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In quantum information theory, the classical capacity of a quantum channel is the maximum rate at which classical data can be sent over it error-free in the limit of many uses of the channel. Holevo, Schumacher, and Westmoreland proved the following lower bound on the classical capacity of any quantum channel Ndisplaystyle mathcal N:

χ(N)=maxρXAI(X;B)N(ρ)displaystyle chi (mathcal N)=max _rho ^XAI(X;B)_mathcal N(rho )

where ρXAdisplaystyle rho ^XA is a classical-quantum state of the following form:

ρXA=∑xpX(x)|x⟩⟨x|X⊗ρxA,displaystyle rho ^XA=sum _xp_X(x)vert xrangle langle xvert ^Xotimes rho _x^A,

pX(x)displaystyle p_X(x) is a probability distribution, and each ρxAdisplaystyle rho _x^A is a density operator that can be input to the channel Ndisplaystyle mathcal N.

Achievability using sequential decoding

We briefly review the HSW coding theorem (the
statement of the achievability of the Holevo information rate I(X;B)displaystyle I(X;B) for
communicating classical data over a quantum channel). We first review the
minimal amount of quantum mechanics needed for the theorem. We then cover
quantum typicality, and finally we prove the theorem using a recent sequential
decoding technique.

Review of quantum mechanics

In order to prove the HSW coding theorem, we really just need a few basic
things from quantum mechanics. First, a quantum state is a unit trace,
positive operator known as a density operator. Usually, we denote it
by ρdisplaystyle rho , σdisplaystyle sigma , ωdisplaystyle omega , etc. The simplest model for a quantum channel
is known as a classical-quantum channel:

x↦ρx.displaystyle xmapsto rho _x.

The meaning of the above notation is that inputting the classical letter xdisplaystyle x
at the transmitting end leads to a quantum state ρxdisplaystyle rho _x at the receiving
end. It is the task of the receiver to perform a measurement to determine the
input of the sender. If it is true that the states ρxdisplaystyle rho _x are perfectly
distinguishable from one another (i.e., if they have orthogonal supports such
that Trρxρx′=0displaystyle mathrm Tr ,leftrho _xrho _x^prime right=0 for x≠x′displaystyle xneq x^prime ), then the channel is a noiseless channel. We are interested in situations
for which this is not the case. If it is true that the states ρxdisplaystyle rho _x all
commute with one another, then this is effectively identical to the situation
for a classical channel, so we are also not interested in these situations.
So, the situation in which we are interested is that in which the states
ρxdisplaystyle rho _x have overlapping support and are non-commutative.

The most general way to describe a quantum measurement is with a
positive operator-valued measure (POVM). We usually denote the elements of a POVM as
Λmmdisplaystyle leftLambda _mright_m. These operators should satisfy
positivity and completeness in order to form a valid POVM:

Λm≥0    ∀mdisplaystyle Lambda _mgeq 0 forall m

∑mΛm=I.displaystyle sum _mLambda _m=I.

The probabilistic interpretation of quantum mechanics states that if someone
measures a quantum state ρdisplaystyle rho using a measurement device corresponding to
the POVM Λmdisplaystyle leftLambda _mright, then the probability p(m)displaystyle pleft(mright) for obtaining outcome mdisplaystyle m is equal to

p(m)=TrΛmρ,displaystyle pleft(mright)=textTrleftLambda _mrho right,

and the post-measurement state is

ρm′=1p(m)ΛmρΛm,displaystyle rho _m^prime =frac 1pleft(mright)sqrt Lambda _mrho sqrt Lambda _m,

if the person measuring obtains outcome mdisplaystyle m. These rules are sufficient for us
to consider classical communication schemes over cq channels.

Quantum typicality

The reader can find a good review of this topic in the article about the typical subspace.

Gentle operator lemma

The following lemma is important for our proofs. It
demonstrates that a measurement that succeeds with high probability on average
does not disturb the state too much on average:

Lemma: [Winter] Given an
ensemble pX(x),ρxdisplaystyle leftp_Xleft(xright),rho _xright with expected
density operator ρ≡∑xpX(x)ρxdisplaystyle rho equiv sum _xp_Xleft(xright)rho _x, suppose
that an operator Λdisplaystyle Lambda such that I≥Λ≥0displaystyle Igeq Lambda geq 0 succeeds with high
probability on the state ρdisplaystyle rho :

TrΛρ≥1−ϵ.displaystyle textTrleftLambda rho rightgeq 1-epsilon .

Then the subnormalized state ΛρxΛdisplaystyle sqrt Lambda rho _xsqrt Lambda is close
in expected trace distance to the original state ρxdisplaystyle rho _x:

EX‖ΛρXΛ−ρX‖1≤2ϵ.displaystyle mathbb E _XleftleftVert sqrt Lambda rho _Xsqrt Lambda -rho _XrightVert _1rightleq 2sqrt epsilon .

(Note that ‖A‖1displaystyle leftVert ArightVert _1 is the nuclear norm of the operator
Adisplaystyle A so that ‖A‖1≡displaystyle leftVert ArightVert _1equiv TrA†Adisplaystyle leftsqrt A^dagger Aright.)

The following inequality is useful for us as well. It holds for any operators
ρdisplaystyle rho , σdisplaystyle sigma , Λdisplaystyle Lambda such that 0≤ρ,σ,Λ≤Idisplaystyle 0leq rho ,sigma ,Lambda leq I:

TrΛρ≤TrΛσ+‖ρ−σ‖1.displaystyle textTrleftLambda rho rightleq textTrleftLambda sigma right+leftVert rho -sigma rightVert _1.

(1)

The quantum information-theoretic interpretation of the above inequality is
that the probability of obtaining outcome Λdisplaystyle Lambda from a quantum measurement
acting on the state ρdisplaystyle rho is upper bounded by the probability of obtaining
outcome Λdisplaystyle Lambda on the state σdisplaystyle sigma summed with the distinguishability of
the two states ρdisplaystyle rho and σdisplaystyle sigma .

Non-commutative union bound

Lemma: [Sen's bound] The following bound
holds for a subnormalized state σdisplaystyle sigma such that 0≤σdisplaystyle 0leq sigma and
Trσ≤1displaystyle Trleftsigma rightleq 1 with Π1displaystyle Pi _1, ... , ΠNdisplaystyle Pi _N being
projectors:
Trσ−TrΠN⋯Π1 σ Π1⋯ΠN≤2∑i=1NTr(I−Πi)σ,displaystyle textTrleftsigma right-textTrleftPi _Ncdots Pi _1 sigma Pi _1cdots Pi _Nrightleq 2sqrt sum _i=1^NtextTrleftleft(I-Pi _iright)sigma right,

We can think of Sen's bound as a "non-commutative union
bound" because it is analogous to the following union bound
from probability theory:

Pr(A1∩⋯∩AN)c=PrA1c∪⋯∪ANc≤∑i=1NPrAic,displaystyle Pr leftleft(A_1cap cdots cap A_Nright)^cright=Pr leftA_1^ccup cdots cup A_N^crightleq sum _i=1^NPr leftA_i^cright,

where A1displaystyle A_1, ldots, ANdisplaystyle A_N are events. The analogous bound for projector
logic would be

Tr(I−Π1⋯ΠN⋯Π1)ρ≤∑i=1NTr(I−Πi)ρ,displaystyle textTrleftleft(I-Pi _1cdots Pi _Ncdots Pi _1right)rho rightleq sum _i=1^NtextTrleftleft(I-Pi _iright)rho right,

if we think of Π1⋯ΠNdisplaystyle Pi _1cdots Pi _N as a projector onto the intersection of
subspaces. Though, the above bound only holds if the projectors Π1displaystyle Pi _1,
..., ΠNdisplaystyle Pi _N are commuting (choosing Π1=|+⟩⟨+|displaystyle Pi _1=leftvert +rightrangle leftlangle +rightvert , Π2=|0⟩⟨0|displaystyle Pi _2=leftvert 0rightrangle leftlangle 0rightvert , and ρ=|0⟩⟨0|displaystyle rho =leftvert 0rightrangle leftlangle 0rightvert gives a counterexample). If the projectors are non-commuting, then Sen's
bound is the next best thing and suffices for our purposes here.

HSW theorem with the non-commutative union bound

We now prove the HSW theorem with Sen's non-commutative union bound. We
divide up the proof into a few parts: codebook generation, POVM construction,
and error analysis.

Codebook Generation. We first describe how Alice and Bob agree on a
random choice of code. They have the channel x→ρxdisplaystyle xrightarrow rho _x and a
distribution pX(x)displaystyle p_Xleft(xright). They choose Mdisplaystyle M classical sequences
xndisplaystyle x^n according to the IID distribution pXn(xn)displaystyle p_X^nleft(x^nright).
After selecting them, they label them with indices as xn(m)m∈[M]displaystyle leftx^nleft(mright)right_min left[Mright]. This leads to the following
quantum codewords:

ρxn(m)=ρx1(m)⊗⋯⊗ρxn(m).displaystyle rho _x^nleft(mright)=rho _x_1left(mright)otimes cdots otimes rho _x_nleft(mright).

The quantum codebook is then ρxn(m)displaystyle leftrho _x^nleft(mright)right. The average state of the codebook is then

EXnρXn=∑xnpXn(xn)ρxn=ρ⊗n,displaystyle mathbb E _X^nleftrho _X^nright=sum _x^np_X^nleft(x^nright)rho _x^n=rho ^otimes n,

(2)

where ρ=∑xpX(x)ρxdisplaystyle rho =sum _xp_Xleft(xright)rho _x.

POVM Construction . Sens' bound from the above lemma
suggests a method for Bob to decode a state that Alice transmits. Bob should
first ask "Is the received state in the average typical
subspace?" He can do this operationally by performing a
typical subspace measurement corresponding to Πρ,δn,I−Πρ,δndisplaystyle leftPi _rho ,delta ^n,I-Pi _rho ,delta ^nright. Next, he asks in sequential order,
"Is the received codeword in the mthdisplaystyle m^textth
conditionally typical subspace?" This is in some sense
equivalent to the question, "Is the received codeword the
mthdisplaystyle m^textth transmitted codeword?" He can ask these
questions operationally by performing the measurements corresponding to the
conditionally typical projectors Πρxn(m),δ,I−Πρxn(m),δdisplaystyle leftPi _rho _x^nleft(mright),delta ,I-Pi _rho _x^nleft(mright),delta right.

Why should this sequential decoding scheme work well? The reason is that the
transmitted codeword lies in the typical subspace on average:

EXnTrΠρ,δ ρXn=TrΠρ,δ EXnρXndisplaystyle mathbb E _X^nlefttextTrleftPi _rho ,delta rho _X^nrightright=textTrleftPi _rho ,delta mathbb E _X^nleftrho _X^nrightright

=TrΠρ,δ ρ⊗ndisplaystyle =textTrleftPi _rho ,delta rho ^otimes nright

≥1−ϵ,displaystyle geq 1-epsilon ,

where the inequality follows from (refeq:1st-typ-prop). Also, the
projectors Πρxn(m),δdisplaystyle Pi _rho _x^nleft(mright),delta
are "good detectors" for the states ρxn(m)displaystyle rho _x^nleft(mright) (on average) because the following condition holds from conditional quantum
typicality:

EXnTrΠρXn,δ ρXn≥1−ϵ.displaystyle mathbb E _X^nlefttextTrleftPi _rho _X^n,delta rho _X^nrightrightgeq 1-epsilon .

Error Analysis. The probability of detecting the mthdisplaystyle m^textth
codeword correctly under our sequential decoding scheme is equal to

TrΠρXn(m),δΠ^ρXn(m−1),δ⋯Π^ρXn(1),δ Πρ,δn ρxn(m) Πρ,δn Π^ρXn(1),δ⋯Π^ρXn(m−1),δΠρXn(m),δ,displaystyle textTrleftPi _rho _X^nleft(mright),delta hat Pi _rho _X^nleft(m-1right),delta cdots hat Pi _rho _X^nleft(1right),delta Pi _rho ,delta ^n rho _x^nleft(mright) Pi _rho ,delta ^n hat Pi _rho _X^nleft(1right),delta cdots hat Pi _rho _X^nleft(m-1right),delta Pi _rho _X^nleft(mright),delta right,

where we make the abbreviation Π^≡I−Πdisplaystyle hat Pi equiv I-Pi . (Observe that we
project into the average typical subspace just once.) Thus, the probability of
an incorrect detection for the mthdisplaystyle m^textth codeword is given by

1−TrΠρXn(m),δΠ^ρXn(m−1),δ⋯Π^ρXn(1),δ Πρ,δn ρxn(m) Πρ,δn Π^ρXn(1),δ⋯Π^ρXn(m−1),δΠρXn(m),δ,displaystyle 1-textTrleftPi _rho _X^nleft(mright),delta hat Pi _rho _X^nleft(m-1right),delta cdots hat Pi _rho _X^nleft(1right),delta Pi _rho ,delta ^n rho _x^nleft(mright) Pi _rho ,delta ^n hat Pi _rho _X^nleft(1right),delta cdots hat Pi _rho _X^nleft(m-1right),delta Pi _rho _X^nleft(mright),delta right,

and the average error probability of this scheme is equal to

1−1M∑mTrΠρXn(m),δΠ^ρXn(m−1),δ⋯Π^ρXn(1),δ Πρ,δn ρxn(m) Πρ,δn Π^ρXn(1),δ⋯Π^ρXn(m−1),δΠρXn(m),δ.displaystyle 1-frac 1Msum _mtextTrleftPi _rho _X^nleft(mright),delta hat Pi _rho _X^nleft(m-1right),delta cdots hat Pi _rho _X^nleft(1right),delta Pi _rho ,delta ^n rho _x^nleft(mright) Pi _rho ,delta ^n hat Pi _rho _X^nleft(1right),delta cdots hat Pi _rho _X^nleft(m-1right),delta Pi _rho _X^nleft(mright),delta right.

Instead of analyzing the average error probability, we analyze the expectation
of the average error probability, where the expectation is with respect to the
random choice of code:

1−EXn1M∑mTrΠρXn(m),δΠ^ρXn(m−1),δ⋯Π^ρXn(1),δ Πρ,δn ρXn(m) Πρ,δn Π^ρXn(1),δ⋯Π^ρXn(m−1),δΠρXn(m),δ.displaystyle 1-mathbb E _X^nleftfrac 1Msum _mtextTrleftPi _rho _X^nleft(mright),delta hat Pi _rho _X^nleft(m-1right),delta cdots hat Pi _rho _X^nleft(1right),delta Pi _rho ,delta ^n rho _X^nleft(mright) Pi _rho ,delta ^n hat Pi _rho _X^nleft(1right),delta cdots hat Pi _rho _X^nleft(m-1right),delta Pi _rho _X^nleft(mright),delta rightright.

(3)

Our first step is to apply Sen's bound to the above quantity. But before doing
so, we should rewrite the above expression just slightly, by observing that

1=EXn1M∑mTrρXn(m)displaystyle 1=mathbb E _X^nleftfrac 1Msum _mtextTrleftrho _X^nleft(mright)rightright

=EXn1M∑mTrΠρ,δnρXn(m)+TrΠ^ρ,δnρXn(m)displaystyle =mathbb E _X^nleftfrac 1Msum _mtextTrleftPi _rho ,delta ^nrho _X^nleft(mright)right+textTrlefthat Pi _rho ,delta ^nrho _X^nleft(mright)rightright

=EXn1M∑mTrΠρ,δnρXn(m)Πρ,δn+1M∑mTrΠ^ρ,δnEXnρXn(m)displaystyle =mathbb E _X^nleftfrac 1Msum _mtextTrleftPi _rho ,delta ^nrho _X^nleft(mright)Pi _rho ,delta ^nrightright+frac 1Msum _mtextTrlefthat Pi _rho ,delta ^nmathbb E _X^nleftrho _X^nleft(mright)rightright

=EXn1M∑mTrΠρ,δnρXn(m)Πρ,δn+TrΠ^ρ,δnρ⊗ndisplaystyle =mathbb E _X^nleftfrac 1Msum _mtextTrleftPi _rho ,delta ^nrho _X^nleft(mright)Pi _rho ,delta ^nrightright+textTrlefthat Pi _rho ,delta ^nrho ^otimes nright

≤EXn1M∑mTrΠρ,δnρXn(m)Πρ,δn+ϵdisplaystyle leq mathbb E _X^nleftfrac 1Msum _mtextTrleftPi _rho ,delta ^nrho _X^nleft(mright)Pi _rho ,delta ^nrightright+epsilon

Substituting into (3) (and forgetting about the small
ϵdisplaystyle epsilon term for now) gives an upper bound of

EXn1M∑mTrΠρ,δnρXn(m)Πρ,δndisplaystyle mathbb E _X^nleftfrac 1Msum _mtextTrleftPi _rho ,delta ^nrho _X^nleft(mright)Pi _rho ,delta ^nrightright

−EXn1M∑mTrΠρXn(m),δΠ^ρXn(m−1),δ⋯Π^ρXn(1),δ Πρ,δn ρXn(m) Πρ,δn Π^ρXn(1),δ⋯Π^ρXn(m−1),δΠρXn(m),δ.displaystyle -mathbb E _X^nleftfrac 1Msum _mtextTrleftPi _rho _X^nleft(mright),delta hat Pi _rho _X^nleft(m-1right),delta cdots hat Pi _rho _X^nleft(1right),delta Pi _rho ,delta ^n rho _X^nleft(mright) Pi _rho ,delta ^n hat Pi _rho _X^nleft(1right),delta cdots hat Pi _rho _X^nleft(m-1right),delta Pi _rho _X^nleft(mright),delta rightright.

We then apply Sen's bound to this expression with σ=Πρ,δnρXn(m)Πρ,δndisplaystyle sigma =Pi _rho ,delta ^nrho _X^nleft(mright)Pi _rho ,delta ^n and the sequential
projectors as ΠρXn(m),δdisplaystyle Pi _rho _X^nleft(mright),delta , Π^ρXn(m−1),δdisplaystyle hat Pi _rho _X^nleft(m-1right),delta , ..., Π^ρXn(1),δdisplaystyle hat Pi _rho _X^nleft(1right),delta . This gives the upper bound
EXn1M∑m2[Tr(I−ΠρXn(m),δ)Πρ,δnρXn(m)Πρ,δn+∑i=1m−1TrΠρXn(i),δΠρ,δnρXn(m)Πρ,δn]1/2.displaystyle mathbb E _X^nleftfrac 1Msum _m2left[textTrleftleft(I-Pi _rho _X^nleft(mright),delta right)Pi _rho ,delta ^nrho _X^nleft(mright)Pi _rho ,delta ^nright+sum _i=1^m-1textTrleftPi _rho _X^nleft(iright),delta Pi _rho ,delta ^nrho _X^nleft(mright)Pi _rho ,delta ^nrightright]^1/2right.
Due to concavity of the square root, we can bound this expression from above
by

2[EXn1M∑mTr(I−ΠρXn(m),δ)Πρ,δnρXn(m)Πρ,δn+∑i=1m−1TrΠρXn(i),δΠρ,δnρXn(m)Πρ,δn]1/2displaystyle 2left[mathbb E _X^nleftfrac 1Msum _mtextTrleftleft(I-Pi _rho _X^nleft(mright),delta right)Pi _rho ,delta ^nrho _X^nleft(mright)Pi _rho ,delta ^nright+sum _i=1^m-1textTrleftPi _rho _X^nleft(iright),delta Pi _rho ,delta ^nrho _X^nleft(mright)Pi _rho ,delta ^nrightrightright]^1/2

≤2[EXn1M∑mTr(I−ΠρXn(m),δ)Πρ,δnρXn(m)Πρ,δn+∑i≠mTrΠρXn(i),δΠρ,δnρXn(m)Πρ,δn]1/2,displaystyle leq 2left[mathbb E _X^nleftfrac 1Msum _mtextTrleftleft(I-Pi _rho _X^nleft(mright),delta right)Pi _rho ,delta ^nrho _X^nleft(mright)Pi _rho ,delta ^nright+sum _ineq mtextTrleftPi _rho _X^nleft(iright),delta Pi _rho ,delta ^nrho _X^nleft(mright)Pi _rho ,delta ^nrightrightright]^1/2,

where the second bound follows by summing over all of the codewords not equal
to the mthdisplaystyle m^textth codeword (this sum can only be larger).

We now focus exclusively on showing that the term inside the square root can
be made small. Consider the first term:

EXn1M∑mTr(I−ΠρXn(m),δ)Πρ,δnρXn(m)Πρ,δndisplaystyle mathbb E _X^nleftfrac 1Msum _mtextTrleftleft(I-Pi _rho _X^nleft(mright),delta right)Pi _rho ,delta ^nrho _X^nleft(mright)Pi _rho ,delta ^nrightright

≤EXn1M∑mTr(I−ΠρXn(m),δ)ρXn(m)+‖ρXn(m)−Πρ,δnρXn(m)Πρ,δn‖1displaystyle leq mathbb E _X^nleftfrac 1Msum _mtextTrleftleft(I-Pi _rho _X^nleft(mright),delta right)rho _X^nleft(mright)right+leftVert rho _X^nleft(mright)-Pi _rho ,delta ^nrho _X^nleft(mright)Pi _rho ,delta ^nrightVert _1right

≤ϵ+2ϵ.displaystyle leq epsilon +2sqrt epsilon .

where the first inequality follows from (1) and the
second inequality follows from the gentle operator lemma and the
properties of unconditional and conditional typicality. Consider now the
second term and the following chain of inequalities:

∑i≠mEXnTrΠρXn(i),δ Πρ,δn ρXn(m) Πρ,δndisplaystyle sum _ineq mmathbb E _X^nlefttextTrleftPi _rho _X^nleft(iright),delta Pi _rho ,delta ^n rho _X^nleft(mright) Pi _rho ,delta ^nrightright

=∑i≠mTrEXnΠρXn(i),δ Πρ,δn EXnρXn(m) Πρ,δndisplaystyle =sum _ineq mtextTrleftmathbb E _X^nleftPi _rho _X^nleft(iright),delta right Pi _rho ,delta ^n mathbb E _X^nleftrho _X^nleft(mright)right Pi _rho ,delta ^nright

=∑i≠mTrEXnΠρXn(i),δ Πρ,δn ρ⊗n Πρ,δndisplaystyle =sum _ineq mtextTrleftmathbb E _X^nleftPi _rho _X^nleft(iright),delta right Pi _rho ,delta ^n rho ^otimes n Pi _rho ,delta ^nright

≤∑i≠m2−n[H(B)−δ] TrEXnΠρXn(i),δ Πρ,δndisplaystyle leq sum _ineq m2^-nleft[Hleft(Bright)-delta right] textTrleftmathbb E _X^nleftPi _rho _X^nleft(iright),delta right Pi _rho ,delta ^nright

The first equality follows because the codewords Xn(m)displaystyle X^nleft(mright) and
Xn(i)displaystyle X^nleft(iright) are independent since they are different. The second
equality follows from (2). The first inequality follows from
(refeq:3rd-typ-prop). Continuing, we have

≤∑i≠m2−n[H(B)−δ] EXnTrΠρXn(i),δdisplaystyle leq sum _ineq m2^-nleft[Hleft(Bright)-delta right] mathbb E _X^nlefttextTrleftPi _rho _X^nleft(iright),delta rightright

≤∑i≠m2−n[H(B)−δ] 2n[H(B|X)+δ]displaystyle leq sum _ineq m2^-nleft[Hleft(Bright)-delta right] 2^nleft[Hleft(B

=∑i≠m2−n[I(X;B)−2δ]displaystyle =sum _ineq m2^-nleft[Ileft(X;Bright)-2delta right]

≤M 2−n[I(X;B)−2δ].displaystyle leq M 2^-nleft[Ileft(X;Bright)-2delta right].

The first inequality follows from Πρ,δn≤Idisplaystyle Pi _rho ,delta ^nleq I and exchanging
the trace with the expectation. The second inequality follows from
(refeq:2nd-cond-typ). The next two are straightforward.

Putting everything together, we get our final bound on the expectation of the
average error probability:

1−EXn1M∑mTrΠρXn(m),δΠ^ρXn(m−1),δ⋯Π^ρXn(1),δ Πρ,δn ρXn(m) Πρ,δn Π^ρXn(1),δ⋯Π^ρXn(m−1),δΠρXn(m),δdisplaystyle 1-mathbb E _X^nleftfrac 1Msum _mtextTrleftPi _rho _X^nleft(mright),delta hat Pi _rho _X^nleft(m-1right),delta cdots hat Pi _rho _X^nleft(1right),delta Pi _rho ,delta ^n rho _X^nleft(mright) Pi _rho ,delta ^n hat Pi _rho _X^nleft(1right),delta cdots hat Pi _rho _X^nleft(m-1right),delta Pi _rho _X^nleft(mright),delta rightright

≤ϵ+2[(ϵ+2ϵ)+M 2−n[I(X;B)−2δ]]1/2.displaystyle leq epsilon +2left[left(epsilon +2sqrt epsilon right)+M 2^-nleft[Ileft(X;Bright)-2delta right]right]^1/2.

Thus, as long as we choose M=2n[I(X;B)−3δ]displaystyle M=2^nleft[Ileft(X;Bright)-3delta right], there exists a code with vanishing error probability.

See also

• Entanglement-assisted classical capacity


• Quantum capacity


• Quantum information theory


• Typical subspace

References

• 
  Holevo, Alexander S. (1998), "The Capacity of Quantum Channel with General Signal States", IEEE Transactions on Information Theory, 44 (1): 269–273, arXiv:quant-ph/9611023, doi:10.1109/18.651037.mw-parser-output cite.citationfont-style:inherit.mw-parser-output .citation qquotes:"""""""'""'".mw-parser-output .citation .cs1-lock-free abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .citation .cs1-lock-subscription abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registrationcolor:#555.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration spanborder-bottom:1px dotted;cursor:help.mw-parser-output .cs1-ws-icon abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center.mw-parser-output code.cs1-codecolor:inherit;background:inherit;border:inherit;padding:inherit.mw-parser-output .cs1-hidden-errordisplay:none;font-size:100%.mw-parser-output .cs1-visible-errorfont-size:100%.mw-parser-output .cs1-maintdisplay:none;color:#33aa33;margin-left:0.3em.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-formatfont-size:95%.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-leftpadding-left:0.2em.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-rightpadding-right:0.2em.


• 
  Schumacher, Benjamin; Westmoreland, Michael (1997), "Sending classical information via noisy quantum channels", Phys. Rev. A, 56: 131–138, Bibcode:1997PhRvA..56..131S, doi:10.1103/PhysRevA.56.131.


• 
  Wilde, Mark M. (2017), Quantum Information Theory, Cambridge University Press, arXiv:1106.1445, Bibcode:2011arXiv1106.1445W, doi:10.1017/9781316809976.001
  


• 
  Sen, Pranab (2012), "Achieving the Han-Kobayashi inner bound for the quantum interference channel by sequential decoding", IEEE International Symposium on Information Theory Proceedings (ISIT 2012), pp. 736–740, arXiv:1109.0802, doi:10.1109/ISIT.2012.6284656.


• 
  Guha, Saikat; Tan, Si-Hui; Wilde, Mark M. (2012), "Explicit capacity-achieving receivers for optical communication and quantum reading", IEEE International Symposium on Information Theory Proceedings (ISIT 2012), pp. 551–555, arXiv:1202.0518, doi:10.1109/ISIT.2012.6284251.