Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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denotes the norm of linear operators, and

k%k

:= tr

ﬃﬃﬃﬃﬃﬃﬃ



is the trace norm on the space of

trace-class operators B



(H).

We use base two logarithms throughout, and we

write ld x := log

x and exp

x := 2

Quantum Channel Capacity

The intuitive concept underlying quantum channel

capacity is made rigorous in the following

definition:

Definition 1 A positive number R is called achiev-

able rate for the quantum channel T : A!B with

respect to the quantum channel S : A

iff for any

pair of integer sequences (n



)

2N

and (m



)

2N

with

lim

 !1



= 1 and lim

 !1



R we have

lim

!1

inf

D;E

kDT

n



E  S

m



¼ 0 ½1

the infimum taken over all encoding channels E and

decoding channels D with suitable domain and

range. The channel capacity Q(T, S)ofT with

respect to S is defined to be the supremum of all

achievable rates. The quantum capacity is the special

case Q(T):= Q(T,id

), with id

being the ideal

qubit channel.

In this article, we mainly concentrate on

channels between finite-dimensional systems. This

is enough to bring out the basic ideas. Many of the

concepts and results discussed here can be general-

ized to Gaussian channels, which play a central

role as building blocks for quantum optical

communication lines (Holevo and Werner 2001,

Eisert and Wolf).

There is considerable freedom in the definition

of quantum channel capacity, at least for ideal

reference channels (Kretschmann and Werner

2004). In particular, the encoding channels E in

eqn [1] may always be restricte d to isometric

embeddings.

In addition, it is not necessary to check an infinite

number of pairs of sequences (n



)

2N

and (m



)

2N

when testing a given rate R,asDefinition 1 would

suggest. Instead, it is enough to find one such pair

which achieves the rate R infini tely often,

lim

 !1



= R.

Without affecting the capacity, the cb-norm kTk

may be replaced by the unstabilized operator norm

kTk or by fidelity measures, which are in general

much easier to compute. In particular, one might

choose the minimum fidelity,

FðTÞ:¼ min

k k¼1

h jTðj ih jÞj i½2

or even the average fidelity,



FðTÞ:¼

h jTðj ih jÞj id ½3

Unfortunately, this equivalence is restricted to

capacities with noiseless reference channel S = id.

In the vicinity of other (nonideal) channels, equiva-

lence of the stabilized and unstabilized error criteria

may be lost. Of course, the comparison of channels

is ultimately based on the comparison of a state to

its image, and here the pure states are the worst

case. Hence, the remarkable insensitivity of the

quantum capacity to the choice of the error criterion

stems from the observation that the compar ison

between an arbitrary state and a pure state is rather

insensitive to the criterion used.

Instead of requiring the error quantity in eqn [1] to

approach zero in the large block limit  !1, one

might feel tempted to impose that the errors vanish

completely for some sufficiently large block length,

since this is the standard setup in the theory of

quantum error correction (see Quantum Error Correc-

tion and Fault Tolerance). While it is true that errors

can always be assumed to vanish exponentially in eqn

[1], requiring perfect correction may completely change

the picture: if a channel has some small positive

probability for depolarization, the same also holds for

its tensor powers, and no such channel allows the

perfect transmission of even one qubit. Hence, the

capacity for perfect correction will vanish for such

channels, while the standard capacity (in accordance

with Definition 1) will be close to maximal, Q(T) 1.

The existence of perfect error-correcting codes thus

gives lower bounds on the channel capacity, but is not

required for a positive transfer rate.

In the other extreme, one might sometimes feel

inclined to tolerate (small) finite errors in the

transmission. For some ">0, we define Q

(T)

exactly like the quantum capacity in Definition 1,

but require only that the error quantity in eqn [1]

falls below " for some sufficiently large .

Obviously, Q

(T)  Q(T) for any quantum

channel T. We also have lim

" !0

(T) = Q(T)

(Kretschmann and Werner 2004). In the classical

setting, even a strong converse is known: if ">0is

small enough, one cannot achieve bigger rates by

allowing small errors, that is, C

(T) = C(T). It is still

undecided whether an analogous prop erty holds for

the quantum capacity Q(T).

Related Capacities

This article is chiefly concerned with the quantum

capacity of a quantum channel. A variety of other

Capacity for Quantum Information 425

capacities have been derived from Definition 1 by

either amending the channel S to be simulated, or

allowing Alice and Bob to make use of additional

resources. Their interrelations are reviewed in Bennett

et al. (2004)

Much interest has been devoted to the hybrid

problem of transmitting classical information undis-

torted over noisy quantum channels. The classical

capacity C(T)ofaquantumchannelT is discussed in

the article Quantum Channels: Classical Capacity of

this Encyclopedia. It is obtained by choosing the ideal

one-bit channel rather than the one-qubit channel as

the standard of reference in Definition 1.Encoding

channels E and decoding channels D are then

restricted to preparations and measurements, respec-

tively. Since a quantum channel can also be employed

to send classical information, we have C(T)  Q( T).

There are, obviously, examples in which this

inequality is strict: the entanglement-breaking channel

T(%) =

hjj%jjijjihjj is composed of a measurement

in the orthonormal basis {jji}

, followed by a prepara-

tion of the corresponding basis states. It destroys all

the entanglement between the sender and a reference

system, implying Q(T) = 0. Yet all the basis states jji

are transmitted undistorted, which is enough to

guarantee that C(T) = 1.

Definition 1 also applies to purely classi cal

channels, and thus to the setting of Shannon’s

information theory. A classical channel T between

two d-level systems is completely specified by the

d  d matrix (T

)

x, y = 1

of transition probabilities.

For these channels the cb-norm difference is just

(twice) the maximal error prob ability:

kid  Tk

= 2 sup

{1  T

}

which is the standard error criterium for classical

information transfer.

Dense coding and teleportation suggest that

entanglement is a powerful resource for information

transfer. It doubles the classical channel capacity of

a noiseless channel, and it allows to send quantum

information over purely classical channels. Surpris-

ingly, the entanglement-assisted capacities are often

simpler and better behaved than their unassisted

counterparts. Unlike the classical and quantum

capacities proper, they are relatively easy to calcu-

late using finite optimization procedures, and there

has recently been significant progress in under-

standing the simulation rates for nonideal channels

in this scenario (see Capacities Enhanced by

Entanglement).

The quantum channel capacity is unaffected by

entanglement-breaking side channels. In particular,

classical forward communication alone cannot

enhance it. However, unlike in the purely classical

case, both the quantum and classical channel

capacity (but not the entanglement-assisted capacity)

may increase under classical feedback.

Elementary Properties

The capacity of a composite channel T

 T

cannot

be bigger than the capacity of the channel with the

smallest bandwidth. This in turn suggests that

simulating a concatenated channel is in general easier

than simulating any of the individual channels. These

relations are known as bottleneck inequalities:

QðT

T

; SÞ minfQðT

; SÞ; QðT

; SÞg ½4

QðT; S

S

ÞmaxfQðT; S

Þ; QðT; S

Þg ½5

Instead of running T

and T

in succession, we may

also run them in parallel. In this case, the capacity

can be shown to be superadditive,

QðT

T

; SÞQðT

; SÞþQðT

; SÞ½6

For the standard ideal channels, we even have

additivity. The same holds true if both S and one

of the channels T

, T

are noiseless, the third

channel being arbitrary. However, results on the

activation of bound-entangled states seem to suggest

that the inequality in eqn [6] may be strict for some

channels (see Entanglement).

Finally, the two-step coding inequality tells us that

by using an intermediate channel in the coding

process we cannot increase the transmission rate:

QðT

; T

ÞQðT

; T

ÞQðT

; T

Þ½7

Applying eqn [7] twice with T

= id and T

= id

immediately yields upper and lower bounds on the

channel capacity with nonideal reference channel,

QðT

 QðT

; T

ÞQðT

ÞQðid; T

Þ½8

The evaluation of the lower bound in eqn [8] then

requires efficient protocols for simulating a noisy

channel T

with a noiseless resource.

There are special cases in which the quantum

channel capacity can be evaluated relatively easily,

the most relevant one being the noiseless channel id

where by the subscript n we denote the dimension of

the underlying Hilbert space. In this case, we have

Qðid

; id

Þ¼

ld n

ld m

½9

The lower bound Q(id

,id

)  ldn=ldm is immedi-

ate from counting dimensions. To establish the

upper bound, we use the fact that a noiseless

quantum channel cannot simulate itsel f with a rate

426 Capacity for Quantum Information

exceeding unity: Q(id

,id

) 1. This is just the

upper bound we want to prove for the special case

n = m, and it can be extended to the general case

with the help of the two-step coding inequality [7]:

Q(id

,id

) Q(id

,id

) Q(id

,id

) 1, implying

Q(id

,id

) 1=Q(id

,id

) ld n=ld m, where in the

last step we have applied the lower bound with the

roles of n and m interchanged.

Combining eqn [9] with the two-step coding

inequality [7], we see that for any channel T

QðT; id

Þ¼

ld m

ld n

QðT; id

Þ½10

which shows that quantum channel capacities relative

to noiseless channels of different dimensionality only

differ by a constant factor. Fixing the dimensionality

of the reference channel then only corresponds to a

choice of units. Conventionally, the ideal qubit

channel id

is chosen as a standard of reference, as

in Definition 1 above, thereby fixing the unit ‘‘bit.’’

The upper bound on the capacity of ideal channels

can also be obtai ned from a general upper bound on

quantum capacities (Holevo and Werner 2001),

which has the virtue of being easily calculated in

many situations. It involves the transposition map,

which we denote by , defined as matrix transposi-

tion with respect to some fixed orthonormal basis.

The transposition is positive but not completely

positive, and thus does not describe a physical

channel (see Channels in Quantum Information

Theory). We have kk

= d for a d-level system.

For any channel T and small ">0,

QðTÞQ

ðTÞld k Tk

¼: Q



ðTÞ½11

where Q

is the finite error capacity introduced in

the section ‘‘Quantum channel capacity.’’

The upper bound Q



(T) has some remarkable

properties, which make it a capacity-like quantity in

its own right. For example, it is exactly additive,



ðS TÞ¼Q



ðSÞþQ



ðTÞ½12

for any pair S, T of channels, and it satisfies

the bottleneck inequality:



ðSTÞ min{Q



ðSÞ; Q



ðTÞ}

Moreover, it coincides with the quantum capacity on

ideal channels, Q



(id

) = Q(id

) = ld n, and it vanishes

whenever T is completely positive. In particular, if

id T maps any entangled state to a state with positive

partial transpose, we have Q



(T) = 0.

State–Channel Duality

Quantum capacity is closely related to the distillable

entanglement, which is the optimal rate m/n at

which n copies of a given bipartite quantum state %

shared betw een Alice and Bob can be asymptotically

converted into m maximally entangled qubit pairs

(see Entanglement). Similar to the quantum capa-

city, the definition involves the large block limit

n, m !1 and an optimization over all conceivable

distillation protocols. These may consist of several

rounds of local quantum operations and (forward or

two-way) classical communication. The one-way

and two-way distillable entanglement of % will be

denoted by D

(%) and D

(%), respectively.

Suppose that Alice and Bob are connected by a

quantum channel T and run such a one-way distilla-

tion protocol on (many copies of) the state

:= (T id)jihj,whereji:= (1=

ﬃﬃﬃﬃﬃﬃ

)

ji, ii

is maximally entangled on H

H

.Ifthedistillation

yields maximally entangled qubits at positive rate R,

Alice may apply the standard teleportation scheme to

send arbitrary quantum states to Bob undistorted at

that same rate R. Like the distillation protocol itself,

teleportation requires classical forward communica-

tion, which however does not affect the channel

capacity (cf. the section ‘‘Related capacities’’). Thus,

Q(T)  D

). If two-way distillation is allowed, we

have Q

(T)  D

) for the capacity Q

(T)assisted

by two-way classical side communication.

Conversely, if Alice and Bob use a bipartite

quantum state % shared between them as a substitute

for the maximally entangled state ji in the

standard teleportation protocol, they will implement

some noisy quantum channel T

. If this channel

allows to transfer quantum information at nonvan-

ishing rate R, Alice may share maximall y entangled

states with Bob at that same rate R. Consequently,

(%)  Q(T

) and D

(%)  Q

These relations (Bennett et al. 1996) allow to

bound channel capacities in terms of distillable

entanglement and vice versa. If the two maps

T 7!%

and % 7!T

are mutually inverse, we even

have D

(%) = Q(T

) and D

(%) = Q

). In this

case, the duality % ÐT

is the physical implementa-

tion of Jamiolkowski’s isomorphism between bipar-

tite states and channels (see Channels in Quantum

Information Theory). This has been shown

(Horodecki et al. 1999) to hold for isotropic states,

which are invariant under the group of all

U U

transformations, where

U is the complex conjugate

of the unitary U. The corresponding channels are

partly depolarizing.

In general, T

6¼T. However, the so-called con-

clusive teleportation allows us to implement T at

least probabilistically, resulting in the relation

QðTÞD

ð%

ÞQðTÞ½13

Capacity for Quantum Information 427

The duality [13] can be applied to show that both

the unassisted and the two-way quantum capacities

are continuous in any open set of channels

having nonvanishing capacities (Horodecki and

Nowakowski 2005).

Coding Theorems

Computing channel capa cities straight from Defini-

tion 1 is a tricky business. It involves optimization in

systems of asymptotically many tensor factors, and

can only be performed in special cases, like the

noiseless channels in the section ‘‘Elementary prop-

erties.’’ Coding theorems aspire to reduce this

problem to an optimization over a low-di mensional

space. They usually come in two parts: the converse

provides an upper bound on the channel capacity

(typically in terms of some entropic expression),

while the direct part consists of a coding scheme

that attains this bound. By Shannon’s celebrated

coding theorem, the classical capacity of a classical

noisy channel can be obtained from a maximization

of the mutual information over all joint input–

output distributions.

For the quantum channel capacity, the relevant

entropic quantity is the coherent information,

ðT;%Þ :¼ HTð%ÞðÞHTidðj

jÞ



½14

where H denotes the von Neum ann entropy:

H(%) = tr% ld%, and

H

is a purifica-

tion of the density operator % 2A. The coherent

information does not increase under quantum

operations, I

(S  T, %) I

(T, %) for any quantum

channel S and state % 2A. This is the data

processing inequality (Barnum et al. 1998), which

shows that the regularized coherent information

provides an upper bound on the quantum channel

capacity: if Alic e and Bob have a coding scheme for

the channel T with capacity Q(T), n channel uses

allow them to share a maximally entangled state of

size exp

nQ(T). The coherent information of this

state equals nQ(T), and was no larger prior to

Bob’s decoding.

Recently, Devetak (2005) developed a coding

scheme to show that this bound is in fact attainable.

Different proofs were outlined by Lloyd and Shor.

Theorem 1 For every quantum channel T,

QðTÞ¼lim

n!1

max

n

;%ðÞ ½15

Unlike the classical or quantum mutual information,

coherent information is strictly superadditive for

some channels (DiVincenzo et al. 1998). Hence,

taking the limit n !1 in eqn [15] is indeed required,

and in general the evaluation of the capacity formula

[15] still demands the solution of asymptotically large

variational problems. This should be contrasted with

the entanglement-assisted capacities C

(T) = 2Q

(T)

(where a simple nonregularized coding theorem is

known to hold, see Capacities Enhanced by Entan-

glement) and the capacity for classical information

C(T) (where additivity is conjectured but not proved,

see Quantum Channels: Classical Capacity). Even a

maximization of the single-shot coherent information

(T, %) appears to be a difficult optimization

problem, since this quantity is neither convex nor

concave and may have multiple local maxima (Shor

2003). Thus, even for simple-looking systems like the

qubit depolarizing channel, so far we only have upper

and lower bounds on the quantum channel capacity,

but do not yet know how to compute its exact value.

We now sketch Devetak’s proof of Theorem 1,

assuming only some familiarity with Holevo–

Schumacher–Westmoreland (HSW) random codes

for the classical channel capacity (see Quantum

Channels: Classical Capacity). It is easily seen from

Stinespring’s dilation theore m (see Channels in

Quantum Information Theory) that a noiseless

quantum channel provides perfect security against

eavesdropping. This is one of the characteristic traits

of quantum mechanics and lies at the heart of

quantum cryptography. In his proo f, Devetak

showed a way to turn this around and upgrade

coding schemes for private classical information to

quantum channel codes.

The relation between quantum information trans-

fer over a channel T : A!B and privacy against

eavesdropping is best understood in terms of the

companion channel T

: A!E. T

arises from a

given Stinespring isometry V : H

H

T T

by interchanging the roles of the output

system B and the environment E:

ð%Þ¼tr

V%V



Ð T

ð%Þ¼tr

V%V



½16

The channel T

describes the information flow into

the environment E, a system we assume to be under

complete control of a potential eavesdropper, Eve

say. The setup for private classical information

transfer (including the definition of rates and capa-

city) is then exactly the same as for the classical

channel capacity (see Quantum Channels: Classical

Capacity), but the protocols now have to satisfy the

additional requirement that T

releases (almost) no

information to the environment. This can be achieved

by randomizing over 

exp

n (T

,{p

, %

}) code

words of a standard HSW code of total size

exp

n (T

,{p

, %

}), where { p

, %

} is the quantum

ensemble from which a set of random code words

428 Capacity for Quantum Information

{

k, l

}



, 

k = 1, l = 1

is generated. The appearance of

the Holevo bound

ðT;fp

gÞ:¼H

Tð%





Tð%



½17

in the dimension of both these code spaces can be

understood from the size of the relevant typical

subspaces (Devetak and Winter 2004).

The randomization guarantees that the remaini ng



 exp

n((T

)  (T

)) code words are almost

indistinguishable to Eve:



l¼1

n









"; 8j; k ¼ 1; ...;

½18

The net transfer rate for private classical informa-

tion is then R (T

) (T

), whi ch is just the total

transfer rate for the channel Alice ! Bob reduced by

the transfer rate Alice ! Eve.

Remarkably, if % =

j is a decomposi-

tion of % 2Ainto pure states, the private transfer

rate exactly equals the coherent information,

ðT

;%Þ¼HT

ð%ÞðÞHT

ð%ÞðÞ

¼ ðT

ÞðT

Þ½19

The so-called entropy exchange

ð%ÞðÞ= HT

idðj



quantifies the extent to which a formerly pure

ancilla state becomes mixed via interaction with

the signal states. Equation[19] then nicely reflects

the intuition that for high-rate quantum information

transfer the signal states should not entangle too

much with the environment. In fact, for an almost

noiseless channel the entropy exchange nearly

vanishes, and the optimized coherent information

almost attains the maximal value 1, while for nearly

depolarizing channels we have I

, %) H(%) 0.

So far, we have sketched a protocol for private

classical information transfer. Devetak’s coherenti-

fication allows to pass from the transmission of

classical messages to the transmission of coherent

superpositions. This technique has also been ap plied

to obtain entanglement distillation protocols from

secret key distillation, and offers a unified view on

the secret classical resources and their quantum

counterparts (Devetak and Winter 2004, Devetak

et al. 2004).

In order to transfer quantum information, Alice

will only need to send one half of a maximally

entangled state of dimensionality exp

, %).

As described in the previou s section, teleportation

then allows her to transfer arbitrary quantum states

from a subspace of that size.

Given a set of pure state code words

{j’



, 

k = 1, l = 1

of a private classical information

protocol, for entanglement transfer Alice prepares

the input state

ji

ﬃﬃﬃﬃﬃ



k¼1

jki



ﬃﬃﬃﬃﬃ



l¼1

j’

½20

where A

denotes a reference system that Alice keeps

in her lab. On his share of the resulting output state

j

Bob will then employ the corresponding

measurement operators {M

}



, 

k, l = 1

to implement the

coherent measurement

j’i

ﬃﬃﬃﬃﬃﬃﬃﬃ

j’i

jkli

which places the measurement outcomes into some

reference system B

B

. Any measurement which

identifies the output with high probability only

slightly disturbs the output state, and thus Bob’s

coherent measurement leaves the total system in an

approximation of the state

j

i¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ



;

k¼1;l¼1

jki

jli

j’

½21

in which Eve and Bob are still entangled. A

completely depolarizing channel T

would directly

yield a factorized output state BE here. Although

the randomization in eqn [18] does not necessarily

result in complete depolarization, there is a controlled

unitary operation which Bob may apply to effectively

decouple Eve’s system, resulting in the output state

(1=

ﬃﬃﬃﬃﬃ



)

jkki

E, which is the maximally

entangled state of size 

 exp

, %) required

for teleportation. The direct part of the capacity

theorem then follows by applying the above coding

scheme to large blocks and maximizing over (pure)

input ensembles, concluding the proof.

Devetak’s proof of the coding theorem seems to

indicate that the private classi cal capacity C

(T)

equals the quantum capacity Q(T) for every

quantum channel T. However, for the coherentifica-

tion protocol, we have restricted the private coding

schemes to pure state input ensembles, and thus we

can only conclude that Q(T) C

(T). The exis tence

of bound-entangled states with positive one-way

distillable secret key rate (Horodecki et al. 2005)

implies that this inequality can be strict. A general

procedure does exist to retrieve (almost) all the

information from the output of a noisy quantum

channel that releases (almost) no information to the

environment. But this requires a stronger form of

privacy than eqn [18].

Capacity for Quantum Information 429

Quantum Channels with Memory

This article has so far been restricted to memory-

less quantum channels, i n which successive chan-

nel inputs are acted on i ndependently. Messages of

n symbols are then processed by the tensor

product channel T

n

,asinDe finition 1 and

illustrated in Figure 1. In many real-world applica-

tions, the assumption of having uncorrelated noise

cannot be justified, and memory e ffects need to be

taken into account. For a quantum channel T with

are conveniently modeled (Bowen and Mancini

2004) by introducing an additional memory

system M, so t hat now T : MA!BM is a

completely positive and trace-pre serving map with

two input systems and two output systems. Long

messages with n signal states will then be

processed by the concatenated channel

: MA

M. In such a concatenation,

the memory system is passed on from one channel

application to the next, and thus introduces

(classical or quantum) correlations between con-

secutive register inputs.

Remarkably, this relatively simple model can be

shown (Kretschmann and Werner 2005) to encom-

pass every reasonable physical process: every sta-

tionary channel S : A

which turns an infinite

string of input states (on the quasilocal algebra A

)

into an infinite string of output states on B

and

satisfies the causality constraint is in fact a con-

catenated memory channel. Causality here means

that the outputs of the stationary channel S at given

time t

do not depend on inputs at times t > t

Figure 2 illustrates the structure theorem for causal

stationary qua ntum channels. In general, it produces

not only the memory channel T with memory

algebra M, but also a map R describing the

influence of input states in the remote past.

Intuitively, such a map is often not needed, because

memory effects decrease in time: the memory

channel T is called forgetful if outputs at a large

time t depend only weakly on the memory initializa-

tion at time zero. In fact, memory effects can be

shown to die out even exponentially. The set of

these channels is open and dense in the set of

quantum memory channels. Hence, generic memory

channels are forgetful.

The capacity of memory channels is defined in

complete analogy to the memoryless case, replacing

the n-fold tensor product T

n

in Definition 1 by

the n-fold concatenation T

. The coding theorems

for (private) classical and quantum information

can then be extended from the memoryless case

to the very important clas s of forgetful channels

(Kretschmann and Werner 2005).

Nonforgetful channels call for universal coding

schemes, which apply irrespec tive of the initializa-

tion of the input memory. Such schemes are

presently known only for very special cases.

Acknowledgmnts

The author thanks the members of the quantum

information group at TU Braunschweig for their

careful reading of the manuscript and many helpful

suggestions. He also gratefully acknowledges the

funding from Deutsche Forschung sgemeinschaft

(DFG).

See also: Capacities Enhanced by Entanglement;

Channels in Quantum Information Theory; Entanglement;

Positive Maps on C



-Algebras; Quantum Channels:

Classical Capacity; Quantum Error Correction and Fault

Tolerance; Source Coding in Quantum Information Theory.

Further Reading

Barnum H, Nielsen MA, and Schumacher B (1998) Information

transmission through a noisy quantum channel. Physical

Review A 57: 4153 (quant-ph/9702049).

Bennett CH, Devetak I, Shor PW, and Smolin JA (2004)

Inequalities and separations among assisted capacities of

quantum channels, quant-ph/0406086.

Bennett CH, DiVincenzo DP, Smolin JA, and Wootters WK

(1996) Mixed-state entanglement and quantum error correc-

tion. Physical Review A 54: 3824 (quant-ph/9604024).

Bowen G and Mancini S (2004) Quantum channels with a finite

memory. Physical Review A 69: 012306 (quant-ph/0305010).

Devetak I (2005) The private classical information capacity and

quantum information capacity of a quantum channel. IEEE

Transactions on Information Theory 51: 44 (quant-ph/0304127).

Devetak I, Harrow AW, and Winter A (2004) A family of

quantum protocols. Physical Review Letters 93: 230504

(quant-ph/0308044).

Devetak I and Winter A (2004) Relating quantum privacy and

quantum coherence: an operational approach. Physical

Review Letters 93: 080501 (quant-ph/0307053).

DiVincenzo DP, Shor PW, and Smolin JA (1998) Quantum

channel capacities of very noisy channels. Physical Review A

57: 830 (quant-ph/9706061).

Eisert J and Wolf MM Gaussian quantum channels. In Cerf N,

Leuchs G, and Polzik E (eds.) Quantum Information with

SR=

tr tr

Time Time

Figure 2 By the structure theorem, a causal automaton S can

be decomposed into a chain of concatenated memory channels

T plus some input initializer R. Evaluation with the partial trace tr

means that the corresponding output is ignored.

430 Capacity for Quantum Information

Continuous Variables of Atoms and Light. London: Imperial

College Press (in preparation)(quant-ph/0505151).

Holevo AS and Werner RF (2001) Evaluating capacities of

bosonic Gaussian channels. Physical Review A 63: 032312

(quant-ph/9912067).

Horodecki M, Horodecki P, and Horodecki R (1999) General

teleportation channel, singlet fraction, and quasidistillation.

Physical Review A 60: 1888 (quant-ph/9807091).

Horodecki P and Nowakowski ML (2005) Simple test for

quantum channel capacity, quant-ph/0503070.

Horodecki K, Pankowski L, Horodecki M, and Horodecki P

(2005) Low dimensional bound entanglement with one-way

distillable cryptographic key, quant-ph/0506203.

Kretschmann D and Werner RF (2004) Tema con variazioni:

quantum channel capacity. New Journal of Physics 6: 26

(quant-ph/0311037).

Kretschmann D and Werner RF (2005) Quantum channels with

memory. Physical Review A 72: 062323 (quant-ph/0502106).

Shor PW (2003) Capacities of quantum channels and how to find

them. Mathematical Programming 97: 311 (quant-ph/0304102).

Capillary Surfaces

R Finn, Stanford University, Stanford, CA, USA

Historical and Conceptual Background

A capillary surface is the interface separating two

fluids that lie adjacent to each other and do not mix.

Examples of such surfaces are the upper surface of

liquid pa rtially filling a vertical cylinder (capillary

tube), the surface of a liquid drop resting in

equilibrium on a tabletop (sessile drop) and the

surface of a liquid drop hanging from a ceiling

(pendent drop); further instances are the surface of a

falling raindrop, the bounding surface of the liquid

in the fuel tank of a spaceship, and the interface

formed by a fluid mass rotating within another fluid.

This last example extends to the problem of rotating

stars.

Interfaces separating fluids and solids share some

of the physical attributes of capillary surfaces, and

the study of wetted portions of rigid ‘‘support

surfaces’’ becomes essential for describing global

behavior of capillary configurations. However, some

significant distinctions appear that change the

formal structure of the problems, and must be

accounted for in the theory.

Phenomena governed by capillarity pervade all of

daily life, and most are so familiar as to escape

special notice. By contrast, throughout the eigh-

teenth century and presumably earlier, great atten-

tion centered on the rise of liquid in a narrow glass

circular-cylindrical tube dipped vertically into a

liquid reservoir (Figure 1); this striking event had a

dramatic impact that confounded intuition. Clarifi-

cation of the behavior became one of the major

problems challenging the scientific world of the

time, and was not achie ved during that period. The

term ‘‘capillary,’’ adapted from the Latin ‘‘capillus’’

for hair, was applied to the phenomenon since it was

observed only for tubes with very fine openings; the

more general usage adopted in the definition above

derives from the recognition of a class of phenomena

with a common physical basis.

The first recorded observations concerning

capillarity seem due to Aristoteles c. 350

BC.He

wrote that ‘‘a broad flat body, even of heavy

material, w ill float on water, however a narrow

thin one such as a needle will always sink.’’ Any

reader with access to a needle and a glass of water

will have little difficulty refuting the assertion.

Remarkably, the error in reasoning seems not to

have been pointed out for almost 2000 years,

when Galileo addressed the problem in his

Discorsi, about 1600. The only substantive studies

till that time are apparently those of Leonardo da

Vinci a hundred years earlier. Leonardo intro-

duced reasoning close in spirit to that of current

literature; however, the Calculus was not available

to him, and he was not in a position t o develop his

ideas in quantitative ways.

Young’s Contribution

The later discovery of the Calculus pr ovided a

driving impetus guiding many new studies during

the eighteenth century. But despite the enormity of

that wea pon, it did not on its own suffice, and initial

quantitative success had to await two initiatives

Figure 1 Capillary tube in infinite reservoir, in downward

gravity field.

Capillary Surfaces 431

taken by Thomas Young in 1805. Young based his

studies on the concept of surface tension that had

been introduced by von Segner half a century earlier.

Segner hypothesized that every curve on a fluid/fluid

interface S experiences on both its sides an orthogo-

nal force  per unit length, which (for given

temperature) depends only on the materials and is

directed into the tangent planes on the respective

sides. The presence of such forces can be indicated

by simple experiments. They become clearly evident

in the case of thin (soap) films spanning a frame, in

which case there is an easily observed orthogonal

pull on the frame, see the section ‘‘Du al inte rpreta-

tion of  : distincti on be tween fluids and solids.’’

Young made two basic conceptual contributions

(Y1, Y2):

Y1. Relation of pressure jump across a free interface

to mean curvature and surface tensi on.

Consider a piec e of surface S in the shape of a

spherical bowl of radius R, separating two immisci-

ble fluid media, as in Figure 2. In equilibrium, any

pressure difference p across S must be balanced by

a tension  on its rim .IfS projects to a disk of

(small) radius r on the plane tangent to S at the

symmetry point, we are led to

r

p ’ 2r sin # ½1

where # is inclination of S at the rim, relative to the

plane. We thus find at the base point

p ¼ 2

d sin #

¼ 2

½2

Young then went on to consider a general S, without

symmetry hypothesis. Letting 1=R

,1=R

denote the

planar curvatures at a point in S of two normal

sections in orthogonal directions, he asserted that

p ¼ 2



 2H ½3

where H is the mean curvature of S at the point.

Although Young provided no formal justification for

this step, we can establish it with the aid of a general

formula from differential geometry that was not

known in his lifetime:

2HN dS ¼

n ds ½4

where N is a unit normal on S, and n is unit

conormal (as indicated in Figure 2)on. Multi-

plying both sides of [4] by , the right-hand side

becomes the net surface tension force on S. Since

that must equal the net balancing pressure force, we

obtain

p  2HðÞN dS ¼ 0 ½5

Letting the diameter of S tend to zero, the assertion

follows.

We emphasize here the im plicit assumption above,

that  is a constant depending only on the particular

materials, and not on the shape of S. This author

knows of no source in which that is clearly

established, although experiments and experience

provide some a posteriori justification. See the

further comments under Y2, and later in sections

‘‘Gauss’ contribu tion: the energy method ’’ an d

‘‘Dual interpre tation of  : distinc tion between fluids

and solids. ’’

Y2. The capillary contact angle .

Young asserted that there are surface t ensions for

solid/fluid interfaces analogous to those just intro-

duced, and again depending only on the materials.

This assertion is erroneous, as was s uggested in

writings of Bikerman and of others, and more

recently established in a definitive example by Finn.

Using his premise, Young attempted to characterize

the contact angle  made by the fluid surface with a

rigid boundary, by requiring that the net tangential

component of the three surface tension vectors

vanish at the triple interface; this leads to t he often

employed but incorrect ‘‘Young diagram,’’ see

Figure 3,andtherelation

cos  ¼



 



½6

Figure 2 Pressure change across fluid element, balanced by

surface tension.

Liquid

Gas

Solid

Figure 3 Young diagram; balance of tangential forces.

Residual normal force remains.

432 Capillary Surfaces

for cos  in terms of the magnitudes of the three

‘‘surface tensions.’ ’ Young concluded that the

contact angle depends only on the materials, and

in no other way on the conditions of the problem.

This basic assertion is by a fortuitous acciden t

correct, as follows from the contribution by

Gauss described below; it underlies all modern

theory.

Using Y1 and Y2, Young produced the first

verifiable prediction for the rise height u

the circular capillary tube of Figure 1.He

assumed the interface to be spherical, so that H

is constant and a = cos =H. He assumed vanish-

ing outside pressure. According to classic laws of

hydrostatics, p = gu

= 2H by Y1, where  is

fluid density; there follows the c elebrated re la-

tion, presented entirely in words in his 1805

article:



2 cos 

a

;¼

g



½7

Young scorned the mathematical method, and

made a point of deriving and publishing his

results on capillarity without use of any mathe-

matical symbols. This personal idiosyncrasy

causes his publications to be something of a

challenge to read.

The Laplace Contribution

In 1806, Laplace published the first analytical expres-

sion for the mean curvature of a surface u(x, y), and

showed that the expression can be written as a

divergence. He obtained the equation

div Tu  2H; Tu 

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1 þru

½8

Thus, if H is known from geometrical or physical

considerations, as it is for the capillary tube in

the example just considered, one finds a second-

order (nonlinear) equation for the surface height

of any solution as a graph. The equation is

elliptic for any function u(x, y)insertedintothe

coefficients, however not uniformly so; the parti-

cular nonuniformity leads to some striking and

unusual behavior of its solutions, as we shall see.

With the aid of [8], Laplace improved the Young

estimate [7] to



2 cos 

a



cos 



1  sin



cos



!"#

a ½9

Both Young and Laplace proposed their for-

mulas for ‘‘narrow tubes’’, but neither gave any

quantitative indication of what ‘‘narrow’’ should

signify. No te that whenever 0  <=2, [9]

becomes negative when the nondimensional Bond

Number B = a

exceeds 8; since u is known to be

positive in the indicated r ange for , [9] provides

no information in that case, whereas [7] is still of

some value. Nevertheless, [9] is asymptotically

exact a nd consists of the first two terms of the

formal expansion in powers of a;thatwasfirst

proved by D Siegel in 1980, almost 200 years

following the discovery of the formulas. In 1968,

P Concus extended the formal expansion for the

height to the entire traverse 0 < r < a. F Brulois

(1981) and independently E Miersemann (1994)

proved the expansion to be a symptotic to every

order. Explicit bounds for the rise height above

and below, making quantitative the notion of

‘‘narrow,’’ were obtained by Finn.

Laplace supplied the first detailed mathematical

investigations into the behavi or of capillary surfaces,

applying his ideas to many specific exampl es. His

underlying motivation apparently derived at least

partly from astronomical problems, and he pub-

lished his contributions in two ‘‘Supple´ments’’ to the

tenth volume of his Me´canique Ce´leste.

Gauss’ Contribution: The Energy Method

Young and Laplace both based their reasonings

on force-balance arguments, which at best were

unclear and at worst conceptually wrong. In

1830, Gauss took up the problem anew from a

variational point of view, using the Johann

Bernoulli principle of virtual work. To do so, he

attempted to characterize both surface energies

and bulk fluid energies in terms of postulated

particle attrac tions a nd repul sions. In an aston-

ishing 30 pages, he essentially introduced founda-

tions of modern potential theory, of measure

theory, and of thermodynamics. He ended up

with elaborate e xpressions that could not readily

be applied, and which at least to some extent he

did not use. He asserted, for example, that the

bulk internal energy would be propor tional to

volume, which for an incompressible fluid is

constant under admissible deforma tions, and o n

that basis he ignored the bulk energy term

completely. His procedures then led him, in an

independent and more convincing way, to the

identical equation and boundary condition that

had been produced by his predecessors. It must,

of course, be remarked that his justification for

ignoring the bulk energy term would not b e

correct for a compressible liquid (see the section

‘‘Com pressibility’’), a nd it is open to some

Capillary Surfaces 433

question for the central m otivating problem of a

capillary tube dipped into an infinite liquid bath,

in which eve nt there is no volume cons traint.

The material that follows is guided by the ideas of

Gauss; however, I have found it advantageous to

replace his elaborate hypotheses on particle attrac-

tions and repulsions by a simpler phenomenological

reasoning as to the nature of the energy terms to be

expected.

To fix ideas, we consider a semi-infinite cylinder

of general section  and of homogeneous material,

closed at the bottom, situated vertically in a down-

ward gravity field g per unit mass, and partly filled

with an incompressible liquid of density  covering

the bottom (a more exact discussion, taking account

of compressibility, is indicated below in the section

‘‘Compr essib ility’’). We assume an equilibr ium fluid

configuration with the liquid bounded abov e by an

ideally thin interfa ce S : u(x, y) (see Figure 4). We

distinguish the energy terms that occur:

1. Surface energy. This is the energy required to

create the surface interface S. We can characterize it

by noting that fluid particles within or exterior to the

liquid are attracted equally to neighboring particles in

all directions; however, at the surface S there is a

differential attraction, to particles of the exterior

medium (such as air) above, or to the liquid below

(see Figure 5). Thus, particles in the interface are

pulled orthogonally to S. In general, for a liquid–gas

interface, significant work will be done only on the

liquid and those particles will be pulled toward the

liquid; otherwise, the liquid would evaporate across

the interface and disappear. The work done in that

(infinitesimal) motion is proportional to the area of S,

so that for the surface energy E

we obtain

¼ 



ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1 þru

dx ½10

The constant  has the dimensions of force per unit

length, and turns out to be the surface tension of the

interface. We note from [10] its dual interpretation

as areal energy density on S, arising from formation

of that surface. This alternative interpretation lends

conceptual support to the supposition that  is

consta nt on S. See the sect ion ‘‘Dual interpre tation

of  : dist inction be tween fluids and solids.’’

Implicit in the above discussion are deep

premises about the nature of the forces acting

within the fluid. Essentially these forces must be

perceptible only at infinitesimal distances, and

grow rapidly with decreasing distance. Forces

both of attraction and of repulsion must be

present. The recognition of the need for such

forces can be traced back to Newton. Quantita-

tive postulates as to their precise nature were

introduced by van der Waals in the late nine-

teenth century, and the topic remains stil l in

active study. Since these forces appear at mole-

cular distance levels, their introduction leads

inevitably to questions of statistical mechanics.

Additionally, our discussion of work done in

forming the surface implicitly assumes a compres-

sible transition layer there, in conflict with our

treatment of S as an ideally thin interface

bounding an incompressible fluid. In these senses,

it is striking that [10] – which is in accord w ith

classical constructi ons – could be obtained via

global qualitative postulates concerning a con-

tinuum in static equilibrium, in which the specific

nature of the forces is not introduced.

Rayleigh me asured the thic kness of the surface

interface between water and air t o be of mole-

cular size, thus providing experimental justifica-

tion for the procedure adopted.

2. Wetting energy. A similar discussion applies at

the interface separa ting the liquid and solid at the

cylinder walls; however, this time the net attraction

can be in either direction, as particles from neither

medium can migrate significantly into the other. For

the wetting energy E

, we write, with  the

boundary of ,

¼



u ds ½11

Figure 4 Liquid in cylindrical capillary tube, of general section .

Reproduced with permission from the American Institute of

Aeronautics and Astronautics.

Figure 5 Attractions on a fluid element: (1) interior to the fluid;

(2) on the surface interface.

434 Capillary Surfaces