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Second order asymptotics of visible mixed quantum source coding via universal codes

The simplest example of a quantum information source with memory is a mixed source which emits signals entirely from one of two memoryless quantum sources with given a priori probabilities. Considering a mixed source consisting of a general one-parameter family of memoryless sources, we derive the second order asymptotic rate for fixed-length visible source coding. Furthermore, we specialize our main result to a mixed source consisting of two memoryless sources. Our results provide the first example of second order asymptotics for a quantum information-processing task employing a resource with memory. For the case of a classical mixed source (using a finite alphabet), our results reduce to those obtained by Nomura and Han [IEEE Trans. on Inf. Th. 59.1 (2013), pp. 1-16]. To prove the achievability part of our main result, we introduce universal quantum source codes achieving second order asymptotic rates. These are obtained by an extension of Hayashi's construction [IEEE Trans. on Inf. Th. 54.10 (2008), pp. 4619-4637] of their classical counterparts.


Introduction
Source coding (or data compression) is essential for efficient storage and transmission of information. Hence, evaluating the optimal rate of data compression is a fundamental problem in information theory. In classical information theory, the simplest class of sources is composed of so-called i.i.d. or stationary, memoryless sources, the name 'memoryless' arising from the fact that there is no correlation between successive signals emitted by such a source. Although these sources play a prominent role in information theory, in real-world applications the assumption of sources being memoryless is not necessarily justified. This is why it is important to study data compression for sources with memory. The simplest example of such a source is a mixed source. It can be constructed from two i.i.d. sources as follows. One associates a priori probabilities, say t and (1 − t), to the two sources respectively. Then the mixed source is one for which all successive signals are emitted from the first source with probability t, or from the second source with probability (1 − t). The memory of the mixed source can be trivially seen to be governed by a two-state Markov chain which is aperiodic but not irreducible, and hence such a source is non-ergodic (see e.g. [19]).
Optimal rates of reliable data compression for the above sources and their quantum analogues were originally evaluated under the requirement that the error incurred in the compressiondecompression scheme vanishes in the asymptotic limit. The optimal asymptotic rate for a classical i.i.d. source is given by its Shannon entropy [24], whereas the corresponding rate for a quantum memoryless source is given by its von Neumann entropy [23]. The optimal rates for mixed sources were derived by employing the so-called Information Spectrum Approach 1 (ISA) by Han [8] in the classical case, and in [4] in the quantum case. It was shown to be given by the maximum of the Shannon (resp. von Neumann) entropies of the two underlying classical (resp. quantum) memoryless sources.
More recently, a more refined asymptotic analysis of data compression for memoryless sources under the (more reasonable) requirement of a non-zero error threshold ε ∈ (0, 1) was done. The quantity analysed was the minimum compression length, which we denote by log M n ≡ log 2 M n . In the classical case this is the minimum number of bits needed to compress signals emitted by n uses of the source so that they can be recovered with an error of at most ε upon decompression. In the quantum case, it is the minimum dimension of the compressed Hilbert space compatible with the given error threshold. The second order asymptotic expansions of the minimum compression length for both the classical and quantum cases were proved to be of the form Here, the coefficient a of the leading order term constitutes the first order asymptotics of the minimum compression length, and, as expected, is given by the optimal asymptotic rate. The coefficient b is a function of both the source and the allowed error threshold ε. It constitutes the second order asymptotics and is hence referred to as the second order rate. It is given by − √ V Φ −1 (ε), where Φ −1 denotes the inverse of the cumulative distribution function of the standard normal distribution, and V denotes the information variance of the source. The asymptotic expansion (1) was evaluated for fixed-length source coding in the classical case by Hayashi [12] and in the quantum case (for the visible setting) in [5].
Obtaining second order asymptotic expansions for any information-processing task employing resources with memory is a more challenging task. The first foray into this task was made by Polyanskiy, Poor and Verdú [21], who obtained second order expansions for the capacity of a classical mixed channel (see also [27]). In [18], Nomura and Han evaluated second order optimal rates for fixed-length source coding for a classical mixed source (see also [12]). In this paper, we consider fixed-length source coding for a mixed source constructed analogously from two quantum memoryless sources, and obtain optimal second order rates in the visible setting. In the classical case, our result reproduces the optimal rates of Nomura and Han in the finite-alphabet setting. The mathematical quantity playing a key role in our derivations is the information spectrum entropy, which is obtained from the information spectrum relative entropy defined in [5]. The latter is a one-shot version of the spectral divergence rates [9,13,17], which are the fundamental quantities of the ISA. We obtain one-shot bounds, that is bounds on the minimum compression length for a single use of a mixed source, in terms of the information spectrum entropy, and then employ the second order asymptotic expansion of the latter to obtain the optimal second order rates.
The optimal asymptotic rate provides a sharp threshold for data compression of memoryless sources in the following sense: compressing at a lower rate results in the compressiondecompression scheme failing with certainty (i.e. the probability of error tends to 1 in the asymptotic limit) whereas schemes with higher rates succeed with certainty (i.e. the probability of error tends to 0). This property of the optimal asymptotic rate of memoryless sources is referred to as the strong converse property. It is, however, not satisfied in general for a mixed source: there is a separation between the strong converse rate (that is, the maximum asymptotic rate below which failure is certain) and the optimal asymptotic rate. The former is given by the minimum of the entropies of the two memoryless sources constituting the mixed source, whereas (as mentioned previously) the latter is given by the maximum of the entropies. This violation of the strong converse property for mixed sources, which was first established via the ISA [8,4], follows directly from our one-shot bounds.
The paper is organized as follows. After setting the notation for the paper and stating some useful lemmas in Section 2, we introduce the definitions of the relevant entropic quantities and their properties in Section 3. The task of fixed-length quantum source coding in the visible setting is explained in detail in Section 4. We define the minimum compression length for a given error threshold ε ∈ (0, 1) for a single use of an arbitrary source, and for n ≥ 1 uses of a mixed source. We also define the optimal asymptotic rate and strong converse rate for a mixed source. Our main results are given in Section 5. These comprise bounds on the minimum compression length for a single use (n = 1) of a mixed source (for a given error threshold ε) in terms of the information spectrum entropy, as well as its second order asymptotic expansion for sufficiently large n. In evaluating the latter, we consider three different cases depending on the relative values of the entropies and quantum information variances of the two underlying memoryless sources. These cases are analogous to the ones considered by Nomura and Han [18] in obtaining optimal rates for a classical mixed source. In Section 5.3 we recover these optimal rates in the finite-alphabet setting. The proofs of our results are given in Section 6. In Section 7 we show how the results [8,4] of the ISA for the optimal asymptotic rate for a mixed source constructed from two arbitrary (i.e. not necessarily memoryless) sources can be recovered from our one-shot bounds. In Appendix A we include a self-contained derivation of the second order asymptotic expansion of the information spectrum entropy. We make use of the following lemma.

Mathematical preliminaries
Lemma 2.1. [20] For operators A, B ≥ 0 and 0 ≤ P ≤ 1, The inverse of the cumulative distribution function (cdf) of a standard normal random variable is defined by We make use of the following lemma.

Information spectrum relative entropies
The following entropic quantities, first defined in [5], play a key role in our proofs.
(i) The information spectrum relative entropies are defined as (ii) The information spectrum entropies H ε s (ρ) and H ε s (ρ) are entropies obtained from D ε s (ρ σ) and D ε s (ρ σ) respectively as follows: The two information spectrum relative entropies of Definition 3.1(i) are related by D ε s (ρ σ) = D 1−ε s (ρ σ) (see Proposition 3.3(ii) below), and consequently, we also have that H ε s (ρ σ) = H 1−ε s (ρ σ). The information spectrum relative entropies are variants of a quantity of the same name, introduced by Tomamichel and Hayashi [26], which is denoted as D ε s (ρ σ) and is given by Definition 3.2 below. However, as proved in [5], D ε s (ρ σ) and D ε s (ρ σ) have the particular advantage of satisfying the data-processing inequality, which is considered to be an essential property of a relative entropy.
Definition 3.2. For ε ∈ (0, 1), ρ ∈ D(H) and σ ∈ P(H), Further, one obtains the following entropy from it: The following proposition lists some useful properties of the entropic quantities defined above.
(v) H ε s (ρ) and H ε s (ρ) are quasi-concave, that is, for states ρ i ∈ D(H) and a probability distribution and the same assertion holds for H ε s (ρ).
Proof. For a proof of (ii) and (iii), see Proposition 4.3 in [5].
(i) The upper bound is Proposition 4.3(i) of [5]. For the lower bound, observe that we have by Lemma 12 of [26], where is the hypothesis testing relative entropy defined by Wang and Renner [28]. The above inequality, together with the bound D ε−δ H (ρ σ) + log δ ≤ D ε s (ρ σ) given by Proposition 4.7(i) of [5] yields the claim.
(iv) Let P = { i p i ρ i ≥ 2 γ i p i σ i }, then by the linearity of the trace where the inequality follows from Lemma 2.1.
By substituting this on the right hand side of (2) we infer that γ = max i D ε s (ρ i σ i ) is feasible for D ε s ( i p i ρ i i p i σ i ), and hence the claim follows. By (ii), the same assertion also holds for D ε s (ρ σ). (v) This follows from (iv) and Definition 3.1(ii).
Our main results (Theorem 5.1 and Theorem 5.2) rely on the second order asymptotic expansion of the information spectrum entropy H ε s (ρ ⊗n ) for any ε ∈ (0, 1) and sufficiently large n. This expansion is given in terms of the quantum relative entropy D(ρ σ) and the quantum information variance V (ρ σ), which are defined as follows: The von Neumann entropy is expressible as S(ρ) = −D(ρ 1).
Note that s(ρ) is equal to the standard deviation of the probability distribution formed by the eigenvalues of ρ. ⋄ The second order asymptotic expansion of H ε s (ρ ⊗n ) is given in terms of the above quantities as follows.
Theorem 3.5. Let ρ ∈ D(H). For an arbitrary ε ∈ (0, 1) and sufficiently large n, the second order asymptotic expansion of H ε s (ρ ⊗n ) is given by This follows from the second order asymptotic expansion of D ε s (ρ ⊗n σ ⊗n ) given by Proposition 4.9 of [5]. However, it can also be proved directly. For the sake of completeness, we include this proof in Appendix A.

Fixed-length visible quantum source coding
A quantum information source is characterized by an ensemble E = {p i , |ψ i } of pure states |ψ i , where the signals |ψ i are emitted by the source with corresponding probabilities p i . We refer to E as the source ensemble and the associated density matrix (or ensemble average state) ρ = i p i ψ i as the source state.
In fixed-length quantum source coding the aim is to store the information emitted by the source in a compressed state ρ c ∈ D(H c ) with dim H c < dim H, such that it can be later decompressed to yield a state which is sufficiently close to the source state ρ, with respect to some chosen distance measure. In the one-shot setting, in which one considers a single use of the source, it is natural to allow a non-zero error in the compression-decompression scheme. Hence, a one-shot source-coding protocol is characterized by a parameter ε which denotes the maximum allowed value of the distance between the source state and the state which is obtained after decompression.
There are two different settings [1,10,29] for the compression part of the protocol outlined above: visible and blind. In this paper we only consider the visible setting. 2 There, the compressor (say, Alice) knows the identity of the signals ψ i . In fact, on each use of the source Alice receives classical information in the form of an index i labelling the signal ψ i emitted by the source. She then uses an arbitrary map V : {i} → D(H c ) to encode the signal ψ i in a state V(i) ∈ D(H c ). Note that the encoding map V (which we refer to as visible encoding) is not necessarily a quantum operation, since Alice simply prepares a quantum state V(i) on receiving the index i. In particular, V can be a non-linear map. This is in contrast to the blind setting of source coding, where the encoder does not have any knowledge about the pure states ψ i and is therefore required to apply a quantum operation E to the source state ρ. Here, we restrict the discussion to the visible setting.
In the decompression phase of the protocol, the compressed signal V(i) is subjected to a quantum operation D : D(H c ) → D(H). Now, consider two memoryless quantum information sources with source ensembles E 1 = {q i , |ϕ i } and E 2 = {r j , |χ j } and source states ρ 1 = i q i ϕ i and ρ 2 = j r j χ i respectively. As the sources are memoryless, there is no correlation in the successive signals emitted by them. Consequently, n uses of the two sources are respectively characterized by the source states ρ ⊗n 1 and ρ ⊗n 2 and source ensembles E n Let us consider a mixed source which is constructed out of these two memoryless sources as follows. We associate a-priori probabilities, t and (1 − t), to the two sources respectively. Then the mixed source is the one for which all successive signals are emitted from the first source with probability t, or from the second source with probability (1 − t). Hence, n uses of the mixed source are characterized by the source state and the source ensemble Henceforth, we denote a mixed source simply by the triple (ρ 1 , ρ 2 , t) and refer to the probability t as the mixing parameter.
In particular, for a single use of the mixed source (i.e. in the one-shot case) the source state and source ensemble are simply given by and Our aim is to derive a second order asymptotic expansion for the minimum compression length with a given error threshold, for fixed-length visible quantum source coding of a mixed source. Here, minimum compression length refers to the minimum dimension of the compressed Hilbert space. The precise definitions are given below. Since we only discuss the visible source coding setting in this paper, we will henceforth suppress the attribute 'visible' in all definitions.
We choose the ensemble average fidelity as the figure of merit in our analysis of fixed-length quantum source coding, which we define for a general quantum information source as follows: This leads to the following definitions in the one-shot setting: (ii) The ε-error one-shot minimum compression length m (1),ε (ρ) is defined by The corresponding strong converse compression length m The central quantity of interest in this paper is the minimum compression length for n uses of a mixed source, for a given error threshold ε ∈ (0, 1). This is defined as follows: For n uses of a mixed source (ρ 1 , ρ 2 , t) (with source state given by (3)), the ε-error minimum compression length for any ε ∈ (0, 1) is given by The optimal asymptotic rate and the strong converse rate for fixed-length source coding for a mixed source (ρ 1 , ρ 2 , t) are then defined as follows: Definition 4.4. For fixed-length source coding of a mixed source (ρ 1 , ρ 2 , t), the source state for n uses of which is denoted by ρ (n) and given by (3), (i) The optimal asymptotic rate is given by (ii) The strong converse rate is given by

Main results
Our main results comprise (i) bounds on the ε-error minimum compression length for a single use of the mixed source, and (ii) second order asymptotic expansions of the ε-error minimum compression length. The first of these are referred to as one-shot bounds and given in Theorem 5.1. For the derivation of the second order asymptotics, we follow the analysis of Nomura and Han [18] for a classical mixed source, distinguishing between three cases based on the relative values of the von Neumann entropies and quantum information variances of the underlying memoryless sources, and the relation between error threshold ε and mixture parameter t. These cases are elucidated in Section 5.2, and the corresponding second order asymptotic expansions are given in Theorem 5.2 therein.

One-shot bounds
Theorem 5.1. Let ρ = tρ 1 + (1 − t)ρ 2 be the source state for a single use of a mixed source, where t ∈ (0, 1) and ρ i ∈ D(H) for i = 1, 2. The minimum compression length m (1),ε (ρ) satisfies the following bounds: (i) Achievability: For ε ∈ (0, 1) and η such that 0 < η < ε, The one-shot bounds of Theorem 5.1 allow us to recover the optimal asymptotic rate and the strong converse rate for an arbitrary sequence of mixed sources, in terms of entropies defined in the ISA. This was first obtained in the classical case by Han [8], and in the quantum case in [4]. This result is stated in Proposition 7.5, which is preceded by a detailed discussion of the definitions of these rates and the relevant entropic quantities.

Second order asymptotics
For the discussion of the second order asymptotics of the ε-error minimum compression length m n,ε (ρ 1 , ρ 2 , t) for n uses of a mixed source with corresponding source state ρ (n) = tρ ⊗n 1 + (1 − t)ρ ⊗n 2 with t ∈ (0, 1) and ρ i ∈ D(H), we adhere to the discussion in [18] by considering the following three cases, abbreviating S i ≡ S(ρ i ) and s i ≡ s(ρ i ) for i = 1, 2: where the assumptions s 1 < s 2 in Case 1 and S 1 > S 2 in Case 2 and 3 are without loss of generality. We state the asymptotic expansion of the minimum compression length m n,ε (ρ 1 , ρ 2 , t) in each of the three cases in the following theorem, which constitutes our main result: Theorem 5.2. Consider a mixed source (ρ 1 , ρ 2 , t) with ρ 1 , ρ 2 ∈ D(H) and t ∈ (0, 1). Assume (without loss of generality) that either S 1 = S 2 and s 1 < s 2 or S 1 > S 2 , and let ε ∈ (0, 1). Then for sufficiently large n, the second order asymptotic expansion of the minimum compression length m n,ε (ρ 1 , ρ 2 , t) is given by the following expressions: These results immediately yield the optimal asymptotic rates in the i.i.d. case: Consider a mixed source (ρ 1 , ρ 2 , t) with ρ 1 , ρ 2 ∈ D(H) and mixing parameter t ∈ (0, 1). Then the optimal asymptotic rate of source coding is given by the maximum of the von Neumann entropies of the two sources, i.e.
On the other hand, the strong converse rate is given by the minimum of the two, that is

Relation to classical mixed-source coding
Let us briefly discuss mixed-source coding in the classical setting, as studied by Nomura and Han [18]. Consider two memoryless information sources characterized by random variables Y 1 , Y 2 taking values y ∈ Y (where Y is a countably infinite alphabet) with probabilities P Y 1 (y) and P Y 2 (y) respectively. Then a mixed source with mixing parameter t is denoted by The compression-decompression scheme consists of the encoding (or compression) map ϕ n : Y n → U Mn and decoding (or decompression) map ψ n : U Mn → Y n . Here, U Mn = {1, . . . , M n }, where M n ∈ N and log M n denotes the compression length. The probability of error incurred as a result of the compression-decompression scheme is given by ε n := P{Y n = ψ n (ϕ n (Y n ))}. Then C n := (ϕ n , ψ n , M n , ε n ) defines a code. A real number R is said to be an (a, ε)-achievable rate, if there exists a sequence of codes {C n } n∈N such that lim sup n→∞ ε n ≤ ε and lim sup The (a, ε)-fixed-length source coding rate is defined as L f (a, ε|Y) := inf{R|R is (a, ε)-achievable}.
Note that the definition of L f (a, ε|Y) incorporates both the first and second order asymptotic rates for fixed-length source coding; a statement L f (a, ε|Y) = R implies that the second order rate for the source Y is equal to R given that the first order rate is a and the maximum probability of error is ε. Nomura and Han derived the following expressions for the source coding rates for a mixed source Y with mixing parameter t. Here H(Y i ) and σ i denote the Shannon entropy and standard deviation of the random variable Y i respectively.
Our results on the second order asymptotics for fixed-length quantum source coding for a mixed source, as stated in Theorem 5.2, employ the same distinction between the three cases. For the case of a classical mixed source with a finite source alphabet Y, Theorem 5.2(ii) and (iii) directly yield the results of Nomura and Han given by Theorem 5.4(ii) and (iii). To see this, note that for the classical mixed source, S i = H(Y i ) and s i = σ i , for i = 1, 2.
The result of Theorem 5.4(i), though applicable for all ε ∈ (0, 1), does not yield an explicit expression for the second order rate c 1 . To circumvent this problem, we first assume in the quantum setting without loss of generality that s 1 < s 2 (see Case 1 in Section 5.2). For this case, by restricting ε to the interval 0, 1 2 , we are able to derive a closed expression for the second order asymptotic rate which is given by −s 2 Φ −1 (ε) (Theorem 5.2(i)). For the range ε ∈ 1 2 , 1 , however, we are only able to obtain an achievable second order asymptotic rate, given by −s 1 Φ −1 (ε) via Proposition 6.4. Note that for the particular case of a classical mixed source, our results for ε ∈ 0, 1 2 and ε ∈ 1 2 , 1 lie in the allowed ranges obtained from Theorem 5.4(i). This is because (7) implies that 6 Proofs of our main results

One-shot bounds
Proof of Theorem 5.1(i). We consider the encoding map where P = {ρ ≥ 2 −γ 1} for some γ ∈ R, the compressed Hilbert space is defined by H c := Im P , and ψ i denotes a pure state in the source ensemble E m (given by (5)) of the mixed source (ρ 1 , ρ 2 , t), with associated probability p i . In this case, we have log M = log dim H c = log Tr P ≤ γ.
Furthermore, we choose the trivial embedding H c ֒→ H as the decoding map D. The ensemble average fidelity satisfiesF We define the projectors Q i,α := {ρ i ≥ 2 −α 1} for i = 1, 2 and arbitrary α ∈ R, and let W α denote the projector onto Im Q 1,α ∪ Im Q 2,α . Then, starting from (9) we obtain where the second inequality follows from Lemma 2.1, and the third inequality follows from the fact that Q i,α ≤ W α . We have Tr Q i,α ≤ 2 α , and therefore 3 Tr W α ≤ 2 · 2 α . This yields the following lower bound on the ensemble average fidelity: and hence, for the choice γ = α − log η + 1, we havē Thus, (V, D, M ) is an ε-admissible code, and by (8) we obtain For a mixed source ρ = tρ 1 + (1 − t)ρ 2 with ρ i ∈ D(H) and t ∈ (0, 1), the quasi-concavity of H ε s (ρ) (Proposition 3.3(v)) implies that In contrast, the upper (achievability) bound in Theorem 5.1(i) involves the maximum of H ε s (ρ 1 ) and H ε s (ρ 2 ), and our goal is to derive a "matching" lower (converse) bound, i.e. a bound which is also given in terms of the maximum of the information spectrum entropies of the two sources. However, one can find examples of a mixed source (ρ 1 , ρ 2 , t) for which H In other words, using the fact that H ε s (ρ 1 ) > H cε s (ρ 1 ) (cf. Proposition 3.3(iii)) the information spectrum entropy of the source ρ 1 can be 'pushed down' below H ε s (ρ 1 ) by replacing the parameter ε by cε. The case H ε s (ρ 1 ) ≤ H ε s (ρ 2 ) is handled analogously. Employing this idea, we arrive at the following proposition. Proposition 6.1. Let ρ 1 , ρ 2 ∈ D(H), t ∈ (0, 1) and set ρ = tρ 1 + (1 − t)ρ 2 . For a fixed 0 < ε ≤ min{t, 1 − t}, we have for any x ∈ 1 t , 1 ε and y ∈ 1 1−t , 1 ε .

Second order asymptotics
Throughout the discussion, we abbreviate ρ n ≡ ρ ⊗n . For ρ 1 , ρ 2 ∈ D(H), we use the shorthand notation S i ≡ S(ρ i ) and s i ≡ s(ρ i ). The following lemma plays a crucial role in the proof of Theorem 5.2: Lemma 6.3. Let ρ 1 , ρ 2 ∈ D(H) be density matrices.
(i) If S 1 > S 2 , then for any ε 1 , ε 2 ∈ (0, 1) there exists an n 0 such that for all n ≥ n 0 we have (ii) If S 1 = S 2 and s 1 < s 2 , then for any ε ∈ 0, 1 2 there exists an n 0 such that for all n ≥ n 0 we have H (iii) If S 1 = S 2 and s 1 < s 2 , then for any δ 1 , δ 2 ∈ (0, 1) with δ 1 < δ 2 there exists an n 0 such that for all n ≥ n 0 we have Proof. (i) By Theorem 3.5, we can expand H ε 1 s (ρ n 1 ) and H ε 2 where c i ∈ O(log n) for i = 1, 2. Setting we observe that k 1 > 0 by assumption. A priori, k 2 can be of either sign; if k 2 ≤ 0 for all n, it follows immediately that H ε 1 . Assume therefore that k 2 > 0. Then we can choose n 0 such that k 1 > k 2 √ n for all n ≥ n 0 , yielding where c i ∈ O(log n) for i = 1, 2 and Φ −1 (ε) < 0 for ε ∈ 0, 1 2 , such that Φ −1 (ε) (s 2 − s 1 ) < 0 by assumption. There exists an n 0 such that we have for all n ≥ n 0 that In (iii), we have since s 2 > s 1 by assumption. Furthermore, by the monotonicity of Φ −1 , we have Φ −1 (δ 2 ) > Φ −1 (δ 1 ). The assertion now follows as in (ii).

Case 1:
We first prove an achievability result in this case, i.e. an upper bound on the minimum compression length m n,ε (ρ 1 , ρ 2 , t).
Proposition 6.4. Consider a mixed source (ρ 1 , ρ 2 , t) for which S 1 = S 2 and s 1 < s 2 . For ε ∈ 0, 1 2 , η ∈ (0, ε), and sufficiently large n, we then have On the other hand, for ε ∈ 1 2 , 1 and sufficiently large n, we have Proof. Let us first assume that ε ∈ 0, 1 2 . We consider the encoding map where ψ i denotes a pure state in the ensemble E m of (5) characterizing the mixed source (ρ 1 , ρ 2 , t), and P = {ρ ≥ 2 −γ 1} for some γ ∈ R, setting H c := Im P . Further, we choose the trivial embedding H c ֒→ H as the decoding map D. Then, we obtain log M := log dim H c = log Tr P ≤ γ.
By (10) of the proof of Theorem 5.1(i), the ensemble average fidelity satisfies the following bound:F Let us consider (16) for n uses of the mixed source (ρ 1 , ρ 2 , t), that is, we consider the input state ρ (n) = tρ n 1 + (1 − t)ρ n 2 . According to Lemma 6.3(ii), we can find n 0 such that for all n ≥ n 0 . Setting α = H ε−η s (ρ n 2 ) and recalling definition (4) of the mixed source ensemble E (n) m , we then obtain from (16) and Definition 3.1(ii) of the information spectrum entropy that, for the choice γ = α − log η + 1, Hence, we arrive at log M ≤ H ε−η s (ρ n 2 ) − log η + 1, which yields the inequality (14).
In order to prove this proposition, we need the following result, which gives an upper bound on the ensemble average fidelity: Proof. Originally proved in [10]. For a proof in our notation, see Proposition 5.7 in [5].
Proof of Proposition 6.5. Fix η, δ > 0 and ε ∈ (0, 1). We set δ 2 = ε + η and δ 1 = ε + η − δ ′ for δ ′ > 0 such that δ 1 < δ 2 . Then, Lemma 6.3(iii) implies that for sufficiently large n we have the relation for sufficiently large n. Furthermore, let Q be a projector satisfying Tr Qρ (n) = max{Tr P ρ (n) | P is a projection on H ⊗n with Tr P = M n }, for an arbitrary M n ∈ N. By Lemma 6.6 we have the following estimate: for any γ ∈ R. The second equality follows from the linearity of the trace, and the second inequality holds by Lemma 2.1 and the fact that Tr Q = M n . Choosing γ = H ε+η s (ρ 2 ), we now assume that log M n < γ + log η.
Replacing ε by ε + δ finally yields the assertion of the proposition.
This concludes the first case: Proof of Theorem 5.2(i). For δ, η > 0, we obtain the following bounds on m n,ε (ρ 1 , ρ 2 , t) by Proposition 6.4 and Proposition 6.5: Setting both η and δ equal to 1 √ n and applying Theorem 3.5 together with Lemma 2.2 yields the result.

Case 2:
We use Lemma 6.3 to derive the following achievability result for n uses of a mixed source: Proposition 6.7. Let (ρ 1 , ρ 2 , t) be a mixed source such that S 1 > S 2 . Further, for ε ∈ (0, t) let η, δ > 0 be such that ε − δ − η > 0. For sufficiently large n, we then have Proof. For any arbitrary α ∈ R we have from (10) that where V and D are defined as in the proof of Theorem 5.1 in Section 6.1. According to Lemma 6.3(i), for any given δ ′ > 0 and sufficiently large n, Substituting α = H (ε−η)/t s (ρ n 1 ) on the RHS of (21), we then infer that, for the choice γ = α − log η + 1,F Replacing ε by ε − δ then proves the claim.
We now have all the tools needed to derive the second order asymptotic expansion of the minimum compression length m n,ε (ρ 1 , ρ 2 , t) in Case 2.
Proof of Theorem 5.2(ii). Using Proposition 6.7, we obtain the upper bound after setting δ = η = 1 √ n and using Theorem 3.5 and Lemma 2.2. For the lower bound, we note that Theorem 5.1(ii) together with Definition 4.3 yield m n,ε (ρ 1 , ρ 2 , t) ≥ max H which together with (22) proves the claim.

Case 3: S 1 > S 2 , t < ε
In Case 3 we assume that t < ε, that is, the error threshold ε is greater than the probability of receiving a signal from the source (with source state) ρ 1 . Hence, we simply ignore the source ρ 1 and only use the second source ρ 2 in the encoding protocol. This leads to the following one-shot achievability result: Proposition 6.8. Let (ρ 1 , ρ 2 , t) be a mixed source. For ε ∈ (t, 1) and ε ′ = ε−t 1−t , we have Proof. We set P = {ρ 2 ≥ 2 −γ 1} and H c := Im P , and choose the encoding map where ψ i denotes a pure state in the ensemble E m given by (4), along with the trivial embedding H c ֒→ H as the decoder D. From this, we obtain log M = log dim H c = log Tr P ≤ γ.
where the first equality follows from (9) in the proof of Theorem 5.1(i) in Section 6.1. Hence, the triple (V, D, M ) constitutes an ε-admissible code, and the result follows from (23).
Proof. Since S 1 > S 2 by assumption, Lemma 6.3(i) implies that for sufficiently large n we have the relation for arbitrary δ ′ ∈ (0, 1) and f (ε) = ε+η−t 1−t . Assume now that where η > 0 and γ ∈ R is to be determined. We recall the following bound on the ensemble average fidelity, given by (19) in the proof of Proposition 6.5: s (ρ n 2 ), we obtain from (24), (25), and Definition 3.1(ii) of the information spectrum entropy that upon setting δ = tδ ′ . Hence, we have proved that if a code satisfiesF (E Replacing ε by ε + δ now yields the assertion of the proposition. This leads to the second order asymptotic expansion of m n,ε (ρ 1 , ρ 2 , t) in Case 3: Proof of Theorem 5.2(iii). The upper bound is obtained from Proposition 6.8 by using Theorem 3.5 and Lemma 2.2.
For the lower bound, our goal is to use Proposition 6.9 with the choices η = 1 √ n and δ = 1 √ n such that we arrive at To this end, we need to make sure that the assertion of Lemma 6.3(i) is still valid for these choices of η and δ, i.e.
Next we consider the case in which S 1 = S 2 . Using Theorem 5.2(i) we infer that the optimal asymptotic rate is given by For the strong converse compression length in this case, we have by Proposition 6.5 and by Proposition 6.4. In both cases, the second order asymptotic expansion for the ε-error minimum compression length is obtained by applying Theorem 3.5 together with Lemma 2.2. The claim R * (ρ 1 , ρ 2 , t) = S 1 = S 2 follows by dividing both bounds by n and taking the limit n → ∞.
Hence, Corollary 5.3 also holds for the case S 1 = S 2 .

Optimal asymptotic rates in the Information Spectrum Approach
The Information Spectrum Approach (ISA) provides a unifying mathematical framework for obtaining asymptotic rate formulae for various different tasks in information theory. The power of the approach lies in the fact that it does not rely on any particular structure or property of the resources used in the tasks. It was introduced by Han and Verdú [9] in classical information theory, and generalized to the quantum setting by Hayashi, Nagaoka and Ogawa [11,13,17,20]. The ISA was further developed in [3]. Its relation to the so-called smooth entropy framework (see e.g. [22,25]), which is an alternative approach to treating information-processing tasks employing arbitrary resources, was established in [6]. The main quantities in the ISA are the spectral divergence rates:  The spectral entropy rates S(ρ) and S(ρ) can be recovered from the information spectrum entropies H ε s (ρ) and H ε s (ρ) respectively, in the following way: Proposition 7.2. Letρ = {ρ n } n∈N be an arbitrary sequence of states with ρ n ∈ D(H ⊗n ). Then the following relations hold:

Retrieving the optimal asymptotic rates from the one-shot bounds
First, we need to adapt the definitions of the optimal asymptotic rate and strong converse rates given in Definition 4.4 to the information spectrum setting: Let t ∈ (0, 1). Given two arbitrary sequences of source statesσ = {σ n } n∈N andω = {ω n } n∈N acting on a sequence of Hilbert spaces H = {H n } n∈N , we consider a sequence of mixed sourceŝ ρ = {ρ n } n∈N , where ρ n := tσ n + (1 − t)ω n , for all n ∈ N. In [4], the optimal asymptotic rate for such a sequence of mixed sources was shown to be given by the maximum of the quantum sup-spectral entropies ofσ andω. Moreover, its strong converse rate was shown to be given by the minimum of the quantum inf-spectral entropies ofσ andω. The one-shot bounds of Theorem 5.1, together with Proposition 7.2, allow us to retrieve these results. This is done in Proposition 7.5 below.

Conclusions and open questions
We derive the second order asymptotic expansion for the minimum compression length in fixedlength quantum source coding for a mixed source (in the visible setting). To our knowledge, this is the first example of a second order asymptotic analysis of the optimal rate for a quantum information-processing task which uses a resource with memory. Previously, such analyses in the quantum setting have been restricted to memoryless (or i.i.d.) resources [26,16,14,15,5,2].
The mixed source that we consider is denoted as (ρ 1 , ρ 2 , t) where ρ 1 , ρ 2 are the density matrices characterizing two memoryless sources and t ∈ (0, 1) is the mixing parameter. All successive signals of the mixed source are emitted from the source ρ 1 with probability t, or from the source ρ 2 with probability (1 − t). We first derive one-shot bounds for the minimum compression length for a single use of such a source, in terms of the information spectrum entropy. In obtaining the second order asymptotics of the minimum compression length, for any given value of the error threshold ε ∈ (0, 1), we distinguish between three cases, based on the relative values of the von Neumann entropies and information variances of the sources ρ 1 and ρ 2 , as well as the relation between the values of the chosen error threshold ε and the mixing parameter t. This is in analogy with the second order analysis for a classical mixed source done by Nomura and Han [18]. We derive optimal first and second order asymptotic rates for all three cases, thus giving an exhaustive description of fixed-length source coding for a quantum mixed source (in the visible setting). For a classical mixed source (with a finite alphabet), our results reduce to those of Nomura and Han. In addition, from our one-shot bounds we can retrieve the known optimal asymptotic rate and the strong converse rate for fixed-length source coding for an arbitrary sequence of mixed sources. This was first derived, using the Information Spectrum Approach, by Han [8] for the classical case, and in [4] for the quantum case. For the sake of completeness, we also include a self-contained derivation of the second order asymptotic expansion of the information spectrum entropy, which is the key tool in our analysis.
It would be interesting to extend our methods in order to obtain second order asymptotics for mixed sources composed of more than two memoryless sources. This would require an extension of Proposition 6.1 to this setting. Further, a generalization of the results on classical mixed channels by Polyanskiy et al. [21] to the quantum setting is an appealing open problem.
A Derivation of the second order asymptotic expansion of the information spectrum entropy In this section we present a direct proof of Theorem 3.5. We first derive the second order asymptotic expansion of the original form of the information spectrum entropy (denoted by H ε s (ρ n ) and defined in Definition 3.2) using a simple application of the Central Limit Theorem. We then use this result to obtain the second order asymptotic expansion of H ε s (ρ n ) by exploiting the relation in Proposition 3.3(i) between H ε s (ρ n ) and H ε s (ρ n ). Throughout the discussion, we abbreviate ρ n ≡ ρ ⊗n .
Proposition A.1. Let ρ ∈ D(H) and ε ∈ (0, 1). For sufficiently large n, the second order asymptotic expansion of H ε s (ρ n ) is given by where S(ρ) is the von Neumann entropy of ρ.
Proof. Let ρ = i r i |i i| be the eigenvalue decomposition of ρ, then we associate to ρ the probability distribution {P ρ (i)} i∈I where P ρ (i) := r i for i ∈ I = {1, . . . , d} with d = dim H. For the state ρ n , the eigenvalue decomposition can be written as where i := (i 1 , . . . , i n ) ∈ I n , | i := |i 1 ⊗ · · · ⊗ |i n and r i = n j=1 r i j = n j=1 P ρ (i j ). Hence, the probability distribution {P ρ n ( i)} i∈I n associated to ρ n satisfies P ρ n ( i) := r i = n j=1 P ρ (i j ), (28) that is, the i.i.d. form of ρ n translates to a product distribution on I n . We define the random variable P ρ with probability distribution {P ρ (i)} i∈I , that is, P ρ takes the value P ρ (i) with probability P ρ (i). By (28), the product random variable P n ρ is distributed according to n j=1 P ρ (i j ) = r i . Hence, Tr[ρ n {ρ n ≤ 2 γ 1}] = i:r i ≤2 γ r i = i: j Pρ(i j )≤2 γ n j=1 P ρ (i j ) = P P n ρ ≤ 2 γ = P log P ρ ≤ γ n , where the random variable log P ρ is also distributed according to the probability distribution {P ρ (i)} i∈I . Consequently, we obtain H ε s (ρ n ) = − sup{γ | Tr[ρ n {ρ n ≤ 2 γ 1}] ≤ ε} = −n sup{γ | P{log P ρ ≤ γ} ≤ ε}.
The mean and variance of the random variable X := log P ρ are given by µ X := i P ρ (i) log P ρ (i) = −S(ρ) (30a) Considering the random variableX := X 1 + · · · + X n n , where X i d = X = log P ρ are i.i.d. random variables, we see thatX d = log P ρ , where for two random variables A and B the notation A d = B denotes that A and B are equal in distribution. By the Central Limit Theorem [7], the random variable converges in distribution to the normal distribution, that is, where F Yn (z) := P(Y n ≤ z) and Φ(z) are the cumulative distribution functions of Y n and a standard normal random variable respectively. We use this result by observing that the quantity appearing on the right-hand side of (29) is equal to the inverse of the cumulative distribution function of X = log P ρ , that is, sup{γ | P{log P ρ ≤ γ} ≤ ε} = F −1 log Pρ (ε). We therefore have H ε s (ρ n ) = −n sup{γ | P{log P ρ ≤ γ} ≤ ε} = −nF −1 log Pρ (ε) = −nµ X − √ n σ X F −1 Yn (ε) = nS(ρ) − √ n s(ρ)F −1 Yn (ε) where we used (31) in the third equality and (30) in the last equality.
We are now ready to derive the second order asymptotic expansion of the information spectrum entropy H ε s (ρ n ). Proof of Theorem 3.5. Proposition 3.3(i) yields the following bounds on H ε s (ρ n ) for arbitrary δ > 0: We use the expansion given by Proposition A.1 for both bounds in (35),set δ = 1 √ n and apply Lemma 2.