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arXiv:1205.4370

Irreversibility of Entanglement Concentration for Pure State

Wataru Kumagai, Masahito Hayashi

Published May 20, 2012• 2 citations
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Abstract

For a pure state $ψ$ on a composite system $\mathcal{H}_A\otimes\mathcal{H}_B$, both the entanglement cost $E_C(ψ)$ and the distillable entanglement $E_D(ψ)$ coincide with the von Neumann entropy $H(\mathrm{Tr}_{B}ψ)$. Therefore, the entanglement concentration from the multiple state $ψ^{\otimes n}$ of a pure state $ψ$ to the multiple state $Φ^{\otimes L_n}$ of the EPR state $Φ$ seems to be able to be reversibly performed with an asymptotically infinitesimal error when the rate ${L_n}/{n}$ goes to $H(\mathrm{Tr}_{B}ψ)$. In this paper, we show that it is impossible to reversibly perform the entanglement concentration for a multiple pure state even in asymptotic situation. In addition, in the case when we recover the multiple state $ψ^{\otimes M_n}$ after the concentration for $ψ^{\otimes n}$, we evaluate the asymptotic behavior of the loss number $n-M_n$ of $ψ$. This evaluation is thought to be closely related to the entanglement compression in distant parties.

DOI
10.48550/arxiv.1205.4370

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For a pure state ψψψ on a composite system HA⊗HB\mathcal{H}_A\otimes\mathcal{H}_BHA​⊗HB​, both the entanglement cost EC(ψ)E_C(ψ)EC​(ψ) and the distillable entanglement ED(ψ)E_D(ψ)ED​(ψ) coincide with the von Neumann entropy H(TrBψ)H(\mathrm{Tr}_{B}ψ)H(TrB​ψ). Therefore, the entanglement concentration from the multiple state ψ⊗nψ^{\otimes n}ψ⊗n of a pure state ψψψ to the multiple state Φ⊗LnΦ^{\otimes L_n}Φ⊗Ln​ of the EPR state ΦΦΦ seems to be able to be reversibly performed with an asymptotically infinitesimal error when the rate Ln/n{L_n}/{n}Ln​/n goes to H(TrBψ)H(\mathrm{Tr}_{B}ψ)H(TrB​ψ). In this paper, we show that it is impossible to reversibly perform the entanglement concentration for a multiple pure state even in asymptotic situation. In addition, in the case when we recover the multiple state ψ⊗Mnψ^{\otimes M_n}ψ⊗Mn​ after the concentration for ψ⊗nψ^{\otimes n}ψ⊗n, we evaluate the asymptotic behavior of the loss number n−Mnn-M_nn−Mn​ of ψψψ. This evaluation is thought to be closely related to the entanglement compression in distant parties.
In quantum information theory, it is widely believed that entanglement concentration for bipartite pure states is asymptotically reversible. In order to examine this, we give a precise formulation of the problem, and show a trade-off relation between performance and reversibility, which implies the irreversibility of entanglement concentration. Then, we regard entanglement concentration as entangled state compression in an entanglement storage with lower dimension. Because of the irreversibility of entanglement concentration, an initial state can not be completely recovered after the compression process and a loss inevitably arises in the process. We numerically calculate this loss and also derive for it a highly accurate analytical approximation.
In quantum information theory, it is widely believed that entanglement concentration for bipartite pure states is asymptotically reversible. In order to examine this, we give a precise formulation of the problem, and show a trade-off relation between performance and reversibility, which implies the irreversibility of entanglement concentration. Then, we regard entanglement concentration as entangled state compression in an entanglement storage with lower dimension. Because of the irreversibility of entanglement concentration, an initial state cannot be completely recovered after the compression process and a loss inevitably arises in the process. We numerically calculate this loss and also derive for it a highly accurate analytical approximation.
Recently, two of the authors showed that entanglement concentration is irreversible. However, it is still not clear what kind of LOCC conversion is reversible. We derive the necessary and sufficient condition for reversibility of LOCC conversion between two bipartite pure entangled states in an asymptotic setting. Simultaneously, we evaluate how many copies of the initial state is to be lost to overcome irreversibility of LOCC conversion. Our result is useful for designing how to store entangled states via LOCC operations without error.
The manipulation of quantum entanglement is fundamentally irreversible: some mixed entangled states require pure entanglement for their preparation, although no pure entanglement can be recovered from them by local operations and classical communication. This irreversibility is known to persist even under the maximal class of operations that do not generate entanglement, revealing a fundamental distinction between entanglement theory and thermodynamics. We construct cases for which any attempt to restore reversibility necessarily incurs an error that increases exponentially with the number of copies. Technically, we demonstrate a strict separation between the exponential strong-converse distillable entanglement and the exponential strong-converse entanglement cost. Our result resolves a conjecture posed by Lami and Regula (Nat. Phys. 19, 184-189 (2023)) and strengthens it by showing that the irreversibility of entanglement persists even at the level of polynomially (in the number of copies) growing error. We further derive a semidefinite-programming lower bound on the exponential strong-converse cost under non-entangling operations. Finally, for the class of completely PPT-preserving operations, we construct analytically solvable families of antisymmetric states exhibiting the exponential strong-converse irreversibility. Remarkably, to our knowledge, no analogous separation between exponential strong converse cost and the analogous distillable entanglement is currently known even under the more restrictive class of LOCC operations.
We investigate the irreversibility of entanglement distillation for a symmetric (d+1)(d+1)(d+1)-parameter family of mixed bipartite quantum states acting on Hilbert spaces of arbitrary dimension $d\ifmmode\times\else\texttimes\fi{}d$. We prove that in this family the entanglement cost is generically strictly larger than the distillable entanglement, so that the set of states for which the distillation process is asymptotically reversible is of measure zero. This remains true even if the distillation process is catalytically assisted by pure-state entanglement and every operation is allowed, which preserves the positivity of the partial transpose. It is shown that reversibility occurs only in cases where the state is a tagged mixture. The reversible cases are shown to be completely characterized by minimal uncertainty vectors for entropic uncertainty relations.
Hoping to simplify the classification of pure entangled states of multi $(m)\ensuremath{-}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{t}\mathrm{e}$ quantum systems, we study exactly and asymptotically (in n)n)n) reversible transformations among nthn\mathrm{th}nth tensor powers of such states (i.e., nnn copies of the state shared among the same mmm parties) under local quantum operations and classical communication (LOCC). For exact transformations, we show that two states whose marginal one-party entropies agree are either locally unitarily equivalent or else LOCC incomparable. In particular we show that two tripartite Greenberger-Horne-Zeilinger states are LOCC incomparable to three bipartite Einstein-Podolsky-Rosen (EPR) states symmetrically shared among the three parties. Asymptotic transformations yield a simpler classification than exact transformations; for example, they allow all pure bipartite states to be characterized by a single parameter---their partial entropy---which may be interpreted as the number of EPR pairs asymptotically interconvertible to the state in question by LOCC transformations. We show that $m\ensuremath{-}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{t}\mathrm{e}$ pure states having an $m\ensuremath{-}\mathrm{w}\mathrm{a}\mathrm{y}$ Schmidt decomposition are similarly parametrizable, with the partial entropy across any nontrivial partition representing the number of standard quantum superposition or ``cat'' states $|{0}^{\ensuremath{\bigotimes}m}〉+|{1}^{\ensuremath{\bigotimes}m}〉$ asymptotically interconvertible to the state in question. For general $m\ensuremath{-}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{t}\mathrm{e}$ states, partial entropies across different partitions need not be equal, and since partial entropies are conserved by asymptotically reversible LOCC operations, a multicomponent entanglement measure is needed, with each scalar component representing a different kind of entanglement, not asymptotically interconvertible to the other kinds. In particular we show that the m=4m=4m=4 cat state is not isentropic to, and therefore not asymptotically interconvertible to, any combination of bipartite and tripartite states shared among the four parties. Thus, although the m=4m=4m=4 cat state can be prepared from bipartite EPR states, the preparation process is necessarily irreversible, and remains so even asymptotically. For each number of parties mmm we define a minimal reversible entanglement generating set (MREGS) as a set of states of minimal cardinality sufficient to generate all $m\ensuremath{-}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{t}\mathrm{e}$ pure states by asymptotically reversible LOCC transformations. Partial entropy arguments provide lower bounds on the size of the MREGS, but for m>2m>2m>2 we know no upper bounds. We briefly consider several generalizations of LOCC transformations, including transformations with some probability of failure, transformations with the catalytic assistance of states other than the states we are trying to transform, and asymptotic LOCC transformations supplemented by a negligible [o(n)][o(n)][o(n)] amount of quantum communication.