兲

s LR

冊冉

s RR ⫽

冑Re i ␤冑Te i ␸冑Te i ␸ ⬘⫺ 冑Re i( ␸ ⫹ ␸ ⬘ ⫺ ␤ )

冊

,
共33兲

共34兲

First, we consider the causality condition in Eq. 共17兲
which requires that the backwards gain ␭ ⬘ vanishes. As weknow the initial state of the detector and have explicit expressions for I and Q, we can directly evaluate the function
S IQ (E) appearing in Eq. 共18兲 in terms of s. A direct calculation can be performed to show that

冕

⬁

⫺⬁

dE

Re 关 S IQ 共 E 兲兴
⫽
E

冕

⬁

⫺⬁

dE

Re 关 F 共 E 兲兴
,
E

共35兲

Im 关 S IQ 共 0 兲兴 ⫽Im 关 F 共 0 兲兴 ,共36兲

where, letting t⬅s RL ,the function F(E) is defined as:e2
F 共 E 兲 ⫽⫺i2␲

In the limit where ␻ →0, N(E,E⫹ប ␻ ) reduces to the wellknown Wigner-Smith delay-time matrix
N共 E 兲 ⫽␮Ld†关 tr s LR
共 ␧ 兲 s LR 共 ␧ 兲兴d␧

兩 i 典 ⫽ 关 ⌸ E L ⭐ ␮ L a L† 共 E L 兲兴关 ⌸ E R ⭐ ␮ R a R† 共 E R 兲兴 兩 vac典 .

共29兲

s 共 E⫹ប ␻ 兲 ⫺s 共 E 兲1s †共 E 兲. 共30兲
2␲i
ប␻

␮R

d␧

where R⫽1⫺T. At zero temperature, the detector is described by a single many-body state 兩 i 典 in which all incident
states in lead ␣ with E⬍ ␮ ␣ are occupied and all other incident states are unoccupied:关 a ␤† n 共 E 兲 N␤ n, ␥ m 共 E,E ⬘ 兲 a ␥ m 共 E ⬘ 兲兴 ,

冋

s LL

s共 E 兲⫽ s
RL

A ␤ n, ␥ m 共 ␣ ;E,E ⬘ 兲 ⫽ ␦ ␤␥ ␦ ␣␤ ␦ nm ⫺ 兵 关 s ␣␤ 共 E 兲兴 † s ␣␥ 共 E ⬘ 兲 其 nm .
共28兲

冕 冕 ⬘兺e2h␮LGiven these definitions, we can now turn to Eqs. 共16兲 and共17兲 and ask what is required of the scattering matrix s inorder to reach the quantum limit. We first focus on the caseN⫽1, where there is a single propagating mode in both contacts. The scattering matrix s is thus 2⫻2, and may be written as关 a ␤† n 共 E 兲 A ␤ n, ␥ m 共 ␣ ;E,E ⬘ 兲 a ␥ m 共 E ⬘ 兲兴 ,

共27兲

Q⫽e

⫽⫺

冕
冕

A. Single-channel case

N

兺

e2
h

where the T j are the transmission eigenvalues of the system.
Without loss of generality, we have assumed that our detector is biased such that the chemical potential of the left reservoir is greater than that of the right reservoir: ␮ L ⫺ ␮ R⫽e 兩 V 兩 ; we also consider the limit of zero temperature.

dE

␤ , ␥ ⫽L,R n,m⫽1

␭⫽⫺

冕

␮L

␮R

dE ⬘ t * 共 E ⬘ 兲冉冊

t 共 E ⬘ ⫹E 兲 ⫺t 共 E ⬘ 兲
.
E
共37兲

Note that Eqs. 共35兲 and 共36兲 are independent of whether I istaken to be I L , I R , or a linear combination of the two. Now,
causality dictates that the scattering matrix s is analytic in theupper half complex plane, and thus so is the function F(E).
The real and imaginary parts of F are thus related by a Hilbert transform, and Eqs. 共20兲, 共35兲, and 共36兲 imply that ␭ ⬘⫽0 for the scattering detector irrespective of the choice of s.
Thus, the causality properties of the scattering matrix s ensure that one of the conditions necessary for reaching thequantum limit is always satisfied. Note that substituting these

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PHYSICAL REVIEW B 67, 165324 共2003兲

A. A. CLERK, S. M. GIRVIN, AND A. D. STONE

expressions for S IQ (E) in Eq. 共19兲 does indeed yield the
expected form of ␭ 关Eq. 共32兲兴. It is also useful to note thatgauge invariance can be used to directly establish24 ␭ ⬘ ⫽0 .
The essence of the argument is that a coupling to the current
关i.e., H int⫽A ␴ z I(x⫽0)] is equivalent to introducing a local
vector potential. The gauge transformation that removes thisterm will only modify the transmission phases in the scattering matrix s 共i.e., ␾ and ␾ ⬘ ) in an energy-independent manner. Using Eq. 共29兲, one can check that 具 Q 典 is independent ofenergy-independent phase changes; thus ␭ ⬘ ⫽0.
Next, we turn to the condition given in Eq. 共16兲, which
requires a certain proportionality between Ĩ and Q̃ in order toreach the quantum limit. Given the state 兩 i 典 that describes thedetector 关Eq. 共34兲兴, the only matrix elements of I and Q
which contribute to the zero frequency noise correlators 关cf.
Eqs. 共4兲兴, involve energy-conserving transitions where ascattering state incident from the left reservoir is destroyed,
while a scattering state incident from the right reservoir iscreated. Since these transitions require an occupied initialstate and an unoccupied final state, they can only occur in theenergy interval ␮ R ⬍E⬍ ␮ L . We are thus interested in thecoefficients of the operators a R† (E)a L (E) appearing in the
expansion of I and Q in this energy interval. The proportionality requirement of Eq. 共16兲 thus results in a necessary condition on s(␧):᭙E ⑀ 关 ␮ R , ␮ L 兴 , 关 s LR 兴 * 共 E 兲 s LL 共 E 兲 ⫽iCNRL 共 E 兲 ,d关 ␤ 共 E 兲 ⫺ ␾ 共 E 兲兴 ⫽0,
dE

dT
共E兲
dE
4␲
.
⫽⫺
T 共 E 兲关 1⫺T 共 E 兲兴CS IQ ⫽iប␭⫹

共39兲

共40兲

Similar conditions for reaching the quantum limit for thisversion of the scattering detector were first developed in Ref.
6 by directly calculating ⌫ meas and ⌫ ␸ 关note that there is asign error in Eq. 共7兲 of Ref. 6 which must be corrected toobtain our Eq. 共39兲兴.25 The fulfilling of these conditions does
not correspond to symmetries usually considered in mesoscopic systems; for example, as we will show, the presence oftime-reversal symmetry is not a necessary requirement. Instead, the conditions of Eqs. 共39兲 and 共40兲 correspond directly to the requirement that there be no missing information in the detector, information that is not revealed in themeasurement of 具 I 典 . We demonstrate this explicitly in what
follows.
1. Phase condition

The first condition 关Eq. 共39兲兴 for reaching the quantum
limit requires that the difference between transmission andreflection phases in the scattering matrix be constant in theenergy interval defined by the voltage. If this condition
holds, changing the state of the qubit will not modulate this

e2
␲

冕

␮L

␮R

冉

dE ⬘ T 共 1⫺T 兲

冊d共␤⫺␸兲 .
dE

共41兲

By definition, the imaginary part of this correlator determines the linear-response coefficient ␭ 关cf. Eq. 共3兲兴 associated with measuring 具 I 典 . In contrast, the real part of thiscorrelator may be interpreted as the linear-response coefficient associated with a measurement where one interferes
reflected and transmitted electrons; the factor of T(1⫺T)
corresponds to the fact that the magnitude of this signal will
be proportional to the amplitude of both the reflected andtransmitted beams. More explicitly, consider the Hermitian
operator I mod , defined by
I mod⫽

共38兲

where C is a real, energy-independent constant. Using Eq.
共33兲, the imaginary and real parts of the above condition
become᭙ E⑀关 ␮R ,␮L兴,

phase difference. Equation 共39兲 thus constrains
information—it ensures that the detector does not extract additional information about the qubit that resides in the relative phase between transmission and reflection. Such information is clearly not revealed in the measurement of 具 I 典 , andwould necessarily lead to additional dephasing over andabove the measurement rate. In principle, this additional information could be extracted by performing an interference
experiment. To be more specific, note that the cross correlator S IQ 关c.f. Eq. 共4c兲兴 is given by

e
ប

冕

␮L

␮R

dE 关 ia R† 共 E 兲 A RL 共 L;E,E 兲 a L 共 E 兲 ⫹H.c.兴 .
共42兲

If one were to now measure I mod , the corresponding linear
response coefficient ␭ mod is precisely the real part of S IQ
关this can be seen by comparing Eqs. 共42兲 and 共27兲兴. The factthat additional information on the state of the qubit is available in the expectation 具 I mod典 implies that the qubit is entangling with the detector faster than the measurement rate associated with 具 I 典 . This remains true even if one does not
explicitly extract this information, as was demonstrated recently in the experiment of Sprinzak et. al.22
Stepping back, we see that the general condition Re S IQ
⫽0 关i.e., the required factor of i on the RHS of Eq. 共16兲兴
needed to reach the quantum limit directly corresponds to therequirement of no missing information discussed in the preceding section. In general, a nonvanishing Re S IQ implies
that additional information about the qubit’s state could beobtained by simultaneously measuring another quantity inaddition to I 共e.g., in our case, the quantity I mod).
Note that in the scattering detector, the symmetry required
to ensure that Eq. 共39兲 holds 共i.e., that the phases ␤ and ␾
coincide兲 is not one that is usually considered in mesoscopic
systems. In particular, the presence of time-reversal symmetry is not necessary to fulfilling the condition of Eq. (39);time-reversal symmetry only implies that ␸ ⫽ ␸ ⬘ , and specifies nothing on the relation between ␸ and ␤ . However, as
pointed out in Ref. 6, a sufficient condition for achieving Eq.
共39兲 is that one has parity symmetry, that is both timereversal symmetry and left-right inversion symmetry 共the latter condition implies that the two reflection phases in s are
identical兲.26 Note that this is not a necessary condition. We
see that the required symmetry here is best understood as
being related to information.

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PHYSICAL REVIEW B 67, 165324 共2003兲

QUANTUM-LIMITED MEASUREMENT AND INFORMATION . . .
2. Transmission condition

We now turn to the second condition 关Eq. 共40兲兴 needed tohave the scattering detector reach the quantum limit, a condition that constrains the energy dependence of the transmission probability T. This condition arises from the requirement that the proportionality between I and Q needed for thequantum limit must hold over the entire energy interval defined by the voltage. In general, energy averaging causes a
departure from the quantum limit—over sufficiently large intervals, the operators I and Q look less and less like one
another. Like Eq. 共39兲, Eq. 共40兲 can also be interpreted as arequirement of no missing information. Here, the requirement is that energy averaging does not result in the loss of
information about the qubit that is encoded in the energy
dependence of T. While such information is not obtained in
the measurement of 具 I 典 关which involves energy averaging,cf. Eq. 共32兲兴, it could be obtained if one measured the entire
function 具 I(V) 典 for 0⭐ 兩 V 兩 ⭐ ␮ L ⫺ ␮ R . As discussed, thepresence of any missing information necessarily implies adeparture from the quantum limit.
Interestingly enough, Eq. 共40兲 may be understood completely classically, even though it formally results from requiring the proportionality of two quantum operators. To doso, we calculate the classical information capacity R 关cf. Eq.
共21兲兴 corresponding to two different possible measurements.
First, imagine we measure the integrated current m⫽ 兰 t0 dt ⬘ I(t ⬘ ), and assume the probability distributionsp(m 兩 ↑) and p(m 兩 ↓) are Gaussian. For weak coupling, one
finds for the capacity:t
R avg⫽⌫ meast⫽
2h

冉 冕
eA

冕

␮L

␮R

⯝

共 ␦␧ 兲t
2h

冉 兺
eA

j

␮L

␮R

d␧

dT 共 ␧ 兲d␧

冊

冊

兺j T 共 ␧ j 兲关 1⫺T 共 ␧ j 兲兴.

⌫ meas⬀

冉

共43兲

共44兲

冊

e 2 dTe2e兩V兩 Ⰶ ,
h dE
h

共46兲

冉 冊冉 冊

共47兲

␭⯝

2
dT
A A A
eV
Ⰶ .
dE
eV h hEven though this smallness of ␭ does not theoretically affectthe approach to the quantum limit, it does severely limit thedetector’s practical value—for very slow measurement rates,
environmental effects on the qubit will become dominant
over back-action effects.
If we now consider finite voltages and fully energydependent scattering, Eq. 共40兲 tells us the condition underwhich energy averaging the transmission does not impede
reaching the quantum limit. The solution to Eq. 共40兲 has theform
T共 E 兲⫽2d␧T 共 ␧ 兲关 1⫺T 共 ␧ 兲兴
dT 共 ␧ j 兲 2d␧How can the problems generally posed by energy averaging be avoided? One possible solution would be to use voltages small enough that the scattering matrix s can be approximated as being linear in energy, that is eV(dT/dE)
Ⰶ1 共this is the approach of Ref. 6兲. However, as the linearresponse coefficient ␭ is given by the energy derivative ofthe transmission 关cf. Eq. 共32兲兴, such a small voltage would
imply both a small signal and essentially no gain. Thechange in current induced by the qubit, ⌬I⫽⫾A␭, would bemuch smaller than the current associated with the coupling
voltage A:

1
1⫹e 4 ␲ (E⫺E 0 )/C

共48兲

.

This form for T(E) implies that there is no extra informationin the energy dependence of T which is lost upon energy
averaging. Amusingly, Eq. 共40兲 corresponds exactly to theenergy-dependent transmission of one channel of an adiabatic quantum point contact.7 The constant E 0 represents thethreshold energy of the channel 共i.e., the transverse mode兲,and the constant C is given byC⫽⫺

2 冑2ប v F

冑dR

共49兲

,

In the last line, we have discretized the energy integrals, i.e.,
partitioned the interval 关 ␮ R , ␮ L 兴 into equal segments oflength ␦ ␧. If we now imagine we could measure each m j⫽ 兰 t0 I j (t), where I j (t) is the contribution to the current fromthe jth energy interval, a similar calculation reveals

where d is the transverse width of the constriction at its center and R is the radius of curvature of the transverse confining potential at the constriction center.

冉

We now consider the situation where there are N channels
in each of the two contacts leading to the reservoirs. It isuseful to write s in terms of its N transmission eigenvalues
T j (E) using the standard polar decomposition:27

共 ␦␧ 兲t
R tot⫽
2h兺j

冊

dT 共 ␧ j 兲 2eAd␧.
T 共 ␧ j 兲关 1⫺T 共 ␧ j 兲兴

共45兲

One can easily check that R tot⭓R avg ; this corresponds to theadditional information that is generally available in the energy dependence of T. A necessary and sufficient condition
for ensuring R tot⫽R avg is precisely the condition in Eq. 共40兲.
On a purely classical level, this condition ensures that no
information is lost when one averages over energy.

B. Multichannel cases共 E 兲⫽

冉

s LL

s LR

s RL

s RR

冊 冉 冊 冉 冑冑
⫽

U

R

V

T

冑T
⫺ 冑R

冊冉 冊
U⬘
V⬘

. 共50兲

Here, U,U ⬘ ,V,V ⬘ are N⫻N energy-dependent unitary matrices, and 冑R and 冑T are diagonal matrices having entries
冑1⫺T j (E) and 冑T j (E), respectively.

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PHYSICAL REVIEW B 67, 165324 共2003兲

A. A. CLERK, S. M. GIRVIN, AND A. D. STONEIn the multichannel case, the backwards gain ␭ ⬘ again
vanishes irrespective of the details of s as a result of theanalytic properties of s. The relevant question then to ask iswhat conditions must be satisfied by s(E) so that the proportionality between I and Q required to reach the quantum limit
关i.e., Eq. 共16兲兴 is achieved. As in the single-channel case, therelevant matrix elements of I and Q involve destroying ascattering state incident from the left and creating an equalenergy state describing an incident wave from the right; theadditional complication now is that these transitions could
result in a change of transverse mode. One thus needs toexamine the coefficients of the operator products†(E)a Lm (E) appearing in the expansion of I and Q, in thea Rnenergy interval 关 ␮ R , ␮ L 兴 . The proportionality condition of
Eq. 共16兲 again yields the requirement that Eq. 共38兲 holds forall energies in this interval; now, however, both the right- andleft-hand sides of this equation are N⫻N matrices:᭙E ⑀ 关 ␮ R , ␮ L 兴 , 关 s LR 共 E 兲兴 † s LL 共 E 兲 ⫽iCNRL 共 E 兲 .

共51兲

Here, C is again an energy-independent real number. Using
the polar decomposition, one can derive from Eq. 共51兲 twonecessary matrix conditions that must hold for all energies in
the interval defined by the voltage:

冑T 共 E 兲 ␾ U 共 E 兲 冑R 共 E 兲 ⫺ 冑R 共 E 兲 ␾ V 共 E 兲 冑T 共 E 兲 ⫽0, 共52兲
dT
共E兲
dE
4␲
⫻1̂.
⫽⫺
T 共 E 兲关 1⫺T 共 E 兲兴C

共53兲These conditions are the multichannel analogs of Eqs. 共39兲and 共40兲. 1̂ denotes the N⫻N unit matrix, and we have introduced the generalized ‘‘phase-derivative’’ Hermitian matrices ␾ U and ␾ V

␾ U 共 ␧ 兲 ⫽⫺iU † 共 ␧ 兲␾ V 共 ␧ 兲 ⫽⫺iV † 共 ␧ 兲

冋
冋

册
册

d
U共 ␧ 兲 ,
dE

共54兲

d
V共 ␧ 兲 .
dE

共55兲

These matrices play the role of the energy derivatives of thephases ␤ and ␾ in the single-channel case. Note the evident
asymmetry in Eq. 共52兲: the polar decomposition matrices U
and V enter, but the matrices U ⬘ and V ⬘ do not. We commenton this in what follows.
1. Phase and channel mixing conditions

The first requirement 关Eq. 共52兲兴 places a stringent requirement on the scattering matrix s. Like the corresponding requirement for the single-channel system, it ensures that there
is no additional information on the state of the qubit available in measurable changes of scattering phases. Again, timereversal symmetry is not necessary to have this condition
hold, as time-reversal symmetry only ensures U⫽U ⬘ andV⫽V ⬘ . However, unlike the single-channel case, even thepresence of parity symmetry 共i.e., the combination of bothtime-reversal symmetry and left-right inversion symmetry兲 is

not sufficient to guarantee that Eq. 共52兲 is satisfied. The presence of parity symmetry would indeed ensure ␾ U ⫽ ␾ V , but
as, in general, 关 冑T, ␾ U 兴 and 关 冑R, ␾ U 兴 ⫽0, this is notenough. In addition to having ␾ U ⫽ ␾ V , one also generally
needs either that ␾ U is diagonal, meaning that the mode
index 共i.e., transverse momentum兲 is conserved during scattering, or that all the transmission eigenvalues T j are identical. We thus see that if the transmissions fluctuate, mode
mixing 共e.g., the nonconservation of transverse energy兲 also
prevents one from reaching the quantum limit of detection.
This can be understood from the point of view of information. If ␾ U and ␾ V matrices are not purely diagonal, information about the qubit could be gained by looking atchanges in how electrons incident in a given mode are partitioned into outgoing modes. Such changes would not be
detectable if all channels had the same transmission. Notethat the matrices U ⬘ and V ⬘ appearing in the polar decomposition of s 关Eq. 共50兲兴 are irrelevant to reaching the quantum limit. As each transverse mode is equally populated withincoming waves in the state 兩 i 典 , there is no information associated with the preferred mode structure for incoming
waves 共i.e., the eigenvectors of U ⬘ and V ⬘ ).
2. Transmission condition

Consider now the condition imposed by Eq. 共53兲, whichconstrains the form of the transmissions T j (␧) of the detector. Similar to the corresponding condition for the singlechannel system, this requirement ensures that there is no additional information available in either the energy or thechannel structure of the 兵 T j (␧) 其 which is lost upon averaging. One obtains a necessary form for the transmissions,similar to what was found in Ref. 6:

T j共 E 兲⫽11⫹e4 ␲ (E⫺E j )/C

.

共56兲

Note that different modes differ from one another only by
their threshold energy E j ; the constant C is the same for eachmode. Again, this form for the transmissions 兵 T j (␧) 其 corresponds exactly to those expected for a multichannel adiabatic
point contact.7 The assumption of adiabaticity implies thattransverse energy is conserved. Thus, if parity symmetry also
holds, we reach the surprising conclusion that a multichannel
adiabatic point contact remains a quantum limited detector
even if the voltage is large enough that several modes contribute to transport. Previous studies have established thatpoint-contact detectors reach the quantum limit in the limitof small voltages, where the energy dependence of scattering
can be neglected.8 –10 We have shown here that in the adiabatic case, the quantum limit continues to hold even at voltages large enough that the energy dependence of scattering isimportant. This is significant from a practical standpoint—
requiring small voltages limits the magnitude of the output
current and thus, the overall scale of the measurement rate,
making the detector more susceptible to environmental effects.

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PHYSICAL REVIEW B 67, 165324 共2003兲

QUANTUM-LIMITED MEASUREMENT AND INFORMATION . . .
3. General expression for noise correlators

For completeness, we give explicit expressions for thenoise correlators. Writing them in terms of energy dependent␮N⫻N matrix kernels „ i.e., S X ⫽ 兰 ␮ L d␧ 关 tr Ŝ X (␧) 兴 …, we ob-

ing for a wider class of input operators Q than that given in
Eq. 共29兲. In general, we may write
Q⫽e

R

tain
2

Ŝ I 共 ␧ 兲 ⫽Ŝ Q 共 ␧ 兲 ⫽2e
T 共 1⫺T 兲 ,
h

共57a兲

冉

e 2ប 共 ⳵ ␧T 兲2⫹2TR 共 ␾ U ⫺ ␾ V 兲 2
2 ␲ 2T 共 1⫺T 兲⫹2 关 ␾ U , 冑TR 兴关 冑TR, ␾ V 兴 ⫹ 关 ␾ U ,T 兴关 T, ␾ U 兴冊⫹ 关 ␾ V ,T 兴关 T, ␾ V 兴 ,共57b兲

Ŝ IQ 共 ␧ 兲 ⫽iប␭ˆ 共 ␧ 兲 ⫹e
共 ⳵ T 兲,h ␧e2
关 TR 共 ␾ U ⫺ ␾ V 兲兴 .
␲

共57c兲
共57d兲

ប2 d
I共 t 兲 .
2
A→0 t→0 A dt

共58兲

While this last expression may seem purely tautological, it isclear that the various contributions to Eq. 共57b兲 for the
charge noise are best understood in terms of information.
Note that the accessible information I could be obtained directly in the present system by calculating the overlap between the detector states corresponding to the two qubit
states. Such a calculation would take the form of an orthogonality catastrophe calculation, similar to that presented inRef. 29.
C. Local potential coupling

In the remaining part of this paper, we consider a moregeneral version of the mesoscopic scattering detector, showing that the main results of the preceding section continue tohold. We relax the condition that the state of the qubit modulates a uniform potential in the scattering region, thus allow-兺␤ , ␥ ⫽L,R

᭙E ⑀ 关 ␮ R , ␮ L 兴 ,A similar expression for the charge noise S Q of a mesoscopic
conductor was first derived by Büttiker.28 Unlike the expression for the current noise S I , which can easily be understoodin terms of partition noise, it would seem at first that there isno simple, heuristic way to interpret the expression for S Q .
However, if we invoke ideas of information, each term in Eq.
共57b兲 acquires a simple meaning. The first term represents
information associated with the energy dependence of thetransmissions; the second represents information associatedwith the energy dependence of phase differences; and the last
three terms represent information associated with the partitioning of electrons into different modes. In general, using
Eqs. 共5兲 and 共25兲, we may define the charge noise in terms of
the accessible information I in the coupled conductor plus
qubit system:
S Q ⫽ lim lim

dEdE ⬘

关 a ␤† n 共 E 兲 W ␤ n, ␥ m 共 E,E ⬘ 兲 a ␥ m 共 E ⬘ 兲兴 ,共59兲where W(E,E ⬘ ) is a 2N⫻2N Hermitian matrix having dimensions of inverse energy. The situation considered in thepreceding section corresponds to choosing W to be N(E,E ⬘ )关Eq. 共30兲兴, which at E⫽E ⬘ is just the Wigner-Smith delaytime matrix. By comparing against the current operator I 关cf.
Eq. 共27兲兴, it is clear that the proportionality condition in Eq.
共16兲 necessary for the quantum limit constrains the diagonalin energy, off diagonal in lead index part of the potential
matrix W:

2

␭ˆ 共 ␧ 兲 ⫽⫺

冕1
关 W 共 E,E 兲兴 RL ⫽i 关 s LR 兴 † 共 E 兲 s LL 共 E 兲 ,C
共60兲

where C is a real constant. We thus see that the required
proportionality between I and Q needed to reach the quantum
limit at zero temperature leaves a large part of the potential
matrix W undetermined 共i.e., terms diagonal in the lead index
and/or off diagonal in energy兲. We now show that by considering a form for W that is drastically different from N, one
can make it easier to reach the quantum limit and have areasonable gain. In particular, one can work at small voltages
without necessarily having a vanishing gain.
We specialize the discussion to a case that in many ways
is the opposite of having global potential coupling. We takethe scattering matrix s to be energy independent over theenergy interval defined by the voltage, and take W to correspond to a local potential W(E,E ⬘ )⫽W over the energies ofinterest. In this case, the scattering matrix s will have one oftwo different energy-independent values depending on thestate of the qubit:s ⫾ ⫽s 0 ⫾eA 共 ⌬s 兲 ,共61兲where s 0 is the scattering matrix at zero coupling (A⫽0).
The matrix W may be directly related to the change in the
scattering matrix, ⌬s 共see Appendix B for a derivation兲:W⫽is †0 共 ⌬s 兲 .

共62兲Note the similarity to the form of W in the global-potential
coupling case 共where W⫽N); now, the energy derivativeds/dE has been replaced by the finite difference ⌬s⬅(s ⫹⫺s ⫺ )/(2eA).
Turning to the conditions needed for the quantum limit,
we find again that the causality properties of the scatteringmatrices s ⫾ ensure ␭ ⬘ ⫽0 always. The remaining proportionality requirement of Eq. 共16兲 places constraints on s ⫾ . These
have an analogous form to Eqs. 共53兲 and 共52兲, but now theenergy derivative d/dE is replaced by the finite difference ⌬„i.e., ⌬X⫽(X 关 s ⫹ 兴 ⫺X 关 s ⫺ 兴 )/(2eA)…:

165324-9⌬T⫽C⫻1̂,
T 共 1⫺T 兲

共63兲

PHYSICAL REVIEW B 67, 165324 共2003兲

A. A. CLERK, S. M. GIRVIN, AND A. D. STONE˜ U 冑R⫺ 冑R ␾˜ V 冑T⫽0,冑T ␾

˜ U ⫽⫺iU † (⌬U) and ␾˜ V ⫽⫺iV † (⌬V) . Importantly,
where ␾the above conditions do not involve any energy averaging, as
we have taken s and W to be energy independent. Nonetheless, there still is a nonvanishing gain ␭ determined by boththe voltage and the ⌬T j :
␭⫽

e 2V
h

兺j ⌬T j .

APPENDIX A: ACCESSIBLE INFORMATION

共64兲

共65兲

Thus, using a local coupling between the qubit and the scattering detector makes it easier to reach the quantum limit andhave a sizeable gain—one can use voltages small enoughthat energy averaging is not a problem, while still having thequbit modulate the transmissions. Note that in the singlechannel case, all that is needed for the quantum limit is thatthe state of the qubit should not change the difference between reflected and transmitted phases: ⌬( ␾ ⫺ ␤ )⫽0. Also
note the various noise correlators are given by Eqs. 共57兲,with the substitution d/dE→⌬.
IV. CONCLUSIONS

We have developed a general set of conditions that areneeded for a detector in the linear-response regime to reachthe quantum limit of detection. One needs both a restrictedproportionality between the input and output operators of thedetector 关cf. Eq. 共16兲兴, and a causal relation between the
output and input 关cf. Eq. 共17兲兴. Applying the concept of accessible information to the detector, one sees that deviations
from the quantum limit imply the existence of missing information residing in the detector, information that is not being
utilized. The general conditions in Eqs. 共16兲 and 共17兲 ensure
the nonexistence of such information. Applying these concepts to the mesoscopic scattering detector, we find that thesegeneral conditions place restrictions on the form of the detector’s scattering matrix. These restrictions do not involve
symmetry properties usually considered in mesoscopic systems, but are rather best understood as following from therequirement of having no missing information. In the mesoscopic scattering detector, missing information may reside in
the relative phase between transmission and reflection, in theenergy or mode structure of the transmission probabilities, orin the partitioning of scattered electrons between different
modes. Surprisingly, we find that an adiabatic point contact
conforms to all the conditions needed for the quantum limit,
even when the voltage is large enough that many modes are
involved in transport, and the energy dependence of scattering is important.
ACKNOWLEDGMENTS

We thank M. Devoret for useful conversations. This work
was partially supported by ARDA through the Army Research Office under Grant No. DAAD19-02-1-0045, by the
NSF under Grants Nos. DMR-0084501 and DMR-0196503,
and by the W. M. Keck Foundation.

In this appendix, we provide a simple proof of Eq. 共24兲for the accessible information I. Given the two states 兩 D ↑ 典
and 兩 D ↓ 典 , the goal is to maximize the classical mutual information R 关defined in Eq. 共21兲兴 over all possible choices ofmeasurements. A given choice of measurement Y corresponds to a choice of basis; the probability distributionsp(y i 兩 ↑) and p(y i 兩 ↓) are determined by the elements of thecorresponding states in this basis. Treating the p(y i 兩 ␴ ) as
independent variables restricted to the interval 关 0,1兴 and using Lagrange multipliers, we minimize R subject to the following constraints:N兺 p 共 y i兩 ␴ 兲 ⫽1,

共A1兲兺 冑p 共 y i兩 ↑ 兲 p 共 y i兩 ↓ 兲 ⫽ 兩 具 D ↑兩 D ↓ 典 兩 ⬅cos ␣ .

共A2兲

i⫽1
N

i⫽1

The second condition, in principle, need only be an inequality, with the left-hand side being greater than or equal to theright-hand side; however, it can be verified that the maximum value of R occurs when it is enforced as an equality.
Also note that without loss of generality, we can choose theinner product appearing in Eq. 共A2兲 to be real and positive,
as R is independent of the relative phase between the states兩 D ␴ 典 . Finally, we have assumed to start that these states haveat most N nonzero components in the chosen basis. Variation
with respect to p(y i 兩 ↑) yields the following condition:ln

p 共 y i兩 ↑ 兲p̄ 共 y i 兲⫹2␭ ↑ ⫹␭

冑

p 共 y i兩 ↓ 兲⫽0,
p 共 y i兩 ↑ 兲

共A3兲

with a similar equation emerging from variation with respect
to p(y i 兩 ↓). ␭, ␭ ↑ , and ␭ ↓ are Lagrange multipliers; p̄(y i )⫽ 关 p(y i 兩 ↑)⫹p(y i 兩 ↓) 兴 /2 is the averaged distribution. Subtracting the ↑ and ↓ equations yields
␭⫽冑p 共 y i 兩 ↓ 兲 p 共 y i 兩 ↑ 兲
p 共 y i兩 ↑ 兲 ⫺ p 共 y i兩 ↓ 兲ln

p 共 y i 兩 ↑ 兲 冑1⫺ ␤ 2i 1⫹ ␤ i⫽ln,
p 共 y i兩 ↓ 兲2␤i1⫺ ␤ i
共A4兲

where we have defined ␤ i via

␤ i⫽p 共 y i兩 ↑ 兲 ⫺ p 共 y i兩 ↓ 兲p̄ 共 y i 兲

.

共A5兲

␤ i may be thought of as the amount of information gained ina measurement, given that the outcome of the measurement
is y i . Now, Eq. 共A4兲 must hold for each ␤ i (i⫽1, . . . ,N);moreover, the function on the right-hand side is symmetric in␤ i and monotone decreasing for 0⭐ ␤ i ⭐1. It thus follows
that for each i,

165324-10

共 ␤ i 兲 2 ⫽constant⫽sin2 ␣ .

共A6兲PHYSICAL REVIEW B 67, 165324 共2003兲

QUANTUM-LIMITED MEASUREMENT AND INFORMATION . . .

The last equality follows from substitution into Eq. 共A2兲.
Further substitution into Eq. 共21兲 for R yields the expressionin Eq. 共24兲; note that the averaged distribution p̄(y i ) and therelevant number of basis elements N do not appear in thisexpression. One can explicitly check that choosing any of thep(y i 兩 ␴ ) to be 0 or 1 results in a lower value of R; thus, Eq.
共24兲 does indeed correspond to the maximum value of R andthus, by definition, to the accessible information I. The condition 共A6兲 required to optimize R implies that the amount of
information gained via measurement is the same for each of
the measurement outcomes y i . Equivalently, each basis element in an optimal basis has the same information content
associated with it. This is similar to requirements obtained tohave the mesoscopic scattering detector reach the quantum
limit; in that case, each channel and each energy were required to have the same information content 关cf. Eq. 共53兲兴.
Note also that there are several distinct choices of bases (i.e.,measurement schemes) which optimize R; this point was notmade in Ref. 14. A particularly simple optimal basis can beconstructed for N⫽2. In this basis, the nonzero componentsof the states 兩 D ␴ 典 are given by兩 D ↑ 典 ⫽ 共 cos ␪ ,sin ␪ 兲 ,兩 D ↑ 典 ⫽ 共 sin ␪ ,cos ␪ 兲 ,共A7兲

where ␪ ⫽ ␲ /4⫹ ␣ /2. By definition, the state (1,0) leads tothe measurement outcome y 1 with perfect certainty, while
the state (0,1) leads to the measurement outcome y 2 withperfect certainty. In geometric terms, the optimal basis given
here is one in which the angle between the two states 兩 D ␴ 典 isbisected by the vector (1,1).
More generally, consider the form of an optimal basis
where N⫽M 共i.e., there are M possible outcomes when ameasurement is made on the state 兩 D ↑ 典 or 兩 D ↓ 典 ). Taking M
to be even for simplicity, and letting 兩 j 典 denote the basis
states, a possible optimal basis is one in which冑典 冑

具 j 兩 D ↑典 ⫽1⫹ 共 ⫺1 兲 j sin ␣
,
M

共A8兲

具 j兩D↓ ⫽1⫺ 共 ⫺1 兲 j sin ␣
.
M

共A9兲

The fact that there are many possible outcomes of a measurement does not degrade from the optimality of mutual information R, as the information associated with each measurement outcome is the same.
APPENDIX B: DERIVATION OF ⌬s

In this appendix, we provide a brief derivation of Eq. 共62兲
which relates the coupling potential matrix W 关cf. Eq. 共59兲兴
to the associated change in the scattering matrix, ⌬s. The
latter quantity determines the noise correlators and gain of
the local-potential coupling version of the mesoscopic scattering detector. Our approach is similar to that used in Ref.
30 to relate the scattering matrix of a quantum dot to its
Hamiltonian.
In what follows, we assume 共as in Sec. II B兲 that the potential matrix W and the zero-coupling scattering matrix s are
independent of energy on the scales of interest. We start by

writing the system Hamiltonian in terms of the scattering
states of problem at zero coupling, assuming the qubit isfrozen in the ↑ state:H⫽ប v F再兺m 冕 dk k ␺ m† 共 k 兲 ␺ m共 k 兲⫹ 共 Ae 兲冎†dk ⬘ 关 ␺ m ⬘ 共 k ⬘ 兲 W m ⬘ m ␺ m 共 k 兲兴 . 共B1兲
冕
兺
m

⬘

We have assumed a linear dispersion near the Fermi energy,
with បk and បk ⬘ representing the deviation of the momentum from the Fermi momentum. We have also neglected thefact that the effective Fermi velocity is channel dependent†( v F drops out of all final expressions兲. The operator ␺ m(k)creates a scattering state incident in the lead and transverse
mode indexed by m. For definiteness, we take our leads 共bothleft and right兲 to be defined only on the half line x⬍0, and to
be confined in the y and z directions. Further, we assume thatthe scattering region is situated on x⬎0. We may write thefull electron field operator in terms of the ␺ m (k) operators,
using the zero-coupling scattering matrix s. Writing xជ
⫽(x,y,z), we have:⌿ 共 xជ 兲 ⫽

dk

⫹

⫽

冋

兺m 冕 冑4 ␲ ␺ m共 k 兲 e i(k ⫹k)x ␾ m共 y,z 兲兺n e ⫺i(k ⫹k)x ␾ n共 y,z 兲 s nm
F

1

m
冑2 兺

⫹

F

1

册

共B2兲

␺ m 共 ⫺x 兲 e ik F x ␾ m 共 y,z 兲

m,n
冑2 兺

␺ m 共 x 兲 e ⫺ik F x ␾ n 共 y,z 兲 s nm .

共B3兲In the last line, we have introduced the operators ␺ m (x),which are the Fourier transforms of the scattering state operators ␺ m (k). Note again that this expression is only valid
for x⬍0, as the leads are only defined on x⬍0. We thus seethat for x⬍0, ␺ m (x) describes an outgoing 共i.e., leftmoving兲 wave, while ␺ m (⫺x) describes an incoming 共i.e.,right-moving兲 wave.
Next, we may express the system Hamiltonian in terms ofthe ␺ m (x) operators. This in turn leads to an equivalent
single-particle Schrödinger equation冋E ˜␺ m 共 E,x 兲 ⫽ប v F i ⳵ x ˜␺ m 共 E,x 兲 ⫹Ae ␦ 共 x 兲册兺n W mn ˜␺ n共 E,x 兲 .
共B4兲

Here, ˜␺ m (E,x) is a wave function that arises when the field
operator ␺ m (x) is expressed in terms of operators corresponding to the eigenmodes of the full Hamiltonian H. Given
the relation of ␺ m (x) to incoming and outgoing waves 关cf.
Eq. 共B3兲兴, we choose the following form for ˜␺ m (x):165324-11

PHYSICAL REVIEW B 67, 165324 共2003兲

A. A. CLERK, S. M. GIRVIN, AND A. D. STONE

˜␺ m 共 E,x 兲 ⫽再e ⫺ikx a in,me ⫺ikxif x⬎0†a out,n if x⬍0,
兺n s mn

共B5兲

where E⫽ប v F k. Substituting this form into Eq. 共B3兲, we seethat the coefficients a in,m and a out,m do indeed correspond
共respectively兲 to the amplitudes of incoming and outgoingwaves.
Integrating Eq. 共B4兲 from x⫽0 ⫺ to x⫽0 ⫹ , interpreting˜␺ (0) as 关 ˜␺ (0 ⫹ )⫹ ˜␺ (0 ⫺ ) 兴 /2, and then using Eq. 共B5兲, wefind the following relation between the amplitude of incoming and outgoing waves:

1

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Note that in Ref. 2, the quantum limit is written as ␹⬍2. This
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the one used here.
12
The proportionality condition expressed in Eq. 共16兲 only involves
a subset of matrix elements, and is thus not incompatible withthe operators Ĩ and Q̃ being Hermitian. In the case where thestate 兩 f 典 ⫽ 兩 i 典 in this equation, one simply needs that the corresponding matrix elements of Ĩ and Q̃ vanish.
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a out,m ⫽兺 s mn

n,n ⬘

⬅

冋 册
i
1⫺ AeŴ2i
1⫹ AeŴ2

a in,n ⬘nn ⬘关 s⫹Ae⌬s 兴 mn ⬘ a in,n ⬘ .
兺
n
⬘

共B6兲

共B7兲

In the last line, we indicate that this relation defines the new
scattering matrix s⫹Ae⌬s that includes effects of the additional potential W. Expanding to lowest order in the dimensionless potential AeW, we find Eq. 共62兲 as advertised.

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More generally, there will be two different detector density matrices ␳ D↑ , ␳ D↓ corresponding to the two qubit states; we consider the case of pure states for simplicity.
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See, e.g., T.M. Cover and J.A. Thomas, Elements of Information
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Note that our Eq. 共40兲 has a constant on the right-hand side, while
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26Note that by left-right symmetry, we mean that the scalar potential V(x,y,z) in the scattering region is a symmetric function ofx, where x is the transport direction. One can still have an absence of time-reversal symmetry if, e.g., there is a vector potential in the scattering region.
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17165324-12