Asymptotic Attainment for Quantum Relative Entropy

Masahito Hayashi

Department of Mathematics, Kyoto University, Kyoto 606-01, Japan e-mail address: masahito@kusm.kyoto-u.ac.jp

Abstract In this paper it was proved that the quantum relative entropy D(σ∥ρ) can be asymptotically attained by Kullback Leibler divergences of probabilities given by a certain sequence of measurements. The sequence of measurements depends on ρ, but is independent of the choice of σ.

In classical statistical theory the relative entropy D(p∥q) is an information quantity which means the statistical efficiency in order to distinguish a probability measure p of a measurable space from another probability measure q of the same measurable space. The states correspond to measures on measurable space. When p, q are discrete probabilities, the relative entropy (called also information divergence) introduced by Kullback and Leibler is defined by [1]:

D(p∥q) := ∑_{i} p_{i} log \frac{p_{i}}{q_{i}} .

In general, when p, q are measures on measurable space Ω, the relative entropy is defined by:

D(p∥q) := ∫_{Ω} log \frac{dp}{dq}(ω)p( dω),

where \frac{dp}{dq}(ω) is Radon-Nikodym derivative of p with respect to q.

Let H be a separable Hilbert space which corresponds to the physical system of interest. In quantum theory the relative entropy was first studied by Umegaki [2]. In quantum theory the states of a system corresponds to positive operators of trace one on H. (These operators are called densities.) The quantum relative entropy of a states ρ with respect to another states σ is defined by:

D(σ∥ρ) := tr σ(log σ − log ρ).

States are distinguished through the result of a quantum measurement on the system. The most general description of a quantum measurement probability is given by the mathematical concept of a positive operator valued measure (POM) [4,5] on the system state space. Generally speaking, if Ω is measurable space, a measurement M satisfies the following:

M(B) = M(B)^{∗}, M(B) ≥ 0, M(∅) = 0, M(Ω) = Id on H, for any B ⊂ Ω.
M(∪_{i}B_{i}) = ∑_{i} M(B_{i}), for B_{i} ∩ B_{j} = (i ≠ j), {B_{i}} is countable subsets of Ω.

A measurement M on H is called simple, if for any B ⊂ Ω, M(B) is projection. In this paper, we treat only measurements whose measurable set is a finite set.

Let tr M(·)ρ be the probability by a measurement M on a quantum system H with respect to a state ρ, then an information quantity we can directly access by a measurement M is not D(σ∥ρ) but D_{M}(σ∥ρ), where D_{M}(σ∥ρ) denotes D(tr M(·)σ∥ tr M(·)ρ). Because the map ρ ↦ tr M(·)ρ is the dual of a unipreserving completely positive map [3], by Uhlmann inequality [6] we have

D_{M}(σ∥ρ) ≤ D(σ∥ρ).  (1)

The equality is attained by a certain measurement M when and only when ρσ = σρ. See for instance [7, Theorem 1.5, Theorem 5.3].

Can the equality of the inequality (1) be asymptotically attained? In order to answer the question we define the quantum i.i.d. condition which is the quantum analogue of the independent and identically distribution condition.

If there exist n samples of the state ρ, the quantum state is described by ρ^{(n)} defined by:

ρ^{(n)} := \underbrace{ρ ⊗ ⋯ ⊗ ρ}_{n} on H^{(n)},

where the composite system H^{(n)} is defined as:

H^{(n)} := \underbrace{H ⊗ ⋯ ⊗ H}_{n} .

In this paper, we consider this condition called the quantum i.i.d. condition. B(H) denotes the set of bounded operators on H, and B(H)^{(n)} denotes \underbrace{B(H) ⊗ ⋯ ⊗ B(H)}_{n} .

Hiai and Petz proved the following theorem [8,9].

Theorem 1 Let ρ, σ be states on H, then there exists a sequence {(l_{n}, M_{n})} of pairs of an integer and a measurement on H^{(l_{n})} such that

\frac{D_{M_{n}}(σ^{(l_{n})}∥ρ^{(l_{n})})}{l_{n}} → D(σ∥ρ) as n → ∞.  (2)

It was proved that there exists a sequence {M_{n}} of measurements satisfying (2), but such a sequence {M_{n}} was not constituted in Ref. [8,9]. It was not answered whether or not such a sequence depends on a state σ. Can we choose a sequence {M_{n}} of measurements satisfying (2) which is independent of σ? The answer is “Yes”. The main theorem of this paper is the following theorem.

Theorem 2 Let ρ be a state on H, then there exists a sequence {(l_{n}, M_{n})} of pairs of an integer and a measurement on H^{(l_{n})} such that

\frac{D_{M_{n}}(σ^{(l_{n})}∥ρ^{(l_{n})})}{l_{n}} → D(σ∥ρ) as n → ∞ ∀σ.  (3)

In this section we consider the relation between a simple measurement and a quantum relative entropy.

We put some definitions for this purpose. A state ρ is called commutative with a simple measurement E(:= {E_{i}}) on H if [ρ, E_{i}] = 0 for any i. For simple measurements E, F, we denote E ≤ F if for any i there exist subsets {A_{i}}_{i} such that E_{i} = ∑_{j∈A_{i}} F_{j}. For a state ρ, E_{ρ} denotes the spectral decomposition of ρ. The conditional expectation E_{E} with respect to a simple measurement E is defined as:

E_{E} : ρ ↦ ∑_{i} E_{i}ρE_{i}.

Theorem 3 Let E be a simple measurement such that w(E) := sup_{i} dim E_{i} < ∞. If states ρ, σ are commutative with a simple measurement E and a simple measurement F satisfies that E, E_{ρ} ≤ F, then we have

D_{F} (σ∥ρ) ≤ D(σ∥ρ) ≤ D_{F} (σ∥ρ) + log w(E).

Note that there exists a simple measurement F such that E, E_{ρ} ≤ F. Proof It is proved by Lemma 1 and Lemma 2. □

Lemma 1 Let σ, ρ be states. If a simple measurement F satisfies that E_{ρ} ≤ F, then

D(σ∥ρ) = D_{F} (σ∥ρ) + D(σ∥E_{F} (σ)).  (4)

Proof Since E_{ρ} ≤ F, F is commutative with ρ. Thus we obtain (4) [7]. □

Lemma 2 Let E, F be simple measurements such that E ≤ F. If a state σ is commutative with E, then

D(σ∥E_{F} (σ)) ≤ log w(E).  (5)

Proof Let a_{i} := tr E_{i}σE_{i}, σ_{i} := \frac{1}{a_{i}} E_{i}σE_{i}, then σ = ∑_{i} a_{i}σ_{i}. Therefore, from joint convexity of quantum relative entropy (See p.20 of [7]),

D(σ∥E_{F} (σ)) ≤ ∑_{i} a_{i}D(σ_{i}∥E_{F} (σ_{i})) ≤ sup_{i} D(σ_{i}∥E_{F} (σ_{i})) ≤ sup_{i} log dim E_{i} = log w(E). (6)
□

From the direct sum decomposition H = H_{1} ⊕ ⋯ ⊕ H_{k}, we can naturally constitute the simple measurement {P_{H_{i}}}, where P_{H_{i}} denotes the projection of H_{i}. When H is finite-dimensional, we denote Ir^{(n)} the simple measurement corresponding to a direct sum decomposition by the irreducible decomposition of the sensor space H^{(n)} of the act of GL(H).

Lemma 3 If H is finite-dimensional, Ir^{(n)} is commutative with σ^{(n)} for any state σ.

Proof If a state σ is faithful, then it is trivial by Schur’s lemma. If a state σ isn’t faithful, then there exists a sequence {σ_{i}} of faithful states such that σ_{i} → σ as i → ∞. Because σ_{i}^{(n)} → σ^{(n)} as i → ∞ and Ir^{(n)} is commutative with σ_{i}^{(n)}, Ir^{(n)} is commutative with σ^{(n)}. □

Theorem 4 Assume that the dimension of H is finite and is denoted by k. Let ρ be a state on H. If a simple measurement M_{n} satisfies that Ir^{(n)}, E_{ρ^{(n)}} ≤ M_{n}, then we obtain the following inequality:

\frac{D_{M_{n}}(σ^{(n)}∥ρ^{(n)})}{n} ≤ D(σ∥ρ) ≤ \frac{D_{M_{n}}(σ^{(n)}∥ρ^{(n)})}{n} + (k − 1)\frac{log(n + 1)}{n} ∀σ.  (7)

Therefore we obtain

\frac{D_{M_{n}}(σ^{(n)}∥ρ^{(n)})}{n} → D(σ∥ρ) as n → ∞ uniformally for σ.  (8)

Proof Since w(Ir^{(n)}) is the dimension of the k-th symmetric tensor space of H, w(Ir^{(n)}) = _{k}H_{n} = \binom{n+k−1}{k−1} = \binom{n+k−1}{n} = _{n+1}H_{k−1} ≤ (n + 1)^{k−1}, where _{k}H_{n} denotes the repeated combination of n from k. Therefore, we have log w(Ir^{(n)}) ≤ (k − 1) log(n + 1). From Theorem 3 and Lemma 3 we have (7). □ Next we consider the infinite-dimensional case.

Theorem 5 Let ρ be a state on H, then there exists a sequence {(l_{n}, M_{n})} of pairs of an integer and a measurement on H^{(l_{n})} such that

\frac{D_{M_{n}}(σ^{(l_{n})}∥ρ^{(l_{n})})}{l_{n}} → D(σ∥ρ) as n → ∞ ∀σ.  (9)

Proof The finite-dimensional case is trivial from Theorem 4. We may assume that the dimension of H is infinite. Because of B(H)’s injectivity, there exists a sequence {α_{n} : B(H_{n}) → B(H)} of unipreserving completely positive maps such that H_{n} is finitedimensional and

lim_{n→∞} D(α_{n}^{∗}(σ)∥α_{n}^{∗}(ρ)) = D(σ∥ρ)  (10)

for any normal states σ, ρ on H such that μρ ≤ σ ≤ λρ for some positive real numbers μ, λ [7]. From (8), for any positive integer n there exist a pair (l_{n}, M′_{n}) of an integer and a simple measurement on H_{n}^{(l_{n})} such that

D(α_{n}^{∗}(σ)∥α_{n}^{∗}(ρ)) − \frac{D_{M′_{n}}(α_{n}^{∗}(σ)^{(l_{n})}∥α_{n}^{∗}(ρ)^{(l_{n})})}{l_{n}} < \frac{1}{n}.  (11)

The completely positive map α_{n}^{(l)} from B(H_{n})^{(l)} to B(H)^{(l)} is defined as α_{n}^{(l)}(A_{1} ⊗ A_{2} ⊗ ⋯⊗A_{l}) = α_{n}(A_{1})⊗α_{n}(A_{2})⊗⋯⊗α_{n}(A_{l}) for ∀A_{i} ∈ B(H). Therefore we have α_{n}^{(l)∗}(σ^{(l)}) = α_{n}^{∗}(σ)^{(l)}. Let M_{n} be α_{n}^{(l_{n})}(M′_{n}), then from (10)(11) we get

\frac{D_{M_{n}}(σ^{(l_{n})}∥ρ^{(l_{n})})}{l_{n}} = \frac{D_{M′_{n}} (α_{n}^{(l_{n})∗} (σ^{(l_{n})})∥ α_{n}^{(l_{n})∗} (ρ^{(l_{n})}))}{l_{n}}
= \frac{D_{M′_{n}} (α_{n}^{∗}(σ)^{(l_{n})}∥ α_{n}^{∗}(ρ)^{(l_{n})})}{l_{n}}
> D (α_{n}^{∗}(σ)∥α_{n}^{∗}(ρ)) + \frac{1}{n}
→ D(σ∥ρ) as n → ∞.

Therefore, we obtain (9). □ Note that the simple measurement M_{n} is independent of σ. Therefore, we could constitute a sequence {M_{n}} of measurements satisfying (3).

Remark 1 Even if ρ_{ϵ} → ρ as ϵ → 0 and M_{n} satisfies the assumption of Theorem 4, the following equation is not always established:

lim_{ϵ→0} lim_{n→∞} \frac{D_{M_{n}}(ρ_{ϵ}^{(n)}∥ρ^{(n)})}{nϵ^{2}} = lim_{n→∞} lim_{ϵ→0} \frac{D_{M_{n}}(ρ_{ϵ}^{(n)}∥ρ^{(n)})}{nϵ^{2}}.  (12)

The left hand of (12) is a important value in quantum local detection, but the right hand of (12) is closely related to quantum estimation.

The following is an example of Remark 1.

Example 1 Let the dimension k of H be 2. Let us define the Pauli matrices σ_{1}, σ_{2} in the usual way:

σ_{1} = (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) , σ_{2} = (\begin{matrix}0 & −i \\ i & 0 \end{matrix}) .

Assume that

ρ = \frac{1}{2}(Id +ασ_{1}), 0 < α < 1
ρ_{ϵ} = \frac{1}{2}(Id +α(cos ϵσ_{1} + sin ϵσ_{2})).

then

lim_{ϵ→0} \frac{D_{M_{n}}(ρ_{ϵ}^{(n)}∥ρ^{(n)})}{ϵ^{2}} = 0
lim_{ϵ→0} lim_{n→∞} \frac{D_{M_{n}}(ρ_{ϵ}^{(n)}∥ρ^{(n)})}{nϵ^{2}} = lim_{ϵ→0} \frac{D(ρ_{ϵ}∥ρ)}{ϵ^{2}} = \frac{1}{4}α log \frac{1 + α}{1 − α} > 0,

where a sequence of measurments {M_{n}} satisfies the assumption of Theorem 4.

Conclusions It was proved that the quantum relative entropy D(σ∥ρ) is attained by the sequence of Kullback-Leibler divergences given by a certain sequence of measurements which is independent of σ. This formula is thought to be important for the quantum asymptotic detection and the quantum asymptotic estimation. To know the quantum asymptotic estimation, see [10]. The realization of the sequence of measurements are left for future study.

Acknowledgments The author wishes to thank Prof. D. Petz for essential suggestions about the extension in the infinite-dimensional case. He wishes to thank Prof. A. Fujiwara and Dr. M. Ishii for several discussions on this topic.

References

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[2] H. Umegaki, Kodai Math. Sem. Rep. 14, 59 (1962).

[3] M. Ozawa, J. Math. Phys. 25, 79 (1984).

[4] C. W. Helstrom, Quantum Detection and Estimation Theory, (Academic Press, New York, 1976).

[5] A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory, (North Holland, Amsterdam, 1982).

[6] A. Uhlmann, Commun. Math. Phys. 54, 21 (1977).

[7] M. Ohya and D. Petz, Qunatum Entropy and Its Use (Springer, Berlin, 1993).

[8] F. Hiai and D. Petz, Commun. Math. Phys. 143, 99-114 (1991).

[9] D. Petz, J. Funct. Anal. 120, 82-97 (1994).

[10] M. Hayashi, Kyoto-Math 97-09 (1997). LANL e-print quant-ph/9704041.