Desurvire E. Classical and Quantum Information Theory: An Introduction for the Telecom Scientist
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22 Quantum data compression
This chapter describes the principle of compression in quantum communication chan-
nels. The underlying concept is that it is possible to convey “faithfully” a quantum
message with a large number of qubits, while transmitting a compressed version of
this message with a reduced number of qubits through the channel. Beyond the mere
notion of fidelity, which characterizes the quality of quantum message transmission,
the description brings the new concept of typicality in the space defined by all possi-
ble “quantum codewords.” The theorem of Schumacher’s quantum compression states
that for a qubit source with von Neumann entropy S, the message compression factor
R has S + ε for the lower bound, where ε is any nonnegative parameter that can be
made arbitrarily small for sufficiently long messages (hence, R ≈ S is the best possible
compression factor). An original graphical and numerical illustration of the effect of
Schumacher’s quantum compression and the evolution of the typical quantum-codeword
subspace with increasing message length is provided.
22.1 Quantum data compression and fidelity
In this chapter, we have reached the stage where it is possible to start addressing the
issues that are central to information theory, namely, “How efficiently can we code
information in a quantum communication channel?” both in terms of economy of means –
the concept of data compression – and accuracy of transmission – the concept of message
integrity or minimal data error, referred to here as fidelity. In QIT, the concepts of
compression and fidelity are, in fact, intimately and nicely related, as described in this
section.
As described in the previous chapter, a quantum communication channel requires
an alphabet of pure states, which are used to encode the classical information con-
tained in message bits, letters, or codewords. In the classical case, a message like
m = 1001110101000110 can be viewed either as a sequence of 16 individual bits that
are transmitted one at a time, or as a single “block” of two-byte size (one byte is an
eight-bit sequence). Similarly, we may conceive of a full quantum message not as a
time sequence of n qubit symbols, but as a “block” represented by the tensor state
|M=|x
1
x
2
...x
n
, with each qubit |x
i
being randomly selected from a given symbol
alphabet
{
|a
k
}
k=1...N
of size N . Since each symbol |a
k
is associated with an occurrence
458 Quantum data compression
probability p
k
, we can define a symbol density operator as:
ρ =
N
k=1
p
k
|a
k
a
k
|. (22.1)
Consistently, the full symbol block is characterized by the message density operator:
ρ
M
= ρ ⊗ ρ ⊗···⊗ρ ≡ ρ
⊗n
. (22.2)
I shall now provide a basic illustration of the above, which will also enable me to introduce
the concept of fidelity. Assume a quantum communication channel with messages based
on a two-symbol alphabet of pure states
{
|a, |b
}
, defined as:
|a=|0=
1
0
|b=
1
√
2
(
|0+|1
)
=
1
√
2
1
1
.
(22.3)
The classical message source, X , is assumed to be uniformly distributed, so the two
quantum symbols have equal probabilities p
a
= p
b
= 1/2. Thus, the symbol density
operator is given by
ρ = p
a
|aa|+p
a
|bb|
=
1
2
10
00
+
1
4
11
11
≡
1
4
31
11
.
(22.4)
The eigenvalues of ρ are easily found to be λ
a
= (1 + 1/
√
2)/2 ≡ cos
2
(π/8) and λ
a
=
(1 − 1/
√
2)/2 ≡ sin
2
(π/8), respectively. The corresponding eigenvectors |λ
a
, |λ
b
are
|λ
a
=
cos
π
8
sin
π
8
,
|λ
b
=
sin
π
8
−cos
π
8
.
(22.5)
and in the eigenstate basis
{
|λ
a
, |λ
b
}
, the density operator ρ has the diagonal form:
ρ =
λ
a
0
0 λ
b
=
cos
2
π
8
0
0sin
2
π
8
. (22.6)
Finally, the VN entropy associated with any message symbol, or per-qubit entropy is
S(ρ) =−λ
a
log λ
a
− λ
b
log λ
b
=−λ
a
log λ
a
− (1 −λ
a
)log(1− λ
a
)
≡ f
cos
2
(π/8)
≡ 0.6008.
22.1 Quantum data compression and fidelity 459
Let us now calculate the overlap between the eigenstates |λ
a
, |λ
b
and the quantum
symbols |a, |b. Using Eqs. (22.3) and (22.5), we obtain:
|λ
a
|a|
2
=|λ
a
|b|
2
= cos
2
π
8
= λ
a
≈ 0.8535
|λ
b
|a|
2
=|λ
b
|b|
2
= sin
2
π
8
= λ
b
≈ 0.1465.
(22.7)
The overlap between the subspaces defined or “spanned” by either |λ
a
or |λ
b
and the
full space defined or spanned by |a, |b is, thus, observed to be significantly greater in
the first case (i.e., 85% vs. 15%, or λ
a
vs. λ
b
).
We may now introduce the concept of fidelity, defined by
F =ψ|ρ|ψ≡p
a
|ψ|a|
2
+ p
b
|ψ|b|
2
, (22.8)
where |ψ is an arbitrary measurement state. Here, since p
a
= p
b
= 1/2, the fidelity
reduces to F = (|ψ|a|
2
+|ψ|b|
2
)/2. From Eq. (22.7), it is clear that the fidelity is
a maximum for the measurement choice |ψ=|λ
a
, regardless of whether |a, |b was
sent by the originator, which gives F = 0.853. Hence, {|ψ} = {|λ
a
} corresponds to
a one-dimensional “likely subspace” with which any message symbol is most strongly
overlapping.
Let us now extend the above notion of “likely subspace” to the case of a message of
three qubits long. The eight equally-likely message outcomes |M are
|M=|aaa, |aab, |aba, |abb, |baa, |bab, |bba, |bbb. (22.9)
Let |ψ=|λ
i
λ
j
λ
k
be an arbitrary measurement state with λ
i
,λ
j
,λ
k
= λ
a
,λ
b
.Using
the property |ψ|M|
2
=|λ
i
λ
j
λ
k
|xyz|
2
=|λ
i
|x|
2
|λ
j
|y|
2
|λ
k
|z|
2
, together with
the result in Eq. (22.7), we obtain the space overlaps applicable to any message |M:
|λ
a
λ
a
λ
a
|M|
2
= λ
3
a
= cos
6
π
8
= 0.6219, (22.10)
|λ
a
λ
a
λ
b
|M|
2
=|λ
a
λ
b
λ
a
|M|
2
=|λ
b
λ
a
λ
a
|M|
2
= λ
2
a
λ
b
= cos
4
π
8
sin
2
π
8
= 0.1067,
(22.11)
|λ
a
λ
b
λ
b
|M|
2
=|λ
b
λ
a
λ
b
|M|
2
=|λ
b
λ
b
λ
a
|M|
2
= λ
a
λ
2
b
= cos
2
π
8
sin
4
π
8
= 0.0183,
(22.12)
|λ
b
λ
b
λ
b
|M|
2
= λ
3
b
= sin
6
π
8
= 0.0031. (22.13)
It is seen that the subspace defined by =
{
|λ
a
λ
a
λ
a
, |λ
a
λ
a
λ
b
, |λ
a
λ
b
λ
a
, |λ
b
λ
a
λ
a
}
is the most likely, while its orthogonal subspace complement,
⊥
={|λ
a
λ
b
λ
b
,
|λ
b
λ
a
λ
b
, |λ
b
λ
b
λ
a
, |λ
b
λ
b
λ
b
}, is the least likely. Such a statement means that mak-
ing any measurement of the message using a state |ψ from the eigenstate basis =
{
|λ
a
, |λ
b
}
⊗3
is more likely to project the state in the subspace than in the subspace
⊥
,
the corresponding probabilities being p() = λ
3
a
+ 3λ
2
a
λ
b
= 0.6219 +3 × 0.1067 =
0.942 ≡ 1 −δ and p(
⊥
) = 3λ
a
λ
2
b
+ λ
3
b
= 3 × 0.0183 +0.0031 = 0.058 ≡ δ, respec-
tively. We may then define E as the projector on the likely four-dimensional subspace
460 Quantum data compression
, according to
E =
|λ
i
λ
j
λ
k
∈
|λ
i
λ
j
λ
k
λ
i
λ
j
λ
k
|
=|λ
a
λ
a
λ
a
λ
a
λ
a
λ
a
|+|λ
a
λ
a
λ
b
λ
a
λ
a
λ
b
|
+|λ
a
λ
b
λ
a
λ
a
λ
b
λ
a
|+|λ
b
λ
a
λ
a
λ
b
λ
a
λ
a
|,
(22.14)
or as expressed in the eigenstate basis :
E =
10000000
01000000
00100000
00010000
00000000
00000000
00000000
00000000
. (22.15)
In the same eigenstate basis, the message’s density operator ρ
M
= ρ
⊗n
takes the diagonal
form:
ρ
M
=
λ
3
a
0000000
0 λ
2
a
λ
b
000000
00λ
2
a
λ
b
00000
00 0λ
2
a
λ
b
0000
0000λ
a
λ
2
b
000
00000λ
a
λ
2
b
00
000000λ
a
λ
2
b
0
0000000λ
3
b
. (22.16)
Based on the definitions in Eqs. (22.15) and (22.16), we clearly have
tr(ρ
M
E) = p() = 1 −δ. (22.17)
We shall now refer to as the typical subspace, which is reminiscent of the typical set of
classical information theory described in Chapter 13. Likewise, any message of the type
|ψ
typ
=|λ
a
λ
a
λ
a
, |λ
a
λ
a
λ
b
, |λ
a
λ
b
λ
a
, |λ
b
λ
a
λ
a
represents the quantum analog of the
typical sequences described in that chapter. We shall now refer to them as typical states.
At this stage, it is only an intuitive notion that as the message length n is increased,
the probability associated with the typical subspace asymptotically increases and
that associated with
⊥
asymptotically decreases, meaning that the uncertainty δ may
become arbitrarily small, and from Eq. (22.17), we may be able to achieve the condition
tr(ρ
M
E) = p() ≈ 1. These observations will be discussed again.
We are now going to exploit the possibility of partitioning the space into typical and
atypical subspaces in order to achieve quantum compression in the message. Assume, for
instance, that we want to reduce the 3-qubit message to a 2-qubit, compressed, message.
The following describes how the originator must proceed. First, the originator should
apply a unitary transformation U , such that the four typical states (indifferently called
22.1 Quantum data compression and fidelity 461
|ψ
typ
) are transformed into states of the form |xy0=|xy⊗|0, and the other four
(atypical) states are transformed into states of the form |xy1=|xy⊗|1. This is just a
specific way of re-encoding the original message |ψ.Call|ψ
=U|ψthe result of this
re-encoding. With the knowledge of the re-encoding transformation U , the recipient is
then able to retrieve the original message state |ψ by performing the reverse operation,
namely |ψ=U
−1
|ψ
.
This above notion of message re-encoding being understood, consider now that the
originator has the option of implementing the following trick. Assume indeed that she
or he performs a measurement on the third qubit of |ψ
:
r
If the outcome of the fuzzy measurement is 0, then the message state (and its post-
measurement) is of the form |xy0, a typical state;
r
If the outcome of the fuzzy measurement is 1, then the message state (and its post-
measurement) is of the form |zk1, an atypical state.
Based on these two fuzzy measurement outcomes, the originator then takes the corre-
sponding actions:
r
If the outcome is 0, the remaining two qubits |xy, which we call |ψ
comp1
, are sent
through the quantum channel;
r
If the outcome is 1, a 2-qubit state |ψ
comp2
is sent through the quantum channel; such
that U
−1
(|ψ
comp2
⊗|0) =|λ
a
λ
a
λ
a
, which is the most likely typical state.
From the recipient’s end, it is understood as a rule that the qubit |0 must be appended
to the incoming message, which, thus, becomes either |ψ
comp1
⊗|0 or |ψ
comp2
⊗
|0. The rule also has it that the inverse transform U
−1
must be applied. From this
“decompression” operation, the recipient, thus, obtains either one of the two states:
|ψ
=U
−1
(|ψ
comp1
⊗|0) = U
−1
U |xy0=|ψ
typ
|ψ
=U
−1
(|ψ
comp2
⊗|0) =|λ
a
λ
a
λ
a
.
(22.18)
On decompression, the recipient is, thus, able to retrieve the original message state
|ψ
typ
, while occasionally getting a “junk” or “best-guess” message |λ
a
λ
a
λ
a
.This
coding trick, along with the agreed rules, has, thus, made it possible for the originator to
transmit the 3-qubit original message to the recipient in a compressed, 2-qubit message,
and for the recipient to uncompress it into a 3-qubit message. But how faithful is this
whole operation? To answer this question, we must express the density operator ˜ρ of
the uncompressed message, after reconstruction by the recipient. The original message,
|ψ, has density operator |ψψ|. Its projection on the typical space is E|ψψ|E
+
=
E|ψψ|E, which provides a first component of ˜ρ, corresponding to the typical message
states |ψ
typ
. The second component of ˜ρ, corresponding to the other atypical message
states, can be heuristically written ˜ρ
junk
=ψ |(I − E)|ψ|λ
a
λ
a
λ
a
λ
a
λ
a
λ
a
|, where I =
I
⊗3
is the identity matrix. Clearly, I − E is the projector on the atypical space
⊥
.
Thus, ψ|(I − E)|ψ equals one if |ψis an atypical state, with corresponding projector
|λ
a
λ
a
λ
a
λ
a
λ
a
λ
a
|, and ψ |(I − E)|ψ is equal to zero otherwise. The transmitted and
462 Quantum data compression
reconstructed message, thus, has the complete density operator
˜ρ = E|ψψ|E +ψ|(I − E)|ψ|λ
a
λ
a
λ
a
λ
a
λ
a
λ
a
|
= E|ψψ |E + ˜ρ
junk
.
(22.19)
From the definition in Eq. (22.8), we obtain the fidelity
˜
F =ψ|˜ρ|ψ
=ψ |E|ψψ|E|ψ+ψ|˜ρ
junk
|ψ
=|ψ|E|ψ|
2
+ψ|(I − E)|ψ|ψ|λ
a
λ
a
λ
a
|
2
= P()
2
+ P(
⊥
) ×|ψ|λ
a
λ
a
λ
a
|
2
≡
(
1 − δ
)
2
+ δ|ψ |λ
a
λ
a
λ
a
|
2
.
(22.20)
Substituting δ = 0.058 and |λ
a
λ
a
λ
a
|ψ|
2
= λ
3
a
= 0.6219 from Eq. (22.10) into
Eq. (22.20), we obtain
˜
F =
(
1 − 0.058
)
2
+ 0.058 × 0.629 ≡ 0.923. (22.21)
The result
˜
F = 0.923 compares well with the fidelity value F = 0.853 obtained earlier.
To recall, this is the fidelity obtained by “guessing” any missing qubit is likely to be
|λ
a
. Thus, the alternate approach of transmitting only the first two qubits as uncoded,
and leaving the recipient the task of “guessing” the third missing one (likely to be |λ
a
)
is F = 0.853. Such a guess-based coding for compression does not have a fidelity as
high as that of the above coding, for which
˜
F = 0.923.
The example has shown that it is possible to compress a quantum message of
n-qubit size (n = 3) into a coded version of m-qubit size (m = 2), with relatively
high fidelity. Here, the compression factor that was achieved is η = m/n = 2/3 =
0.666 ... = 66.66%. The next section will establish that for messages of asymptotically
increasing length, the achievable compression factor is bounded according to
η ≥ S(ρ), (22.22)
where S(ρ) is the VN entropy carried by each of the message symbols. This result
is known as Schumacher’s (quantum coding) theorem. In this example, the per-qubit
entropy is S(ρ) = 0.6008, corresponding to a best compression factor of η = 60.08%.
Therefore, for 3-qubit messages based on the symbol density matrix defined in
Eq. (22.6), there is no code capable of achieving compression down to a single
qubit (η
= 0.333 ...) with any high fidelity. Such a compression factor would require
S(ρ) ≤ 0.333 ...Based on Schumacher’s theorem, the key conclusion is that the lower
the VN entropy per qubit, the higher the message compression potential.Wemay,thus,
view the VN entropy as representing redundant information, meaning susceptible to
compression. In the limiting case S(ρ) = 1(orρ = I /2), no compression is possible.
This density operator corresponds to the qubit of the type |± =
(
|0±|1
)
/
√
2, where
the information randomness is evenly distributed between 0 and 1 cbits, as with a clas-
sical source having the maximal entropy H (X) = 1 bit. This observation is consistent
with the fact that there is no code that is capable of compressing a purely random bit
sequence as generated by a classical source H(X
n
) = n. But as soon as randomness
is no longer evenly distributed between 0 and 1 bits (H (X
n
) < n), there is information
redundancy and, hence, the possibility of reducing the sequence to a size m < n.
22.1 Quantum data compression and fidelity 463
Table 22.1 Probabilities associated with quantum messages of length n = 4, defining the typical space
=
{
|λ
a
λ
a
λ
a
λ
a
, |λ
a
λ
a
λ
a
λ
b
, |λ
a
λ
a
λ
b
λ
b
, |↔
}
with likelihood 1 − δ = 0.9888.
Codeword type Probability p Number N Probability Np 1 − δδ
|λ
a
λ
a
λ
a
λ
a
0.5308 1 0.5308
|λ
a
λ
a
λ
a
λ
b
0.0911 4 0.3643
|λ
a
λ
a
λ
b
λ
b
0.0156 6 0.0937 0.9888
|λ
a
λ
b
λ
b
λ
b
0.0027 4 0.0107
|λ
b
λ
b
λ
b
λ
b
0.0005 1 0.0005 0.0112
16 1.0000
I have stated earlier that as the message length n is increased, the probability associated
with the typical subspace asymptotically increases, and that associated with
⊥
asymptotically decreases. This suggests that given a compression factor η = S(ρ), the
fidelity F asymptotically increases with the message length n. Here, I shall heuristically
verify this property by considering messages longer than three qubits, and evaluating
the corresponding fidelity, while assuming S(ρ) = 0.6008 with the eigenvalues λ
a
=
0.8535,λ
b
= 0.1465.
The probabilities associated with messages of 4-qubit length (n = 4) are defined in
Table 22.1. This table reads as follows. We call “codeword” any 4-qubit |λ
i
λ
j
λ
k
λ
l
with λ
i
,λ
j
,λ
k
,λ
l
= λ
a
,λ
b
. We call “codeword type” any 4-qubit obtained by per-
mutation of the same values λ
i
,λ
j
,λ
k
,λ
l
. Each type has a probability p = λ
i
λ
j
λ
k
λ
l
and a number N given by the associated combinatorics. For instance, the codeword
type |λ
a
λ
a
λ
b
λ
b
has p = λ
2
a
λ
2
b
= 0.0156 and N = C
2
4
= 6, as shown in Table 22.1.
The product Np, also shown in the table, corresponds to the likelihood of all
messages based on a given codeword type. We define the typical space as =
{
|λ
a
λ
a
λ
a
λ
a
, |λ
a
λ
a
λ
a
λ
b
, |λ
a
λ
a
λ
b
λ
b
, |↔
}
, where ↔ stands for all possible permuta-
tions of the index i, j, k, l, corresponding overall to 1 + 4 + 6 = 11 “typical” code-
words. We observe from the table that has a total likelihood of P() = 1 − δ =
0.5308 + 0.3643 + 0.0937 = 0.9888, while for the complementary space
⊥
, the like-
lihood is p(
⊥
) = δ = 0.0112. Similarly to the definition in Eq. (22.20), we obtain the
fidelity:
˜
F
n=4
= P()
2
+ P(
⊥
) ×|ψ
|
λ
a
λ
a
λ
a
λ
a
|
2
=
(
1 − δ
)
2
+ δλ
4
a
=
(
0.9888
)
2
+ 0.0112 ×
(
0.8535
)
4
≡ 0.983.
(22.23)
This result compares favourably with the fidelity
˜
F
n=3
= 0.923 obtained earlier for the
compression of quantum messages of length n = 3, which illustrates the property that
fidelity increases with message length. In the case n = 4, however, the compression
factor is η
n=4
= 3/4 = 0.75, which represents less compression than in the previous
case, where η
n=3
= 2/3 = 0.66. On the other hand, we cannot compress the 4-qubit
message into a 2-qubit one, because
S(ρ) × 4
=
2.40
= 3 bits, meaning that the
minimum length of the compressed message is three qubits.
464 Quantum data compression
Consider, next, the case n = 5, based on the same assumptions for S(ρ), λ
a
,λ
b
.
We have
S(ρ) × 5
=
3.004
= 4 bits, which means that the minimum length of
the compressed message is four qubits. The associated compression factor is η
n=5
=
4/5 = 0.80, to compare with η
n=4
= 3/4 = 0.75 and η
n=3
= 2/3 = 0.66 in the previous
cases. Thus, as we expect the fidelity to be higher with messages of length n = 5, the
compression performance is poorer than in the previous cases.
Consider, next, the case n = 6, based on the same assumptions for S(ρ), λ
a
,λ
b
.
We have
S(ρ) × 6
=
3.608
= 4 bits, which means that the minimum length of the
compressed message is four qubits. Thus, a compression factor of η
n=6
= 4/6 = 0.66
is achievable, which represents compressing the 6-qubit original message into a
4-qubit one. As the next section describes, the compression code requires the recip-
ient to manipulate the transmitted states |ψ
comp1
⊗|00 or |ψ
comp2
⊗|00, with the
first case corresponding to the typical or most likely subspace. I will also clarify how
the dimension of the typical subspace can be defined formally.
22.2 Schumacher’s quantum coding theorem
In this section, I shall formalize Schumacher’s quantum coding theorem. The driving
concept is that it is possible to encode a message with high fidelity when using quantum
states from the typical subspace . The key property of is that it asymptotically reaches
a dimension close to 2
nS(ρ)
. It is useful to look back at Section 13.2 concerning typical
sets. To recall, the typical set represents roughly 2
nH(X)
bit strings of length n, referred
to as typical sequences. Such typical sequences roughly contain nq 1 bits and n(1 −q)
0 bits, with H(X) = f (q) being the source entropy of the sequence bits, assumed to be
generated independently, (H(X
n
) = nH(X)). The fundamental property is that when n
becomes large, any typical sequence asymptotically has the probability 2
−nH(X)
of being
observed. Thus, there is a one-to-one conceptual correspondence between the typical set
of classical bit sequences, of size 2
nH(X )
, and the typical subspace of quantum state
messages, of dimension close to 2
nS(ρ)
.
Consider, now, the quantum message block ρ
M
= ρ
⊗n
of length n defined in
Eq. (22.2), where ρ is the density operator associated with any of the individual mes-
sage symbols, as defined in Eq. (22.1).The2
n
eigenvalues and eigenstates of ρ
M
are
µ
1
,µ
2
...µ
n
and |µ
1
, |µ
2
...|µ
n
, respectively. To recall, the eigenvalues µ
i
represent
the probability that the message is in the state |µ
i
. Formally, the typical subspace is
defined by the set of eigenstates
{
|µ
i
}
for which the eigenvalues µ
i
satisfy the double
inequality:
2
−n(S+ε)
≤ µ
i
≤ 2
−n(S−ε)
, (22.24)
where S ≡ S(ρ) and ε is a given positive real, which can be arbitrary small. We note that
this double inequality is conceptually identical to that in Eq. (13.26), corresponding to
the formal definition of typical sequences.
Let me immediately illustrate this definition of the typical subspace by means of
the n = 3 example used in the previous section. For reading clarity, we recall here the
22.2 Schumacher’s quantum coding theorem 465
Table 22.2 Lower bounds for the parameter ε, defining different
possibilities for the typical subspace and its complement
⊥
.
Typical subspace ()
µ
i
ε and complement (
⊥
)
λ
3
a
0.6219 0.372
1
λ
2
a
λ
b
0.1067
2
λ
2
a
λ
b
0.1067 0.475
λ
2
a
λ
b
0.1067
3
λ
a
λ
2
b
0.0183 1.323
⊥
1
λ
a
λ
2
b
0.0183
⊥
2
λ
a
λ
2
b
0.0183
λ
3
b
0.0031 2.171
⊥
3
density matrix of ρ
M
:
ρ
M
=
λ
3
a
0000000
0 λ
2
a
λ
b
000000
00λ
2
a
λ
b
00000
00 0λ
2
a
λ
b
0000
0000λ
a
λ
2
b
000
00000λ
a
λ
2
b
00
000000λ
a
λ
2
b
0
0000000λ
3
b
, (22.25)
which shows that the 2
n
eigenvalues µ
i
of ρ
M
are of the form µ
i=u,v
= λ
u
a
λ
v
b
with
u + v = n. Taking the base-2 logarithm of Eq. (22.24),wehave
S + ε ≥
1
n
log
1
µ
i
≥ S − ε
↔|
1
n
log
1
µ
i
− S|≤ε.
(22.26)
We note that the inequality is conceptually identical to that in Eq. (13.25), correspond-
ing to the formal definition of typical sequences. Substituting n = 3, S = 0.6008, λ
a
=
cos
2
π/8 = 0.853, λ
b
= sin
2
π/8 = 0.146, µ
1
= λ
3
a
, µ
2
= µ
3
= µ
4
= λ
2
a
λ
b
, µ
5
=
µ
6
= µ
7
= λ
a
λ
2
b
, and µ
8
= λ
3
b
in Eq. (22.26), we obtain the lower bounds for ε listed in
Table 22.2. We observe that ε = 0.372 (eigenvalue µ
1
= λ
3
a
) defines a first, 1D typical
subspace
1
, spanned by |λ
a
λ
a
λ
a
. A second possible boundary, ε = 0.475 (eigen-
values µ
1
and µ
2
= µ
3
= µ
4
= λ
2
a
λ
b
), defines a 4D typical subspace
2
spanned by
{
|λ
a
λ
a
λ
a
, |λ
a
λ
a
λ
b
, |λ
a
λ
b
λ
a
, |λ
b
λ
a
λ
a
}
. A third possible boundary, ε = 0.475 (eigen-
values µ
1
,µ
2
and µ
5
= µ
6
= µ
7
= λ
a
λ
2
b
) defines a 7D typical subspace
3
spanned by
{
|λ
a
λ
a
λ
a
, |λ
a
λ
a
λ
b
, |λ
a
λ
b
λ
a
, |λ
b
λ
a
λ
a
, |λ
a
λ
b
λ
b
, |λ
b
λ
b
λ
a
, |λ
b
λ
a
λ
b
}
.Theremain-
ing subspace, spanned by |λ
b
λ
b
λ
b
, corresponds to the smallest atypical subspace,
⊥
=
⊥
3
.
466 Quantum data compression
Table 22.3 Parameter δ and fidelity
˜
F
corresponding to the typical
subspace choices for and related dimension dim().
dim() p() =
i
µ
i
δ
˜
F
1
1 0.6219 0.378 0.622
2
4 0.9419 0.058 0.923
3
7 0.9969 0.003 0.996
In the previous section, we heuristically used
2
as “the” typical subspace, but it
is now clear that the other subspaces
1
and
3
are also eligible under the intrinsic
boundary definition in Eq. (22.24) or Eq. (22.26) for the corresponding eigenvalues,
as defined by the parameter ε. Given the highest value of ε in the typical subspaces
=
1
,
2
,
3
, we shall call any “codeword” |µ
i
=|λ
j
λ
k
λ
l
∈ as “ε-typical.”
To recall, the probability p() that any message belongs to the typical subspace is
given by the sum of the associated eigenvalues µ
i
. Formally, if E is the projector onto
,wehave
p() =
dim()
i=1
µ
i
= tr(ρ
M
E) ≥ 1 − δ, (22.27)
where δ is a positive real and smaller than unity. Table 22.3 shows the values of the
parameter δ corresponding to the typical subspaces shown in Table 22.2, and the corre-
sponding fidelity as defined in the previous section by
˜
F =
(
1 − δ
)
2
+ δλ
3
a
,Eq.(22.20).
It is seen from the table that the fidelity increases as the parameter δ decreases and
as the dimension of the typical subspace dim() = tr(E), or the number of ε-typical
codewords increases. We note that the case =
1
, which has only one codeword, is
a poor choice, since the fidelity is
˜
F = 62.2%, corresponding to a “useless channel”
almost half of the time. And what valuable information can be transmitted with only
one codeword, independently of this consideration? The best choice is =
3
, which
has seven codewords (out of 2
3
= 8 codeword possibilities), since the fidelity reaches
the maximum;
˜
F = 99.6%. But we have now reached a confusing situation where there
apparently exist several possible choices for the typical subspace, which are all consistent
with previous eligibility criteria. We, thus, need to develop the analysis further, bearing
in mind that we want to show that the dimension of the typical subspace is close to2
nS(ρ)
.
In the above n = 3 example, we have 2
nS(ρ)
= 2
3×0.6008
= 3.488 ≈ 4 = dim(
2
), which
indicates that
2
is, in fact, the correct choice! But to reach such a conclusion, we
must consider messages of asymptotically increasing length, and we will have to forget
the previous n = 3 example, despite its usefulness in demonstrating the possibility of
message compression.
As we have seen earlier, the two parameters ε, δ, defined in Eqs. (22.26) and (22.27),
determine a lower bound for the probability p() = tr(ρ
M
E) of a given message
codeword to be ε-typical. More generally, we may state that given the two parame-
ters ε, δ, there exists a message length n sufficiently long to provide the condition that
p() is at least 1 −δ. To keep the focus, we shall not worry here about formally