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arXiv:quant-ph/0306009 · Journal of Physics A Mathematical and General

Entanglement cost of antisymmetric states and additivity of capacity of some quantum channels

Keiji Matsumoto, Fumitaka Yura

Published Mar 29, 2004• 45 citations
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Abstract

We study the entanglement cost of the states in the antisymmetric space, which consists of ( d − 1) d -dimensional systems. The cost is always log 2 ( d − 1) ebits when the state is divided into bipartite . Combined with the arguments in [6], additivity of channel capacity of some quantum channels is also shown.
DOI
10.1088/0305-4470/37/15/l03

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We study the entanglement cost of the states in the contragredient space, which consists of (d−1)(d-1)(d−1) ddd-dimensional systems. The cost is always log2(d−1)\\log_2 (d-1)log2​(d−1) ebits when the state is divided into bipartite Cdotimes(Cd)d−2\\C^d \\otimes (\\C^d)^{d-2}Cdotimes(Cd)d−2. Combined with the arguments in \\cite{Matsumoto02}, additivity of channel capacity of some quantum channels is also shown.
This report gives a lower bound of entanglement cost for antisymmetric states of bipartite d-level systems to be log_2 (d/(d-1)) ebit (for d=3, E_c >= 0.585...). The paper quant-ph/0112131 claims that the value is equal to one ebit for d=3, since all of the eigenvalues of reduced matrix of any pure states living in N times tensor product of antisymmetric space is not greater than 2^(-N) thus the von Neumann entropy is not less than N, but the proof is not true. Hence whether the value is equal to or less than one ebit is not clear at this moment.
This preprint proves a sharp purity bound for arbitrary complex pure states in tensor products of antisymmetric pair spaces, across the physical cut consisting of the first system from each pair versus the second system from each pair. For every finite number m of pairs and local dimensions at least two, purity is at most 2^(-m) and entanglement entropy is at least m bits, with equality characterized by products of normalized decomposable two-forms. It derives entanglement of formation m for m copies of the normalized antisymmetric state and, using the published regularization theorem, entanglement cost one ebit for every local dimension at least two. The archive provides original formal sources and a precise account of their scope; it does not claim a complete Lean translation of every ordinary LOCC execution. Research directed and curated by Matthew Liston. The mathematics, proofs, formalization and manuscript preparation substantially involve AI systems; paper-specific attribution and checking limits are disclosed in the paper and archive. No independent human proof verification, external endorsement or peer review is claimed. Licensing: original manuscript and documentation CC BY 4.0; original code MIT. The inherited QICLean SchattenNorm helper retains Apache 2.0 and its original attribution. The archive provides the exact file-level license mapping. Manuscript revision: 22 October 2026. Research director's note — Matthew Liston: This project is motivated by philosophical exploration and curiosity about AI-assisted research. My aim in sharing these results is to make them available for scrutiny, correction and further development, so that others can build on whatever proves useful to mathematics and science. Correspondence: atemporalwalk@gmail.com.
We first investigate the one-shot static entanglement cost to simulate a bipartite quantum channel under the set of nonentangling channels. The lower bound on the one-shot static entanglement cost is given by the generalized robustness of the target channel as well as the robust-generating power of the channel, which implies that the cost necessary to generate a bipartite quantum channel might not be retrievable in general. Next, we conceive a set of quantum channels that extends the set of nonentangling channels. We find out that the one-shot static entanglement cost under the extended set of channels is given by the channel's standard log-robustness. This quantity also gives the one-shot dynamic entanglement cost under a set of free superchannels that do not generate a dynamic entanglement resource; the extended set is shown to be equivalent to the set of free superchannels.
Bipartite entanglement is one of the fundamental quantifiable resources of quantum information theory. We propose a new application of this resource to the theory of quantum measurements. According to Naimark's theorem any rank 1 generalised measurement (POVM) M may be represented as a von Neumann measurement in an extended (tensor product) space of the system plus ancilla. By considering a suitable average of the entanglements of these measurement directions and minimising over all Naimark extensions, we define a notion of entanglement cost E_min(M) of M. We give a constructive means of characterising all Naimark extensions of a given POVM. We identify various classes of POVMs with zero and non-zero cost and explicitly characterise all POVMs in 2 dimensions having zero cost. We prove a constant upper bound on the entanglement cost of any POVM in any dimension. Hence the asymptotic entanglement cost (i.e. the large n limit of the cost of n applications of M, divided by n) is zero for all POVMs. The trine measurement is defined by three rank 1 elements, with directions symmetrically placed around a great circle on the Bloch sphere. We give an analytic expression for its entanglement cost. Defining a normalised cost of any d-dimensional POVM by E_min(M)/log(d), we show (using a combination of analytic and numerical techniques) that the trine measurement is more costly than any other POVM with d>2, or with d=2 and ancilla dimension 2. This strongly suggests that the trine measurement is the most costly of all POVMs.