Information spectrum in quantum hypothesis testing

Masahito Hayashi Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan e-mail address: masahito@kusm.kyoto-u.ac.jp

Let H be a Hilbert space which represents a physical system in interest. In quantum theory the states of system correspond to nonnegative operators of trace one on H. ( These operators are called densities and S(H) denotes the set of density operators on H.) A measurement in a quantum system corresponding to a Hilbert space H is described by a partition of unit by nonnegative operators {M_{i}| ∑_{i=1}^{k} M_{i} = I} which is called a positive operator valued measure (POVM). If a POVM M consists of projections, it is called a projection valued measure (PVM). If a state ρ ∈ S(H) measured by a measurement corresponding to {M_{i}| ∑_{i=1}^{k} M_{i} = I}, then data obey the probability P_{ρ}^{M} given by P_{ρ}^{M}(i) := Tr[M_{i}ρ].

We treat the problem of hypothesis testing a null hypothesis ρ ∈ S(H) versus an alternative hypothesis σ ∈ S(H). This problem is to decide which hypothesis is true, and the decision is given by a two-valued quantum measurement {T, I −T }, where T corresponds to the acceptance of ρ and I −T corresponds to the acceptance of σ. We call T (0 ≤ T ≤ I) a test in the sequel. In this talk we consider the asymptotic case: null hypotheses are given by a sequence ρ⃗ = {ρ_{n}}, and alternative hypotheses are given by a sequence σ⃗ = {σ_{n}}, where ρ_{n} and σ_{n} are density operators on a Hilbert space H_{n}. For a sequence T⃗ = {T_{n}} of tests, define the error probability of the first kind and the second kind by α(T_{n}) := 1 − Tr[ρ_{n}T_{n}], β(T_{n}) := Tr[σ_{n}T_{n}] respectively. We define the following.

B(ϵ|ρ⃗ ∥ σ⃗) =^{def} sup \{R | ∃T⃗, lim sup_{n→∞} α(T_{n}) ≤ ϵ and lim sup_{n→∞} \frac{1}{n} log β(T_{n}) ≤ −R\} ,

DM⃗ (ϵ|ρ⃗∥σ⃗) =^{def} sup \{R |lim sup_{n→∞} P_{ρ_{n}}^{M^{n}} \{\frac{1}{n} log \frac{P_{ρ_{n}}^{M^{n}}}{P_{σ_{n}}^{M^{n}}} ≤ R\} ≤ ϵ\} ,

D(ϵ|ρ⃗∥σ⃗) =^{def} sup \{R |lim sup_{n→∞} P_{ρ_{n}}^{σ_{n},ρ_{n}} \{\frac{1}{n} log \frac{\overline{ρ_{n}}}{\overline{σ_{n}}} ≤ R\} ≤ ϵ\} ,

where for self-adjoint traceclass operators X = ∑_{i} x_{i}E(X)_{i}, Y = ∑_{j} y_{j}E(Y)_{j}(spectral decompositions), we define a probability distribution P_{ρ}^{X,Y} which probability variables \overline{X}\overline{Y} obey:

P_{ρ}^{X,Y} (\overline{X} = x_{i}, \overline{Y} = y_{i}) =^{def} Tr[E(X)_{i}E(Y)_{j}ρE(Y)_{j}].

For a PVM M = {M_{i}}, we denote the supremum of the rank of M_{i} by w(M).

Theorem 1 If there exists a sequence E⃗ = \{E^{n}\} such that \frac{log w(E^{n})}{n} → 0 and any element of a PVM E^{n} is commutative with density operators ρ_{n}, σ_{n}, then we have the following:

B(ϵ|ρ⃗∥σ⃗) = sup_{M⃗} DM⃗ (ϵ|ρ⃗∥σ⃗) = D(ϵ|ρ⃗∥σ⃗) = DM⃗′ (ϵ|ρ⃗∥σ⃗) ,

where M⃗′ = {M′^{n}} is a sequence of PVMs such that w(M′^{n}) = 1 and any element of a PVM M′^{n} is commutative with a density operator σ_{n} and any element of a PVM E^{n}.

Theorem 2 Let H be a finite-dimensional Hilbert space. If H_{n} = H^{⊗n}, σ_{n} = σ^{⊗n}, ρ_{n} = ∫_{Φ} ρ_{θ}^{⊗n}μ( dθ), then we have the following:

B(ϵ|ρ⃗∥σ⃗) = sup {R |μ {θ |D(ρ_{θ}∥σ) ≤ R} ≤ ϵ} .