The 24th Symposium on Information Theory and Its Applications (SITA2001) Kobe, Hyogo, Japan, Dec. 4 - 7, 2001

Universal distortion-free entanglement concentration achieving the optimal rate

Masahito Hayashi ^{∗}and Keiji Matsumoto^{†}

Abstract— We propose a universal distortion-free concentration, that is, a protocol such that (1) the protocol works for an arbitrary unknown pure entangled states, (2) we can always get maximally entangled states without any distortion, and (3) The number of outcoming maximally entangled state is probabilistic, and takes optimal value (=von Neuman entropy rate) with probability infinitely close to 1, asymptotically.

It is proved that among all the possible protocols satisfying (1) and (3), our protocol is optimal in view of the speed that the failure probability goes to zero. In the construction of our protocol, invariance by the action of symmetric group is essential. Applications to teleportation and dense coding are also studied. Keywords—quantum information, maximally entangled state, universal distortion-free entanglement concentration, universal approximate entanglement concentration, symmetric group

1 Introduction

In quantum systems, we can perform some information processes which do not appear in classical systems. For example, quantum teleportation, dense coding etc. For them it is necessary to share an entangled state between two systems. If the entangled state is a maximally entangled state, its analysis is very easy. Otherwise, it is not easy [1, 2].

We can produce maximally entangled states from arbitrary entangled states by local operations and classical communications (LOCC) and call such an operation an entanglement concentration. As is proved by Bennett et al[3], when we share the n-tensor product state |ϕ⟩⟨ϕ|^{⊗n} on the total tensor product system H_{A}^{⊗n} ⊗ H_{B}^{⊗n}, we can produce, by local operations, nH(ρ_{A})-qubit maximally entangled states asymptotically with the probability 1, where ρ_{A} := Tr_{B} |ϕ⟩⟨ϕ| and H(ρ_{A}) is the entropy − Tr ρ_{A} log ρ_{A}.

In this paper, we propose a universal distortion-free entanglement concentration in which, we can, independently of ρ_{A}, produce no less than nH(ρ_{A})-qubit maximally entangled states asymptotically with the probability 1 without a distortion and classical communications. (§2) As another method, we can perform an entanglement concentration after the state estimation on ϵn systems. But, if we perform entanglement concentration which depends on the estimated state, the final state is not necessarily a maximally entangled state i.e. the protocol has a distortion because the estimated state does not exactly coincide with the true state. This protocol can be regarded as a universal approximate entanglement concentration. As is proved in §3, our protocol achieves the optimal failure exponent among universal approximate entanglement concentrations which achieve the optimal rate nH(ρ_{A}) for any state. Our universal distortion-free entanglement concentration is constructed by focusing on the invariance w.r.t. the n-the symmetric group. In §4, in order to check that our protocol has no distortion, etc, we discuss an entanglement concentration protocol under a group invariance.

If entangled states are shared, we can send R_{1} bits classical message by sending only R_{2}( < R_{1}) qubits. This type information process is called (super) dense coding. Usually, we consider only R_{1}, but in this setting we regard the maximum of R_{1}−R_{2} as the capacity because the number R_{1} − R_{2} signifies the effect of entanglement. As is discussed in §5, we can prove that the maximum of R_{1} − R_{2} is asymptotically equal to nH(ρ). Even if we do not know the density ρ_{A}, using our universal distortion-free entanglement concentration we can send 2nH(ρ) bits of classical information by sending only nH(ρ) qubits.

An application to quantum teleportation is briefly stated in §6. As is pointed out by Keyl and Werner [4], this group invariant method is applicable to the estimation of spectrum. Concerning this topic, we will discuss another paper[5].

2 Universal distortion-free entanglement

concentration

In this section, we propose a universal distortionfree entanglement concentration. For this purpose, we focus on the invariance of the tensor product state w. r. t. n-th symmetric group. Now, we let d be the maximum of dim H_{A} and dim H_{B} and we can regard H_{A} and H_{B} as d-dimensional space by adding vectors if d > dim H_{A} or d > dim H_{B}. We assume that the state on the tensored total system H_{A}^{⊗n} ⊗ H_{B}^{⊗n} is written by n-tensored state |ϕ⟩⟨ϕ|^{⊗n}, where |ϕ⟩⟨ϕ| is a pure state on the single total system H_{A} ⊗ H_{B}. In order to treat representations of n-th symmetric group S_{n} , we define the Young index n by

n := (n_{1}, …, n_{d}), ∑_{i=1}^{d} n_{i} = n, n_{i} ≥ n_{i+1}.

Young index n uniquely corresponds to the unitary irreducible representation V_{n} of S_{n} and the unitary irreducible representation U_{n} of the d-th special unitary group SU(d). The tensored space H_{A}^{⊗n} is decomposed as (5) by

∗ Laboratory for Mathematical Neuroscience, Brain Science

Institute, RIKEN. 2-1 Hirosawa, Wako, Saitama, 351-0198, Japan † Quantum Computation and Information Project, ERATO,

JST. 5-28-3, Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan

H_{A}^{⊗n} = ⨁_{n} W_{n,A}, W_{n,A} := U_{n,A} ⊗ V_{n,A}
H_{B}^{⊗n} = ⨁_{n} W_{n,B}, W_{n,B} := U_{n,B} ⊗ V_{n,B}.

For the detail, see Weyl [6], Goodman-Wallch[7], Iwahori [8]. The density ρ_{A}^{⊗n} is invariant w.r.t. the representation of the symmetric group S_{n} on the tensored space H_{A}^{⊗n}. This type decomposition does not depends on ρ_{A} and |ϕ⟩⟨ϕ| and depends on the group representation invariance.

Now, we construct a universal distortion-free entanglement concentration as follows. Using this protocol, we can get a maximally entangled state with the size dim V_{n} in the probability Tr W_{n,A}ρ_{A}^{⊗n}. First, we perform the projection measurements {U_{n,A} ⊗ V_{n,A}}_{k} and {U_{n,B} ⊗ V_{n,B}}_{k} on H_{A}^{⊗n} and H_{B}^{⊗n}, i.e. we perform the projection measurement {U_{n_{A},A} ⊗V_{n_{A},A} ⊗U_{n_{B},B} ⊗ V_{n_{B},B}}_{n_{A},n_{B}} on the total system H_{A}^{⊗n} ⊗ H_{B}^{⊗n}. As is discussed in section 4, the two data n_{A} and n_{B} always coincide and we get the data n := n_{A} = n_{B} with the probability Tr W_{n,A}ρ_{A}^{⊗n}. Next, we take the partial trace on U_{n,A}⊗U_{n,B}. As is proved in section 4, the final state is a maximally entangled state on V_{n,A} ⊗ V_{n,B}, is independent of the initial state and depends only on the data n. Concerning the random variable \frac{1}{n} log dim V_{n}, we have the following theorem. Theorem 1 The success probability is evaluated as

lim_{n→∞} \frac{−1}{n} log ∑_{n}\{Tr W_{n}ρ_{A}^{⊗n}| dim V_{n} ≤ 2^{nR}\}

= min{D(q∥p)|H(q) ≤ R}

= max_{s≥1} \frac{(1 − s)R − ψ(s)}{s},

and the failure probability is evaluated as

lim_{n→∞} \frac{−1}{n} log ∑_{n}\{Tr W_{n}ρ_{A}^{⊗n}| dim V_{n} ≥ 2^{nR}\}

= min{D(q∥p)|H(q) ≥ R}

= max_{0 <s≤1} \frac{(1 − s)R − ψ(s)}{s},

where ψ(s) := log Tr ρ_{A}^{s} and the vector p = (p_{1}, …, p_{d}) consists of eigenvalues of ρ_{A} satisfying p_{1} ≥ p_{2} ≥ … ≥ p_{d} and D(q∥p) is the relative entropy. Notice that sup_{s≥1} \frac{(1−s)R−ψ(s)}{s} > 0 when R < H(ρ_{A}). Theorem 1 implies that our protocol asymptotically achieves the bound H(ρ_{A}) with the probability 1. Since the equations min\{D(q∥p)|H(q) ≤ R\} = max_{s≥1} \frac{(1−s)R−ψ(s)}{s} and min\{D(q∥p)|H(q) ≥ R\} = max_{0 <s≤1} \frac{(1−s)R−ψ(s)}{s} follow from Appendix B of Ogawa-Nagaoka[9], the above theorem follows from the following lemmas proved in Appendix of [10].

Lemma 2 There exists a constant number C such that the inequality

|\frac{1}{n} log dim V_{n} − H (\frac{n}{n})| ≤ \frac{2d^{2} + d}{2n} log(n + d) + \frac{C}{n}

holds for any Young index n.

Lemma 3 For any state ρ_{A} on H_{A} and any set R ⊂ R_{+} := {p|p_{1} ≥ p_{2} ≥ … ≥ p_{d} ≥ 0, ∑_{i=1}^{d} p_{i} = 1} ⊂ R^{d} and any ϵ > 0 there exists N such that

lim_{n→∞} \frac{−1}{n} log ∑_{\frac{n}{n} ∈R} Tr W_{n}ρ_{A}^{⊗n} ≥ D(\overline{R}∥p) := max_{q∈\overline{R}} D(q∥p),

where \overline{R} is the closure of R.

3 Optimal exponent of universal approximate entanglement concentrations

We prove that our protocol is optimal among universal approximate entanglement concentrations which achieves the optimal rate nH(ρ_{A}) for any state and consists of local operations. In quantum system, any state evolution by a measurement is described by an instrument consisting of a decomposition C = {C(ω)}_{ω} by CP maps of a trace preserving CP map. When we perform an instrument C = {C(ω)}_{ω} for an initial state ρ, we get the data ω and the final state \frac{C(ω)(ρ)}{Tr C(ω)(ρ)} with the probability Tr C(ω)(ρ). A sequence {(C_{n} = {C_{n}(ω)}_{ω}, H_{n})} of pairs of an instrument consisting of local operations on {H_{A}^{⊗n} ⊗H_{B}^{⊗n}} and a function H_{n} : ω ↦ H_{n}(ω) is called an approximate entanglement concentration of |ϕ⟩⟨ϕ| on H_{A} ⊗ H_{B} if

∑_{ω} Tr C_{n}(ω)(|ϕ⟩⟨ϕ|^{⊗n})

∥\frac{C_{n}(ω)(|ϕ⟩⟨ϕ|^{⊗n})}{Tr C_{n}(ω)(|ϕ⟩⟨ϕ|^{⊗n})} − |ϕ_{H_{n}(ω)}⟩⟨ϕ_{H_{n}(ω)}|∥_{Tr} → 0,

where |ϕ_{H_{n}(ω)}⟩⟨ϕ_{H_{n}(ω)}| is a maximally entangled state with the size H_{n}(ω). In our protocol in §2, the above value is 0 for any Young index n. From the monotonicity of the infimum of the relative entropy D(|ϕ⟩⟨ϕ|, ρ) among non-entangled states ρ on H_{A} ⊗H_{B}, an approximate entanglement concentration {(C_{n}, H_{n})} of |ϕ⟩⟨ϕ| satisfies that

lim sup_{n→∞} ∑_{ω} Tr C_{n}(ω)(|ϕ⟩⟨ϕ|^{⊗n})\frac{H_{n}(ω)}{n} ≤ H(ρ_{A}). (1)

A sequence {(C_{n}, H_{n})} is called a universal approximate entanglement concentration of a state family S := {θ ∈ Θ| |ϕ_{θ}⟩⟨ϕ_{θ}|} on H_{A} ⊗ H_{B} if it is an approximate entanglement concentration of any state |ϕ_{θ}⟩⟨ϕ_{θ}| and satisfies that

lim_{n→∞} ∑_{ω} \{Tr C_{n}(ω)(|ϕ_{θ}⟩⟨ϕ_{θ}|^{⊗n}) |\frac{H_{n}(ω)}{n} ≥ H(ρ_{A,θ}) − ϵ\}
= 1, ∀ϵ > 0, ∀θ ∈ Θ,  (2)

where ρ_{A,θ} := Tr_{B} |ϕ_{θ}⟩⟨ϕ_{θ}|. From (1) and (2), the equation

∑_{ω} \{Tr C_{n}(ω)(|ϕ_{θ}⟩⟨ϕ_{θ}|^{⊗n}) ||\frac{H_{n}(ω)}{n} − H(ρ_{A,θ})| > ϵ\}
= 1, ∀ϵ > 0, ∀θ ∈ Θ  (3)

holds. Thus, we can regard the function \frac{H_{n}(ω)}{n} as a consistent estimator of the parameter H(ρ_{A,θ}) on the state family {θ ∈ Θ|ρ_{A,θ}}. Therefore, we have the following theorem. Theorem 4 A universal approximate entanglement concentration {(C_{n}, H_{n})} of a state family S := {θ ∈ Θ| |ϕ_{θ}⟩⟨ϕ_{θ}|} on H_{A} ⊗ H_{B} satisfies that

lim sup_{n→∞} \frac{−1}{n} log ∑_{\frac{Hn(ω)}{n} <R} Tr C_{n}(ω)(|ϕ_{θ_{0}}⟩⟨ϕ_{θ_{0}}|^{⊗n})

≤ inf_{H(ρ_{A,θ}) <R} D(ρ_{A,θ}∥ρ_{A,θ_{0}})

for any real R > 0 and any state |ϕ_{θ_{0}}⟩⟨ϕ_{θ_{0}}|. From this theorem, we can see that our protocol achieves the optimal failure exponent for the state family such that all ρ_{A,θ} commute each other. This family is included by the state family of all pure states. Proof We define two probabilities p_{n} := ∑_{\frac{Hn(ω)}{n} <R} Tr C_{n}(ω)(|ϕ_{θ_{0}}⟩⟨ϕ_{θ_{0}}|^{⊗n}) and q_{n} := ∑_{\frac{Hn(ω)}{n} <R} Tr C_{n}(ω)(|ϕ_{θ}⟩⟨ϕ_{θ}|^{⊗n}) for any state ρ_{A,θ} satisfying H(ρ_{A,θ}) < R. Since C_{n} consists of local operations, we can regard C_{n} as a POVM on H_{A}^{⊗n}. Thus, using the monotonicity of relative entropy, we have

nD(ρ_{A,θ}∥ρ_{A,θ_{0}}) ≥ q_{n} log \frac{q_{n}}{p_{n}} + (1 − q_{n}) log \frac{1 − q_{n}}{1 − p_{n}} .

Since it follows from (3) that q_{n} → 1, we have

lim sup_{n→∞} \frac{−1}{n} log p_{n} ≤ D(ρ_{A,θ}∥ρ_{A}).

We obtain the desired assertion. □

4 Distortion-free entanglement concen-

tration under group invariance

In order to prove the above mentioned properties of the universal distortion-free entanglement concentration, we construct a distortion-free entanglement concentration under group representation-invariance in a non-asymptotic setting. Let f_{A} and f_{B} be equivalent unitary representations of a group G on finite dimensional spaces H_{A} and H_{B}. Assume that we share the pure state |ϕ⟩⟨ϕ| which is invariant under the tensor representation f_{A} ⊗ f_{B} on the total system H_{A} ⊗ H_{B}, i.e. f_{A}(g) ⊗ f_{B}(g)|ϕ⟩ = |ϕ⟩, ∀g ∈ G. Lemma 5 If f_{A} and f_{B} are irreducible, the invariant vector ϕ is given as

|ϕ⟩ = \frac{1}{\sqrt{d}} ∑_{j=1}^{d} |e_{j,A}⟩ ⊗ |e_{j,B}⟩,

where {e_{j,A}}_{j=1}^{d} and {e_{j,B}}_{j=1}^{d} are CONSs of H_{A} and H_{B} such that f_{is}(e_{j,A}) = e_{j,B}, where f_{is} is the unique isomorphism map from H_{A} to H_{B}, w.r.t. the representation of G. With ambiguity of constant factor, the vector |ϕ⟩ is uniquely defined from the invariance of the representation of G. Then, we call the vector ϕ the invariant maximally entangled state on H_{A} ⊗ H_{B} in the irreducible case. Proof Since H_{A} and H_{B} are equivalent w.r.t. the representation space of G, we can identify the space H_{A} ⊗ H_{B} ≅ H_{A} ⊗ H_{B}^{∗} with the set L(H) of linear transforms on H_{A}. In this identification, the representation of G on H_{A} ⊗ H_{B} is regarded as the adjoint representation on L(H) because f_{A}(g)f_{B}(g)|ϕ_{A}⟩ ⊗ |ϕ_{B}⟩ ≅ f_{A}(g)|ϕ_{A}⟩ ⊗ ⟨ϕ_{B}|f_{B}(g)^{∗} ≅ f_{A}(g)|ϕ_{A}⟩ ⊗ ⟨ϕ_{B}|f_{A}(g)^{−1}, ∀|ϕ_{A}⟩ ∈ H_{A}, ∀|ϕ_{B}⟩ ∈ H_{B}. Therefore, using Schur’s lemma[7], we can prove the desired assertion. □ Since the dimension of H_{A} is finite, there exists a decomposition into irreducible representations of H_{A} as follows:

H_{A} = ⨁_{k} (⨁_{i=1}^{l_{k}} V_{k,i,A}) , H_{B} = ⨁_{k} (⨁_{i=1}^{l_{k}} V_{k,i,B}) ,(4)

where V_{k,i} and V_{k,j} is equivalent w.r.t. the representation of G. Therefore, there are l_{k} spaces equivalent with V_{k,1} w.r.t. the representation of G. We have another type decomposition:

H_{A} = ⨁_{k} U_{k,A} ⊗ V_{k,A}, H_{B} = ⨁_{k} U_{k,B} ⊗ V_{k,B}, (5)

where we let U_{k,A} and U_{k,B} be the vector spaces ⟨e_{k,1,A}, …, e_{k,l_{k},A}⟩ and ⟨e_{k,1,B}, …, e_{k,l_{k},B}⟩. and V_{k,A} and V_{k,B} be a vector space equivalent with V_{k,i,A}, V_{k,i,B} w.r.t. the representation of G. From the invariance of f_{A}⊗f_{B}, we can choose the decomposition (4) satisfying that

|ϕ⟩ = ∑_{k} ∑_{i=1}^{l_{k}} \sqrt{s_{k,i}d_{k}}|ϕ_{k,i}^{P}⟩  (6)

where d_{k} = dim V_{k}, and the vector ϕ_{k,i}^{P} ∈ V_{k,i,A}⊗V_{k,i,B} is the invariant maximally entangled state on V_{k,i,A} ⊗ V_{k,i,B}. Similarly to Lemma 5, this fact follows from Schur’s lemma. The constant factor s_{k} satisfies that

ρ_{A} = ∑_{k} ∑_{i=1}^{l_{k}} s_{k,i}V_{k,i,A},

where we identify the subspace of H_{A} with its projection. We cannot choose the decomposition (4) satisfying (6) from the invariance of f_{A} ⊗ f_{B}. But, we can uniquely construct the decomposition (5) from the invariance of f_{A} ⊗ f_{B}.

Let us construct a distortion-free entanglement concentration as follows. Using this protocol, we can get the maximally entangled state with the size dim V_{k} in the probability Tr ρ_{A}U_{k} ⊗ V_{k}. First, we perform the projection measurements {U_{k,A} ⊗ V_{k,A}}_{k} and {U_{k,B} ⊗ V_{k,B}}_{k} on H_{A} and H_{B}, i.e. we perform the projection measurement {U_{k_{A},A} ⊗ V_{k_{A},A} ⊗ U_{k_{B},B} ⊗ V_{k_{B},B}}_{k_{A},k_{B}} on the total system H_{A} ⊗ H_{B}. It follows from (6) that the data k_{A} and k_{B} coincide with the probability 1. and we get the data k := k_{A} = k_{B} with the probability d_{k} ∑_{i=1}^{l_{k}} s_{k,i} = Tr ρ_{A}U_{k} ⊗ V_{k}. Then, the obtained state on U_{k_{A},A} ⊗ V_{k_{A},A} ⊗ U_{k_{B},B} ⊗ V_{k_{B},B} is written by

\frac{1}{\sqrt{c_{k}}} ∑_{i=1}^{l_{k}} \sqrt{s_{k,i}d_{k}}|ϕ_{k,i}^{P}⟩.

Final, taking the partial trace on U_{k,A} ⊗ U_{k,B}, we get the invariant maximally entangled state on V_{k,A}⊗V_{k,B}, whose size is d_{k} = dim V_{k} because of Lemma 5. Thus, the final state is independent of initial state and depends only on the data k.

5 Dense coding

We formulate the effect of dense coding as follows. We assume that the state on the tensored total system H_{A}^{⊗n} ⊗ H_{B}^{⊗n} is written by n-tensored state |ϕ⟩⟨ϕ|^{⊗n}, where |ϕ⟩⟨ϕ| is a pure state on the single total system H_{A}⊗H_{B}. we call the quadruple Φ^{n} = (M_{n}, N_{n}, C_{•}^{(n)}, X^{(n)}) a code for |ϕ⟩⟨ϕ|^{⊗n} when it consists of a natural number M_{n} (the size of classical information sent), a natural number N_{n} (the size of quantum state sent), a POVM (decoding) X^{(n)} = {X_{i}^{(n)}}_{i=1}^{M_{n}} and a mapping (encoding) C_{•}^{(n)} : {1, …, M_{n}} ∋ i → C_{i}^{(n)}, where C_{i}^{(n)} is a CP map from S(H_{A}^{⊗n}) to S(C^{N_{n}}) and S(H) denotes the set of densities on H. Therefore, the effect of entanglement is characterized by the quantity log \frac{M_{n}}{N_{n}}.

For a code Φ^{(n)} = (M_{n}, N_{n}, C_{•}^{(n)}, X^{(n)}), the average error probability is represented by

E[Φ^{(n)}] = \frac{1}{M_{n}} ∑_{i} Tr(C_{i}^{(n)} ⊗ I)(|ϕ⟩⟨ϕ|^{⊗n})(I − X_{i}^{n}).

Thus, we focus on the capacity defined by

C(|ϕ⟩⟨ϕ|) :=

sup \{lim inf_{n→∞} \frac{1}{n} log \frac{M_{n}}{N_{n}} | \begin{matrix}∃\{Φ^{(n)} = (M_{n}, N_{n}, C_{i}^{(n)}, X^{n})\} \\ s.t. E[Φ^{(n)}] → 0 \end{matrix} \} ,

and calculate it as

C(|ϕ⟩⟨ϕ|) = H(ρ_{B}) = H(ρ_{A}),  (7)

where ρ_{A} := Tr_{B} |ϕ⟩⟨ϕ| and ρ_{B} := Tr_{A} |ϕ⟩⟨ϕ|. The direct part of (7) is obtained from our universal distortionfree entanglement concentration as follows. Using it, we can produce the maximally entangled state with the size dim V_{n}. With the probability 1, we can send classical information with the size dim V_{n}^{2} by sending the quantum state with the size dim V_{n}. In this case M_{n} = dim V_{n}^{2} and N_{n} = dim V_{n}. If R ≤ H(ρ_{A}), the probability of the relation \frac{M_{n}}{N_{n}} = dim V_{n} ≤ 2^{nR} goes to 0 with the exponent sup_{s≥1} \frac{(1−s)R−ψ(s)}{s}. For a proof of the converse part of (7), see Hayashi-Matsumoto[10]. Next, we compare its exponent with another protocol. The Burnashev-Holevo [11] random coding exponent of the pair ({U_{i}}_{i}, P) achieving H(ρ_{A}) is sup_{2≥s≥1} (1 − s)R − ψ(s), which is strictly greater than our exponent sup_{s≥1} \frac{(1−s)R−ψ(s)}{s} when R < H(ρ_{A}) is large enough.

6 Quantum teleportation

In the quantum teleportation, we can send a quantum state with LOCC. In such a setting we maximize the number of teleported qubits only with LOCC. If we send R-qubit of a part of maximally entangled states, we can share R-qubit maximally entangled state. Thus, to share R-qubit maximally entangled state is equivalent with to send R-qubit of maximally entangled states only with LOCC. Therefore, the optimal number of available qubits for quantum teleportation is nH(ρ_{A}), under the assumption that we share the n-tensor product state |ϕ⟩⟨ϕ|^{⊗n} on the total system H_{A}^{⊗n} ⊗ H_{B}^{⊗n}. Even if we do not know the density operator ρ_{A}, using our universal distortion-free entanglement concentration, we can perform nH(ρ_{A}) qubits quantum teleportation asymptotically with the probability 1. In our protocol, it is enough to send the minimum classical communications of the size of 2nH(ρ_{A}) bits.

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