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Optimal Visible Compression Rate For Mixed States Is Determined By Entanglement Of Puri?cation

Masahito Hayashi1, 2, ?

2

ERATO-SORST Quantum Computation and Information Project, JST, Tokyo 113-0033, Japan Superrobust Computation Project, Information Science and Technology Strategic Core (21st Century COE by MEXT), Graduate School of Information Science and Technology, The University of Tokyo Entanglement of puri?cation was introduced by Terhal et al.[6] for characterizing the bound of the generation of correlated states from maximally entangled states with sublinear size of classical communication. On the other hand, M. Horodecki [5] obtained the optimal compression rate with a mixed states ensemble in the visible setting. In this paper, we prove that the optimal visible compression rate for mixed states is equal to the limit of the regularized entanglement of puri?cation of the state corresponding to the given ensemble. This result gives a new interpretation to the entanglement of puri?cation.

1

arXiv:quant-ph/0511267v2 10 Feb 2006

Introduction: Many information quantities appear as the bound of the respective information processing in quantum information theory. Usually, these bounds depend on the information processing of interest. However, recently, Devetak et al. [1] considered the relation among quantum information processing. In their paper, they found remarkable conversions among quantum protocols, and succeeded in simplifying the proofs of several important theorems. Also Bennett et al. [2] obtained the conversion relation among error correction of quantum channel and one-way distillation of mixed entangled state. In this paper, we focus on visible quantum data compression with mixed states and generation of a correlated state from maximally entangled state with classical communication of the sublinear size. Using a conversion relation similar to Bennett et al. [2], we ?nd an interesting conversion relation between the quantum compression with a mixed states ensemble and the state generation from maximally entangled state. Quantum data compression was initiated by Schumacher[3]. As the quantum information source, he focused on the quantum states ensemble (px , Wx )x∈X , in which the quantum state Wx generates with the probability px . He showed that the asymptotic optimal compression rate R(W, p) is equal to the enIn his original Wp = x px Wx of this ensemble. problem, the encoder is restricted to performing a quantum operation. However, M. Horodecki [4] considered another problem, in which the encoder is de?ned as the any map from X to the quantum states. This formulation is called visible, while the former is called blind. He also showed that even in the visible setting if every state Wx is pure, the optimal rate R(W, p) is equal to the entropy rate H(Wp ). However, it had been an open problem to characterize the rate R(W, p) in the mixed states case. M. Horodecki[5] studied this problem, and succeeded in its characterization. However, his characterization contains a limiting expression. Hence, it is an open problem whether it can be characterized tropy H(Wp ) = ? Tr Wp log Wp of the average state

def def

without any limiting expression. On the other hand, Terhal et al.[6] introduced entanglement of puri?cation Ep (ρ) for any partially entangled state ρ. They also consider the generation of the tensor product of any partially entangled state ρ on the composite system HA ?HB from maximal entangled states in the asymptotic form. In particular, they restrict the rate of the classical communication to zero asymptotically. Indeed, when the target state ρ is pure, this optimal rate is H(TrA ρ), which is equal to the optimal rate without any restriction for the rate of the classical communication[7]. Their main result is that the optimal rate with this reE (ρ?n ) striction is equal to limn→∞ p n . Of course, if the entanglement of puri?cation satis?es the additivity, i.e., Ep (ρ) + Ep (σ) = Ep (ρ ? σ), this optimal rate is equal to the entanglement of puri?cation. However, this additivity is still open. In this paper, we give another formula for the optimal visible compression rate R(W, p) as R(W, p) = lim

def

n→∞

1 ? ?n Ep (Wp ), n

(1)

A A ? where Wp = x px |ex ex | ? Wx . In order to prove this equation, we ?rst give an error-free visible compression protocol of the speci?c ensemble related to the state generated by LOCC from a maximally entangled state, ? ?n which is close to Wp . This compression is realized by quantum memory the size of a maximally entangled state and classical communication with the same size as the LOCC operation. Next, we prove this ensemble is close to our target ensemble. Combining them, we prove that the optimal compression rate is less than the regularized en1 ? ?n tanglement of puri?cation limn→∞ n Ep (Wp ) The converse relation is proved from the axiomatic properties of entanglement of puri?cation. Hence, using the relation (1), we clarify the relation between the two problems, the mixed state compression and the state generation from maximally entangled state with classical communication of the sublinear size. Thus, if the additivity of entanglement of puri?cation is proved, the optimal rate of visible

2 compression is equal to entanglement of puri?cation of the state corresponding to the given ensemble. State generation from maximally entangled state with communication of the sublinear size: In state generation from maximally entangled state, our protocol is described by an LOCC quantum operation (TP-CP) κ and the initial maximally entangled state |ΦL ΦL | with the size L. When we generate a partially entangled state ρ on the composite system HA ? HB by this protocol, its performance is evaluated by i) the size L, ii) the quality of the generated state, which is given by ε(ρ, κ, L) = 1 ? F 2 (ρ, κ(|ΦL ΦL |)), √ √ where F (ρ, σ) is the ?delity Tr | ρ σ|, and iii) the size of classical communication, which is denoted by CC(κ). In the asymptotic formulation, the bound with classical communication of the sublinear size is given by Ec (ρ) =

def def

E3 (Continuity) When any sequences of two states {ρn } and {σn } on the system Hn satisfy ρn ?σn 1 → 0, |E (ρn )?Ep (σ )| the convergence plog dim Hn n → 0 holds. E4 (Convergence) The quantity n → ∞.

Ep (ρ?n ) n

converges as

Using these properties, they showed the opposite inequality Ec (ρ) ≤ lim Hence, we obtain the relation Ec (ρ) = lim Ep (ρ?n ) . n→∞ n (5) Ep (ρ?n ) . n→∞ n (4)

Further, they obtained the following property: Ep (ρ) ≤ H(ρA ). (6)

{κn ,Ln }

inf

log Ln ε(ρ?n , κn , Ln ) → 0 lim log CC(κn ) →0 n n

.

The RHS is the in?mum value of lim lognLn under the conditions ε(ρ?n , κn , Ln ) → 0 and log CC(κn ) → 0. Lo & n Popescu [7] calculated this value in the pure states case as follows. Ec (|u u|) = H(TrB |u u|). Further, Terhal et al. [6] introduced the entanglement of puri?cation Ep (ρ) as Ep (ρ) =

def u:TrA2 ,B2 |u u|=ρ

Visible State Compression: In the visible state compression, we consider the compressed quantum system K. The encoder is given by a map τ from X to S(K), and the decoder is represented by a TP-CP map ν from S(K) def to S(H). The triple Ψ = (K, τ, ν) is called a visible code. That is, the information is stored by a quantum memory. Therefore, the error εp (Ψ) and the size |Ψ| of the code Ψ are de?ned as follows: εp (Ψ) =

def x∈X

px 1 ? F 2 (Wx , ν ? τ (x)) ,

|Ψ| = dim K.

def

min

H(TrB |u u|),

Then, the optimal compression rate is given by R(W, p) =

def

where HA2 and HB2 are additional spaces. In the above de?nition, u is a puri?cation of ρ with the reference system HA2 ? HB2 . TrB is the partial trace concerning the original space HB and the additional space HB2 . Using Lo & Popescu’s result, they showed that Ec (ρ) ≥ Ep (ρ). Applying this inequality to ρ?n , they also showed Ec Ep (ρ?n ) . (ρ) ≥ lim n→∞ n (3) (2)

{Ψ(n) }

inf

lim

1 log |Ψ(n) | εpn (Ψ(n) ) → 0 . n

Indeed, if the encoder τ is given as a TP-CP map (quantum operation), the setting is called blind. In order to treat this problem, M. Horodecki [5] focused on the quantity: H ext (W, p) = and showed R(W, p) = lim H ext (W (n) , pn ) . n→∞ n (7)

def

ext Wx :puri?cation of Wx

inf

H(

x

ext px Wx ),

Further, they proved the following properties for entanglement of puri?cation: E1 (Normalization) Ep (ρ) = log d when ρ is a maximally entangled state of dimension d. E2 (Weak monotonicity) Let κ be a operation containing quantum communication with size d. Then, Ep (κ(ρ)) ≤ Ep (ρ) + log d.

The following is the main theorem. Theorem: The optimal compression rate is given by ? R(W, p) = Ec (Wp ) = lim ? def Wp =

x

1 ? ?n Ep (Wp ), n→∞ n

(8)

px |eA eA | ? Wx , x x

3 where the {eA } is CONS indicated by x ∈ X . x ? From the de?nition of Ep (Wp ), we can easily check that ext ? Ep (Wp ) ≤ H (W, p). Using this theorem, we obtain ? ?n Ep (Wp ) H ext (W (n) , pn ) = lim . n→∞ n→∞ n n lim (9) Construction of the code Ψ satisfying (11) and (12): The following construction of Ψ from one-way LOCC operation κ is similar to a simulation of one-way LOCC distillation protocol by quantum error correction[2]. We give an errorfree visible compression protocol of the ensemble TrB (|eA eA |?IB )κ(|ΦL ΦL |) x x with the compression Tr(|eA eA |?IB )κ(|ΦL ΦL |)

x x

|Ψ ? limn→∞ log n n | ≤ Ec (Wp ) + ?. Therefore, we obtain ? R(W, p) ≤ Ec (Wp ).

Further, when all states Wx are pure, we obtain 1 ? ?n limn→∞ n Ep (Wp ) = H(Wp ), which implies H(Wp ) ≤ ? p ). From (6), we have Ep (W ? Ep (Wp ) = H(Wp ). (10)

Proof of direct part: In this paper, the direct part means the existence of the visible compression attaining the limit of the regularized entanglement of puri?cation of the state corresponding to the given ensemble while the converse part does the nonexistence of the visible compression with a smaller rate than the limit of the regularized entanglement of puri?cation of the state corresponding to the given ensemble. The direct part follows the following lemma. We brie?y mention our construction of a code Ψ before going to its detail. First, we choose a one-way LOCC operation κ such that the state κ(|ΦL ΦL |) is ? close to Wp . Assume that we perform the measurement A A ? {|ex ex | ? I}x . When the state is Wp , the ?nal state on HB with the measurement outcome x is Wx . Hence, when the state is κ(|ΦL ΦL |), we can expect that the ′ ?nal state Wx on HB with the outcome x is close to Wx . ′ Further, the ensemble (Wx )x∈X can be compressed to the pair of classical information of the size CC(κ) and Hilbert space of the dimension L in the visible framework without any error. When this compression protocol is described by a code Ψ, this insight is formulated as the following lemma. Lemma: Let κ be a one-way LOCC operation. There exists a code Ψ such that 1 ? εp (Ψ) ≤ 1 ? F 2 (Wp , κ(|ΦL ΦL |)) 2 1 ? + Wp ? κ(|ΦL ΦL |) 1 , 2 |Ψ| = L · CC(κ).

x∈X

size L · CC(κ). Assume that the operation κ has the form κ = i κA,i ? κB,i , where {κA,i }ln is an instrui=1 ment (CP maps valued measure) on HA and κB,i is a TP-CP map on HB for each i. De?ne the probability qx

qx = Tr(|eA eA | ? IB ) x x =

i

def

i

κA,i ? κB,i (|ΦL ΦL |) (13)

Tr κ? ((|eA eA |) ? IB (|ΦL ΦL |)) x A,i x

and the probability pi,x and the state ρi,x as

pi,x = ρi,x

Tr κ? ((|eA eA |) ? IB (|ΦL ΦL |)) x x A,i qx ? A A def TrA κA,i ((|ex ex |) ? IB (|ΦL ΦL |)) = . qx pi,x

def

(11) (12)

(Note that any two-way LOCC operation can be simulated by one-way LOCC when the initial state is pure [8].) Using this lemma, we obtain the direct part as follows. Let κn be a one-way LOCC operation satisfying log CC(κn ) ? ?n → 0, lim F (Wp , κn (|ΦLn ΦLn |)) = 1, n log Ln ? lim ≤ Ec (Wp ) + ? n→∞ n

n→∞

Now, we construct the coding protocol Ψ: When the encoder receives the input signal x, he sends the state ρi,x with the probability pi,x and sends the classical information i. The decoder performs the TP-CP map κB,i dependently of the classical signal i. This protocol gives the visible compression the ensemble TrB (|eA eA |?IB )κ(|ΦL ΦL |) x x , which can be realized Tr(|eA eA |?IB )κ(|ΦL ΦL |)

x x

x∈X

by classical memory of the size CC(κ) and quantum memory with the dimension L. That is, inequality (12) follows from this construction. Next, we prove that TrB (|eA eA |?IB )κ(|ΦL ΦL |) x x px , Tr(|eA eA |?IB )κ(|ΦL ΦL |)

x x

for any ? > 0. Thus, by applying this lemma, there exists a sequence of codes {Ψn } such that εpn (Ψn ) → 0 and

x∈X

the ensemble is close to the This

given ensemble, i.e., show inequality (11).

4 inequality follows from the evaluation: F2

x

Concerning the ?rst term of (15), the inequality 1 (1 ? F 2 (Wx , 2 ≤1 ? F (Wx , ≤1 ? TrB

i

px |eA eA | ? Wx , x x px |eA eA | ? Wx x x

i

κA,i ? κB,i (|ΦL ΦL |) κA,i ? κB,i (|ΦL ΦL |) κA,i ? κB,i (|ΦL ΦL |)

pi,x κB,i (ρi,x )))

i

≤ Tr = Tr

x

pi,x κB,i (ρi,x )) pi,x κB,i (ρi,x )

i

x

i

√ px |eA eA | ? x x Wx

Wx

i

Wx

(17)

=

x

√ px TrB

holds. Hence, (11) follows from (15), (16), and (17). κA,i ? κB,i (|ΦL ΦL |) Proof of converse part: The converse part essentially follows from Conditions E2 and E3 of entanglement of puri?cation. For any ? > 0, we choose a sequence of codes Ψn = (Kn , τn , νn ) such that R = lim

def

· TrA |eA eA | ? IB x x ≤ · =

x

i

√ px TrB

x

Wx κA,i ? κB,i (|ΦL ΦL |) pi,x κB,i (ρi,x )

i

TrA |eA eA | ? IB x x √ px qx TrB Wx

1 log |Ψn | ≤ R(W, p) + ?, n

εpn (Ψn ) → 0.

i

(14)

=

x

√ ( px qx ? px ) TrB px TrB

x

Wx

i

pi,x κB,i (ρi,x ) (15)

Wx . Then, the state ρn = ? satis?es ? ?n ρ F (Wp , ιA ? νn (?n )) = ≥

x∈X n

? ?n The state Wp is described by the i.i.d. distribution n A A n ? ?n {px }x∈X n of {px }x∈X as Wp = ∈X n px |ex ex | ?

(n) def x∈X n

pn |eA eA | ? τn (x) x x x

+

Wx

i

pi,x κB,i (ρi,x ).

x∈X n

(n) pn F (Wx , νn ? τn (x)) x

(n) pn F 2 (Wx , νn ? τn (x)) → 1, x

The above relations can be checked as follows: i) The ?rst inequality follows from a basic inequality F 2 (ρ, σ) ≤ √ √ Tr ρ σ. ii) The second inequality follows from the ma√ trix concavity of t. iii) The equation (14) follows from qx

i

pi,x κB,i (ρi,x ) κB,i (TrA (κ? (|eA A,i x eA |) x ? IB )|ΦL ΦL |)

where ιA is the identical operation on the system HA . Note that τn is the encoding and νn is the decoding. From (6) and Condition E2(weak monotonicity of entanglement of puri?cation), log |Ψn | ≥ H(TrA ρn ) ≥ Ep (?n ) ≥ Ep (ιA ? νn (?n )). ? ρ ρ Hence, Condition E3(continuity of entanglement of puri?cation) yields lim 1 1 ? ?n log |Ψn | ≥ lim Ep (Wp ). n→∞ n n

=

i

= TrA |eA eA | ? IB x x

i

κA,i ? κB,i (|ΦL ΦL |),

where ιB is the identical operation on HB . The second term of (15) is evaluated by √ ( px qx ? px ) TrB √ ( px qx ? px )+ = qx ( ? 1)+ px = px Wx

i

Hence, using (5), we obtain ? R(W, p) ≥ Ec (Wp ) = lim

n→∞

pi,x κB,i (ρi,x ) qx ?1 px p + x

1

x

1 ? ?n Ep (Wp ). n

≤ ≤ ≤

x

x

x

x

1 q?p (qx ? px )+ = 2

1 ? Wp ? κ(|ΦL ΦL |) 1 , 2

(16)

where (t)+ is t when t is positive and it is 0 otherwise. The ?nal inequality follows from the de?nition of the distribution q (13).

Conclusion: We have proved that the bound of visible mixed state compression is equal to the optimal bound of the state generation from maximally entangled state with classical communication of the sublinear size. In particular, inequalities (11) and (12) express the relation between two problems. In the proof of direct part, we have constructed an error-free visible compression protocol of the ensemble corresponding to a state generation protocol from maximally entangled state. The converse part

5 has been proved from the weak monotonicity and continuity of entanglement of puri?cation. This result may indicate that these two problems are essentially equivalent. The obtained relation is essentially based on the relation between the noiseless channel and the maximally entangled state. Hence, a further relation based on this relation can be expected among several information protocols. Further, when there is no restriction concerning the size of classical communication, the optimal rate of the state generation from maximally entangled state (entanglement cost) is closely related to additivity of the channel capacity[9, 10]. Hence, it is interesting to consider the relation between state generation with sublinear-size classical communication and channel problems. The author thanks Professor Hiroshi Imai and ERATO-SORST Quantum Computation and Information Project for supporting this research. He is also grateful for Dr. Andreas Winter to useful discussion on the related topics. He is also indebted to the reviewer for commenting this paper and pointing out the relation with the paper [2]. Help Submitted manuscripts - follow the links to view or make changes and resubmit LL10788 View Optimal Visible Compression Rate For Mixed States Is De

[1] I. Devetak, A.W. Harrow, and A. Winter, “A family of quantum protocols,” Phys. Rev. Lett., 93, 230504 (2004). [2] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, “Mixed State Entanglement and Quantum Error Correction,” Phys. Rev. A, 54, 3824-3851 (1996). [3] B. Schumacher, “Quantum coding,” Phys. Rev. A, 51, 2738-2747 (1995). [4] M. Horodecki, “Limits for compression of quantum information carried by ensembles of mixed states,” Phys. Rev. A, 57, 3364–3369 (1998); quant-ph/9712035 (1997). [5] M. Horodecki, “Optimal compression for mixed signal states,” Phys. Rev. A, 61, 052309 (2000). [6] B. M. Terhal, M. Horodecki, D. W. Leung, and D. P. DiVincenzo, “The entanglement of puri?cation,” J. Math. Phys., 43, 4286 (2002). [7] H.-K. Lo and S. Popescu, “Classical Communication Cost of Entanglement Manipulation: Is Entanglement an Interconvertible Resource?,” Phys. Rev. Lett., 83, 1459 (1999). [8] H.-K. Lo and S. Popescu, “Concentrating entanglement by local actions: Beyond mean values,” Phys. Rev. A, 63 022301 (2001); quant-ph/9707038 (1997). [9] K. Matsumoto, T. Shimono, and A. Winter, “Remarks on additivity of the Holevo channel capacity and of the entanglement of formation,” Comm. Math. Phys., 246(3), 427–442 (2004); quant-ph/0206148 (2002). [10] P. W. Shor, “Equivalence of Additivity Questions in Quantum Information Theory,” Comm. Math. Phys., 246(3), 453–473 (2004). quant-ph/0305035 (2003).

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