1 Introduction: the asymptotic continuity catastrophe↩︎

1.1 Entanglement dilution↩︎

A fundamental discovery of quantum information science is that quantum entanglement is not only one of the most striking features of quantum mechanics [1], but it can be considered from the operational side as a resource, namely, as a fuel that powers quantum technology. Among its many applications, we find quantum teleportation [2], dense coding [3], the violation of Bell inequalities [4], and quantum key distribution [5], [6]. As any other resource, quantum entanglement can be manipulated, i.e., transformed into different forms. The canonical way to do so is by employing local operations and classical communication (LOCC), which constitute the most general operations that distant parties bound to the rule of quantum mechanics and allowed to exchange only classical messages can implement [7].

The most important entanglement manipulation protocol is perhaps entanglement distillation [8]–[10], whose goal is to transform many identical and ideally distributed (IID) copies of a quantum state \(\rho_{AB}\) into as many copies as possible of the ebit, i.e., the two-qubit maximally entangled state shared between \(A\) and \(B\) while making an asymptotically vanishing error. The converse task is, however, equally important from the conceptual [10] as well as possibly practical [11] point of view: it consists of using LOCC assisted by pure entanglement, in the form of ebits, to prepare as many copies of \(\rho_{AB}\) as possible, again with asymptotically vanishing error. It is therefore called entanglement dilution. The duality between entanglement distillation and entanglement dilution offers a striking parallel with classical thermodynamics, where heat can be transferred from hot to cold baths to obtain work, or work can be invested to transfer heat in the opposite direction. Entanglement is, however, more complex than what this intriguing similarity might suggest. In fact, while in classical thermodynamics work and heat can be interconverted reversibly by means of Carnot cycles, entanglement theory harbors the fascinating phenomenon of irreversibility, which shows that different forms of entanglement are in general inequivalent [12]–[17].

One of the pillars of finite-dimensional entanglement theory is the characterization of the optimal rate of ebits that must be consumed per output copy of \(\rho = \rho_{AB}\) in entanglement dilution under LOCC. This rate is called the entanglement cost of \(\rho_{AB}\), denoted by \(\require{physics} E_c\qty(\rho)\). As proved in Ref. [18] in the case of finite-dimensional quantum systems (see also [19]), it is given by \[\require{physics} \begin{align} E_c\qty(\rho) = E_f^\infty\qty(\rho) \relax\lim_{n\to\infty} \frac{1}{n}\, E_f\big(\rho^{\otimes n}\big)\, , \label{HHT} \end{align}\tag{1}\] where \(E_f\), called the entanglement of formation [10], is defined as the convex-roof extension of the entanglement entropy [8], and \(E_f^\infty\) is its regularized form.

1.2 Enter infinite-dimensional systems↩︎

So far so good. But, problematically, the quantum physics in our world is not necessarily as straightforward as the analysis in the finite-dimensional case might suggest. To see why one needs to consider an infinite-dimensional system, consider that even within the paradigm of current quantum technology, we can easily step out of the reassuring finite-dimensional setting. In actual quantum experiments, a “perfect” qubit, i.e., an intrinsically two-dimensional quantum system without any additional degrees of freedom, is hardly ever encountered. Rather, we make qubits by embedding them into an infinite-dimensional system in various ways. This is what happens, e.g., for superconducting qubits, trapped ions, and single photons. In these experimental platforms, a subspace of the overall system spanned by the ground state and the first excited state can be used as a qubit; at the same time, the overall Hilbert space is intrinsically infinite-dimensional.

Regardless of whether quantum information processing is designed to be qubit-based, infinite-dimensional Hilbert space is bound to lurk just above the finite-dimensional subspace where we carry out our computations. Indeed, the effects of state transitions leaking into higher energy levels are common in practical implementations of quantum protocols. With more precise control, in principle, we can potentially take advantage of the higher dimensions of Hilbert space to enhance the performance of quantum information processing.

Also from the practical standpoint, one of the most important applications of entanglement theory is the theory of entanglement distribution in quantum optical networks, which are intrinsically infinite-dimensional objects [20], [21]. The many results involving the calculation or the estimation of capacities of continuous-variable channels, such as the quantum capacities of the pure loss channel [22]–[28], of the thermal attenuator [24], [27], [29]–[33], and of non-Gaussian bosonic dephasing channels [34], as well as the calculation of the classical capacity of phase-insensitive single-mode Gaussian channels [35], [36], are exemplary of this widespread interest. Recently, also the entanglement cost of continuous-variable quantum channels has been thoroughly investigated [37].

Finally, from a more fundamental standpoint, quantum field theory suggests that the elementary constituents of matter are represented by infinite-dimensional quantum systems. Accordingly, entanglement of quantum fields has become the subject of increasing interest [38].

Whether the motivation lies in practical applications or in the fundamental logical consistency of the theory, the question is the same: what are the limitations on entanglement manipulation in infinite-dimensional systems? To answer this question, in recent years, there has been a growing interest in the generalization of results in entanglement theory from finite- to infinite-dimensional quantum systems.

1.3 Main contribution↩︎

The present paper aims to generalize the formula 1 for the entanglement cost to all infinite-dimensional quantum systems with an underlying separable Hilbert space.

Several issues arise when one attempts to extend the original proof, which works only in finite dimension, to the infinite-dimensional case. Let us leave aside for a second the fact that even the definition of entanglement of formation presents some subtleties in the latter case [39], as those can be avoided by restricting to sufficiently well-behaved, e.g., finite-energy, quantum states. The proof of 1 in Ref. [18] is composed of two main parts: the direct part, which consists of constructing an entanglement dilution protocol that achieves an asymptotic rate close to the right-hand side of 1 , and the converse part, whose goal is to show that this is the optimal rate, i.e., that no other protocol can achieve a lower rate.

Neither the direct nor the converse part of the argument in Ref. [18] work as stated in infinite-dimensional systems. Regarding the direct part, the key technique in constructing the entanglement dilution protocol in [18] was strong typicality, which means that if a sequence \(\require{physics} \qty(x_1,\ldots,x_N)\in\mathcal{X}^N\) is sampled in an independent and identically distributed (IID) way from a probability distribution \(\require{physics} p\qty(x)\) over a finite set \(\mathcal{X}\), then each element \(x\in\mathcal{X}\) appears in this sequence approximately \(\require{physics} N p\qty(x)\) times. The problem here lies in the fact that strong typicality holds only for finite sets \(\mathcal{X}\). It is necessary to take into account infinite sets if one wants to generalize the result to infinite-dimensional systems.

An equally serious issue arises in the converse part of the proof. Remarkably, the issue arises even if the target state \(\rho_{AB}\) itself is supported on the tensor product of finite-dimensional local spaces. Indeed, it is in principle possible that some LOCC protocols could aim to obtain an approximation to several copies of \(\rho_{AB}\), i.e., to the state \(\rho_{AB}^{\otimes n}\), by actually preparing an infinite-dimensional state that acts on the full Hilbert space but is nonetheless close to \(\rho_{AB}^{\otimes n}\) in trace distance. Such protocols could have, in principle, a lower entanglement cost than any finite-dimensional protocol. The key technical tool that is employed in the finite-dimensional proof to control the approximation to the target state is the asymptotic continuity of the entanglement of formation \(E_f\) [40]–[42]. This is a particularly strong form of continuity in which the dimension of the underlying space makes an explicit appearance. For infinite-dimensional spaces, however, the statement trivializes. Due to this phenomenon, dubbed the “asymptotic continuity catastrophe” in Ref. [43], it was impossible, prior to our work, to establish a converse bound for entanglement dilution in terms of the regularized entanglement of formation. Some converse bounds in infinite-dimensional systems were available [43], [44], but they did not coincide with the regularized entanglement of formation, and thus were not sufficient to establish 1 .

A somewhat different case is that of Ref. [37], which proves that the entanglement cost of certain bosonic Gaussian channels is lower bounded by their regularized entanglement of formation. This is an example of a converse statement for entanglement dilution obtained in an infinite-dimensional setting. So, why is this proof not affected by the asymptotic continuity catastrophe? In short, the reason has to do with the fact that Ref. [37] studies channels rather than states. In finding lower bounds on the cost in this setting, one can simply choose to input a state \(RA\) with local finite energy on \(R\). Since \(R\) is not touched by the channel nor by its simulation, the two relevant states coming into play can be guaranteed to have finite local energy, which enables Ref. [37] to apply the energy-constrained continuity of the entanglement of formation in Ref. [45]. By contrast, in the case of states we are dealing with here, we may impose some energy constraint on the target state \(\rho_{AB}^{\otimes n}\), but not on the infinite-dimensional state that is actually prepared by the protocol; thus, we cannot use the energy-constrained continuity in Ref. [45] for these states.

In this work, we prove that the entanglement cost of infinite-dimensional states \(\rho_{AB}\) is given by the same formula 1 as in the finite-dimensional case, provided that the quantum entropy of the reduced state of \(\rho_{AB}\) on either \(A\) or \(B\) is finite. For the direct part, we construct an entanglement dilution protocol that works for both finite- and infinite-dimensional systems and achieves a rate as close to the regularized entanglement of formation as desired. To prove that this protocol works as intended, we develop a technique for approximating multiple copies of infinite-dimensional mixed states, which goes beyond a mere application of typicality. The protocol uses one-way LOCC as in the finite-dimensional case, and remarkably, even if \(\rho_{AB}\) is infinite-dimensional, it is implementable with only finitely many bits of one-way classical communication.

To demonstrate the optimality of this protocol (converse part), we prove that even with the use of separable operations (SEP) one cannot achieve a lower entanglement dilution rate. To circumvent the obstacles arising from infinite dimensionality, we show a generalized result on the monotonicity of the entanglement of formation under infinite-dimensional SEP operations, progressing beyond the existing monotonicity results [46]–[49]. Then, to circumvent the asymptotic continuity catastrophe, we develop an alternative argument based on the semi-continuity bound on the entanglement of formation of infinite-dimensional states recently introduced by Ref. [50], rather than the conventional asymptotic continuity in Refs. [40]–[42], which is applicable only to finite-dimensional states, or the continuity relation in Ref. [45], which instead applies only to two energy-constrained states.

The rest of this paper is organized as follows. In Sec. 2 we introduce our terminology and establish some notation. In Sec. 3 we present our main results. To prove them, in Sec. 4 we construct the asymptotically optimal one-way LOCC protocol for entanglement dilution to infinite-dimensional states, and in Sec. 5 we establish the converse statement, demonstrating that not even SEP operations can do better. We conclude by summarizing our findings and describing some open problems in Sec. 6.

2 Preliminaries↩︎

In this section, we fix our terminology and explain our notation. In Sec. 2.1, we summarize the formulation of quantum mechanics in infinite-dimensional systems. In Sec. 2.2, we discuss various notions of topologies that are needed to define the convergence of infinite-dimensional quantum operations. In Sec. 2.3, we introduce several classes of operations considered in our analysis, clearing up some subtleties related to infinite-dimensional systems. In Sec. 2.4 and Sec. 2.5, we formalize the notions of entanglement cost and entanglement of formation. Finally, in Sec. 2.6, we discuss why exactly the finite-dimensional analysis of entanglement cost in Ref. [18] cannot be directly extended to the infinite-dimensional case. Throughout this paper, we let \(\mathbb{N}\) denote the set of non-negative integers and \(\mathbb{N}_+\) the set of positive integers.

2.1 Quantum mechanics↩︎

Quantum systems are represented by complex separable Hilbert spaces \(\mathcal{H}\). A linear operator \(B\) on \(\mathcal{H}\) is said to be bounded if the operator norm \[\require{physics} \label{eq:operator95norm} \left\|B\right\|_\infty\relax\sup\qty{\left\|B\ket{v}\right\|:\ket{v}\in\mathcal{H},\,\left\|\ket{v}\right\|\leq 1}\tag{2}\] is finite, where \(\left\|\ket{v}\right\|\) for \(\ket{v}\in\mathcal{H}\) is the norm induced by the inner product of \(\mathcal{H}\). In what follows, we will denote by \(\require{physics} \mathcal{B}\qty(\mathcal{H})\) the Banach space of bounded operators on \(\mathcal{H}\) equipped with the operator norm \(\|\cdot\|_\infty\). We let \(\require{physics} \mathbb{1}\in\mathcal{B}\qty(\mathcal{H})\) denote the identity operator. A bounded operator \(\require{physics} B\in\mathcal{B}\qty(\mathcal{H})\) is said to be compact if \(B\) maps the unit ball of \(\mathcal{H}\) into a relatively compact set, i.e., if the closure of \(\require{physics} B\qty{\ket{v}\in\mathcal{H}:\left\|\ket{v}\right\|\leq 1}\) is compact in \(\mathcal{H}\). The Banach space of compact operators on \(\mathcal{H}\) equipped with the operator norm will be denoted by \(\mathcal{K}(\mathcal{H})\).

A bounded operator \(\require{physics} B\in\mathcal{B}\qty(\mathcal{H})\) is said to be positive semidefinite if \(\bra{v} B \ket{v} \geq 0\) for any \(\ket{v}\in\mathcal{H}\), which we write as \(B\geq 0\). Let \(\{\ket{n}\}_n\) be any orthonormal basis of \(\mathcal{H}\). Given any positive semidefinite bounded operator \(B\), the convergence of the (possibly infinite) sum \(\sum_n\bra{n} B \ket{n}\), which contains only non-negative terms, as well as its value if it converges, is independent of the choice of \(\{\ket{n}\}_n\) [51]. If the sum converges, the trace of the positive semidefinite operator \(B\) is defined as \(\require{physics} \Tr\qty[B]\relax\sum_n\bra{n} B \ket{n}\). A bounded operator \(\require{physics} T\in \mathcal{B}\qty(\mathcal{H})\) is said to be of trace class if \(\require{physics} \|T\|_1 \relax\Tr\qty[|T|]=\sum_n\bra{n} |T| \ket{n} < \infty\), where \(|T| \relax\sqrt{T^\dagger T}\geq 0\), and \(T^\dagger\) is the adjoint operator of \(T\). Although a trace-class operator is not necessarily positive semidefinite, the above definition of the trace extends naturally to any trace-class operator \(T\), by setting \[\require{physics} \begin{align} \label{eq:definition95trace} \Tr\qty[T]\relax\sum_n\bra{n} T \ket{n}, \end{align}\tag{3}\] which turns out to converge and is independent of the choice of orthonormal basis \(\{\ket{n}\}_n\) [51]. In the finite-dimensional case, the trace defined in 3 coincides with the sum of the diagonal elements in the matrix representation of the finite-dimensional operator. The set of trace-class operators is necessarily compact. We will let \(\require{physics} \mathcal{T}\qty(\mathcal{H})\) denote the Banach space of trace class operators equipped with the trace norm \(\|\cdot\|_1\). For two trace-class operators \(\require{physics} T,T^\prime\in\mathcal{T}\qty(\mathcal{H})\), we will call the quantity \(\frac{1}{2}\left\|T-T^\prime\right\|_1\) the trace distance between \(T\) and \(T^\prime\), and the corresponding topology the trace-norm topology on \(\require{physics} \mathcal{T}\qty(\mathcal{H})\). We will use the trace-norm topology for \(\require{physics} \mathcal{T}\qty(\mathcal{H})\) throughout the paper.

Quantum states are then represented by density operators, i.e., positive semidefinite trace-class operators on \(\mathcal{H}\) with trace \(1\). We let \(\mathcal{B}_+(\mathcal{H})\), \(\mathcal{K}_+(\mathcal{H})\), and \(\mathcal{T}_+(\mathcal{H})\) denote the cones of positive semidefinite bounded, compact, and trace-class operators, respectively. The set of density operators will be denoted instead by \(\require{physics} \mathcal{D}\qty(\mathcal{H})\relax\qty{\rho\in\mathcal{T}_+\qty(\mathcal{H}):\,\Tr\qty[\rho]=1}\). Given a state \(\require{physics} \rho\in \mathcal{D}\qty(\mathcal{H})\) with spectral decomposition \(\rho = \sum_x p(x) \ket{x}\!\!\bra{x}\), the quantum entropy of \(\rho\) is defined by \[\require{physics} \begin{align} S\qty(\rho) \relax-\Tr\qty[ \rho \log_2 \rho ] = - \sum_x p(x) \log_2 p(x)\, , \label{entropy} \end{align}\tag{4}\] where we set \(0\log_2 0\relax 0\) and \(\require{physics} S\qty(\rho) = \infty\) if the series on the right-hand side diverges. The entropy function \(S\) on \(\require{physics} \mathcal{D}\qty(\mathcal{H})\) is: concave, meaning that \(\require{physics} S\qty(p\rho+\qty(1-p)\sigma)\geq pS\qty(\rho)+\qty(1-p)S\qty(\sigma)\) for all \(\require{physics} \rho,\sigma\in\mathcal{D}\qty(\mathcal{H})\) and all \(p\in[0,1]\); and lower semi-continuous, i.e., \(\require{physics} \liminf_{\widetilde{\rho}\to\rho} S\qty(\widetilde{\rho})\geq S\qty(\rho)\), where the limit is taken in the trace-norm topology.

The Hilbert space describing a composite quantum system \(AB\) is the tensor product of the Hilbert spaces representing its subsystems, in formula \(\mathcal{H}_{AB} \relax\mathcal{H}_A\otimes \mathcal{H}_B\). We may use an operator with subscript \(AB\) to represent the state of \(AB\), e.g., \(\require{physics} \rho_{AB}\in\mathcal{D}\qty(\mathcal{H}_A\otimes \mathcal{H}_B)\). The reduced state of \(A\) obtained from the partial trace over \(B\) is denoted by \(\require{physics} \rho_{A} = \Tr_B\qty[\rho_{AB}]\in\mathcal{D}\qty(\mathcal{H}_A)\), and similarly for \(B\). The composite system obtained by joining \(n\) copies of \(AB\) is indicated by \(A^nB^n\).

A quantum state \(\sigma_{AB}\) of \(AB\) is said to be separable if it is in the closed convex hull (in the trace-norm topology) of the set of all tensor-product states \(\omega_A\otimes\tau_B\) for \(\require{physics} \omega_A\in\mathcal{D}\qty(\mathcal{H}_A)\) and \(\require{physics} \tau_B\in\mathcal{D}\qty(\mathcal{H}_B)\). The set of separable states will be denoted by \(\pazocal{S}(A\!:\!B)\); it is formally defined by \[\require{physics} \begin{align} \pazocal{S}(A\!:\!B) \relax\mathop{\mathrm{cl}}\mathop{\mathrm{conv}}\left\{ \omega_A\otimes\tau_B:\;\omega_A\in\mathcal{D}\qty(\mathcal{H}_A),\;\tau_B\in\mathcal{D}\qty(\mathcal{H}_B) \right\} , \label{separable95states} \end{align}\tag{5}\] where \(\mathop{\mathrm{cl}}\) denotes the closure with respect to the trace-norm topology. Rather intuitively, \(\pazocal{S}(A\!:\!B)\) can alternatively be defined as the set of barycenters of all Borel probability measure \(\mu\) on \(\require{physics} \mathcal{D}\qty(\mathcal{H}_A)\times\mathcal{D}\qty(\mathcal{H}_B)\); that is, any separable state \(\sigma_{AB}\) can be written in terms of the (Bochner) integral as \(\require{physics} \sigma_{AB} = \int \dd\mu \big(\omega_A,\tau_B\big)\;\omega_A\otimes\tau_B\) [52], [53]. If there exists a representation of the form \(\sigma_{AB} = \sum\nolimits_{x=1}^{\infty} p(x)\, \omega_A^{(x)} \otimes\tau_B^{(x)}\) then \(\sigma_{AB}\) is said to be countably separable. In the infinite-dimensional case, it is known that separable states that are not countably separable do exist [52], [53].

A state that is not separable is said to be entangled [52], [54]. The simplest example of an entangled state is the two-qubit maximally entangled state, called an entanglement bit or ebit. This is defined by \(\Phi \relax\ket{\Phi}\!\!\bra{\Phi}\), where \[\label{eq:ebit} \ket{\Phi}_{AB}\relax\frac{1}{\sqrt2} \left(\ket{0}_A\otimes\ket{0}_B + \ket{1}_A\otimes\ket{1}_B\right)\in\mathcal{H}_A\otimes\mathcal{H}_B\, .\tag{6}\]

2.2 Topologies on the set of quantum channels↩︎

We use a subset of linear maps \(\require{physics} \pazocal{E}:\mathcal{T}\qty(\mathcal{H})\to\mathcal{T}\qty(\mathcal{H}^\prime)\) to represent transformations between quantum states. Here, \(\mathcal{H}\) and \(\mathcal{H}^\prime\) are the input and output Hilbert spaces, respectively. The linear map \(\pazocal{E}\) is called: positive, if \(\require{physics} \pazocal{E}\qty(T)\geq 0\) for all \(T\geq 0\); completely positive (CP), if \(I_k\otimes \pazocal{E}\) is positive for all \(k\in \mathbb{N}_+\), where \(I_k\) denotes the identity map on the set of \(k\times k\) complex matrices; and trace-preserving (TP), if \(\require{physics} \Tr\qty[\pazocal{E}\qty(T)]=\Tr\qty[T]\) for all \(\require{physics} T\in\mathcal{T}\qty(\mathcal{H})\). A completely positive trace-preserving (CPTP) map is also called a quantum channel. Quantum channels describe the transformations between quantum states allowed within the law of quantum mechanics. For a finite or countably infinite set \(\mathcal{J}\), a family of CP maps \(\require{physics} \qty(\pazocal{E}_j)_{j\in\mathcal{J}}\) summing up a TP map \(\sum_{j\in\mathcal{J}} \pazocal{E}_j\) is called an instrument. Instruments describe physical measurements on quantum systems: the output state corresponding to input state \(\rho\) and measurement outcome \(j\) is simply \(\require{physics} \frac{\pazocal{E}_j(\rho)}{\Tr \pazocal{E}_j(\rho)}\).

Since we are dealing with possibly infinite-dimensional systems, we need to choose a topology on the space of quantum channels. A possible choice is the topology of uniform convergence, induced by the diamond norm [55]. However, it has been observed that this topology is often too fine, and as a consequence, it exhibits some pathological behaviors—for instance, it places physically close channels at nonzero distance [56]. Two alternatives that are more physically sound are the topology induced by the energy-constrained diamond norm [27], [56], [57] and the topology of strong convergence. Fortunately, Ref. [57] has shown that for many physically motivated Hamiltonians, these two alternatives are equivalent, so we adopt the latter without loss of generality. We then say that a sequence¹ \(\big(\pazocal{E}_n\big)_n\) of CP maps \(\require{physics} \pazocal{E}_n: \mathcal{T}\qty(\mathcal{H}) \to \mathcal{T}\qty(\mathcal{H}^\prime)\) converges strongly to a map \(\require{physics} \pazocal{E}: \mathcal{T}\qty(\mathcal{H}) \to \mathcal{T}\qty(\mathcal{H}^\prime)\), and we write \(\pazocal{E}_n \mathrel{ \setbox 2=\stackon{\scriptstyle\! n\rightarrow \infty\!}{\scriptstyle\mathrm{s}} \stackunder[5pt]{ \xrightarrow{\makebox[\dimexpr\wd 2\relax]{\scriptstyle\mathrm{s}}} }{ \scriptstyle\! n\rightarrow \infty\!\, }} \pazocal{E}\), if \[\require{physics} \begin{align} \lim_{n\to\infty} \left\| \pazocal{E}_n(T) - \pazocal{E}(T)\right\|_1 = 0 \qquad \forall\;T\in \mathcal{T}\qty(\mathcal{H})\, . \end{align}\] Under these assumptions, this map \(\pazocal{E}\) must be CP. If the \(\pazocal{E}_n\) are quantum channels, then so must be \(\pazocal{E}\). The closure with respect to the topology of strong convergence is denoted by \(\mathop{\mathrm{cl}}_{\mathrm{s}}(\cdot)\).

So far we have discussed the strong convergence of quantum channels. However, for what follows, we will also care about strong convergence of instruments, i.e., countable collections \(\mathcal{I} = (\pazocal{E}_i)_i\) of CP maps \(\require{physics} \pazocal{E}_i:\mathcal{T}\qty(\mathcal{H}) \to \mathcal{T}\qty(\mathcal{H}^\prime)\) that sum up to a CPTP map \(\sum_i \pazocal{E}_i\). One can easily think of instruments as quantum channels by appending an ancillary output system that records the outcome of the measurement, i.e., by constructing the quantum channel \(\rho \mapsto \sum_i \pazocal{E}_i(\rho) \otimes \ket{i}\!\!\bra{i}\). In this way, we can extend the notion of strong convergence from channels to instruments. We let \(\mathcal{I}_\nu \mathrel{ \setbox 2=\stackon{\scriptstyle\! \nu\to\infty \!}{\scriptstyle\mathrm{s}} \stackunder[5pt]{ \xrightarrow{\makebox[\dimexpr\wd 2\relax]{\scriptstyle\mathrm{s}}} }{ \scriptstyle\! \nu\to\infty \!\, }} \mathcal{I}\) denote the strong convergence of the sequence of instruments \(\big(\mathcal{I}_\nu\big)_\nu\) to \(\mathcal{I}\).

2.3 LOCC for infinite-dimensional systems↩︎

To talk about entanglement manipulation, we need to first discuss a few subtleties associated with the definition of local operations and classical communication (LOCC) in the case of infinite-dimensional systems. The good news is that most of the work done by Ref. [7] to define finite-dimensional LOCC carries over straight away to the infinite-dimensional case. Let us summarise the main steps of Ref. [7] here. For some positive integer \(N\), let \(\mathcal{H}=\mathcal{H}_{A_1}\otimes\cdots\otimes\mathcal{H}_{A_N}\) represent an \(N\)-partite quantum system, where \(\mathcal{H}_{A_n}\) for each \(\require{physics} n\in\qty{1,\ldots,N}\) is the separable Hilbert space associated with the \(n^{\text{th}}\) party. For simplicity of presentation, here we use the same input and output systems \(\mathcal{H}\) in the following definitions of maps, but the general case with different input and output systems follows similarly. We allow for either finite or countably infinite sets as index sets of the instruments in the following definitions. For two instruments \(\require{physics} \mathcal{I}=\qty(\pazocal{E}_i)_{i\in I}\) and \(\mathcal{I}^\prime=\big(\pazocal{E}_{j}^\prime\big)_{j\in J}\), we call \(\mathcal{I}^\prime\) a coarse-graining of \(\mathcal{I}\) if there exists a \(J\)-indexed partition of \(I\), i.e., \(I = \coprod_{j\in J} I_j\), such that for all \(j\in J\), it holds that \(\pazocal{E}_{j}^\prime = \sum_{i\in I_{j}} \pazocal{E}_i\). Despite the somewhat cumbersome definition, the physical meaning is clear: we can implement \(\mathcal{I}^\prime\) by first measuring \(\mathcal{I}\), and then “binning” the outcome into one of the subsets \(I_{j}\). For each party \(n\), an instrument \(\require{physics} \mathcal{I}=\qty(\pazocal{E}_i)_{i}\) is called one-way local with respect to party \(n\) if each of its CP maps has the form \[\require{physics} \begin{align} \pazocal{E}_i=\qty(\bigotimes\nolimits_{n^\prime\neq n}\pazocal{F}_i^{\qty(n^\prime)})\otimes \pazocal{E}_i^{\qty(n)}\, , \end{align}\] where \(\require{physics} \pazocal{E}_i^{\qty(n)}\) is a CP map on \(\require{physics} \mathcal{T}\qty(\mathcal{H}_{A_n})\), and \(\require{physics} \pazocal{F}_i^{\qty(n^\prime)}\) is a CPTP map on \(\require{physics} \mathcal{T}\qty(\mathcal{H}_{A_{n^\prime}})\). The interpretation here is that party \(n\) applies an instrument \(\require{physics} \big(\pazocal{E}_i^{\qty(n)}\big)_i\) on their system, communicates the outcome \(i\) to all the other parties, and each party \(n'\neq n\) post-processes their system with some quantum channel \(\require{physics} \pazocal{F}_i^{\qty(n^\prime)}\) that may depend on \(i\).

An instrument \(\mathcal{I}^\prime\) is LOCC-linked to \(\require{physics} \qty(\pazocal{E}_i)_i\) if, for all \(i\), there exists a one-way local instrument \(\big(\pazocal{F}_{j|i}\big)_{j}\) with respect to some party \(\require{physics} n_i\in\qty{1,\ldots,N}\) such that \(\mathcal{I}^\prime\) is a coarse-graining of the instrument \(\big(\pazocal{F}_{j|i}\circ\pazocal{E}_i\big)_{i,j}\). Physically, after the outcome \(i\) has been obtained and broadcast to all parties, one of them, say party \(n_i\), depending on \(i\), can, in turn, apply another instrument on their system. We say that:

1. \(\mathcal{I}\in\mathrm{LOCC}_1\), if \(\mathcal{I}\) is the coarse-graining of a one-way local instrument with respect to some party;

2. \(\mathcal{I}\in\mathrm{LOCC}_r\) (\(r\geq 2\)), if \(\mathcal{I}\) is LOCC-linked to some \(\mathcal{I}\in\mathrm{LOCC}_{r-1}\);

3. \(\mathcal{I}\in\mathrm{LOCC}_\mathbb{N}\), if \(\mathcal{I}\in\mathrm{LOCC}_r\) for some finite \(r\).

We are now ready to extend the definition of LOCC in Ref. [7] from finite- to infinite-dimensional systems.

To prove the converse part of our main result, we need to extend also the notion of separable operations, originally introduced in Refs. [58], [59] for the finite-dimensional case, to infinite-dimensional systems.

The rationale behind the above definition reflects the phenomenology observed for separable states in infinite-dimensional systems. According to 5 , \(\pazocal{S}(A\!:\!B)\) is defined as the closure of the set of countably separable states with respect to an appropriate topology for states; in the same way, we now construct \(\mathrm{SEP}\) as the closure of \(\mathrm{SEP}_\mathbb{N}\) with respect to an appropriate quantum channel topology. The fact that taking the closure is a necessary step will be apparent from the discussion below.

There is a simple hierarchy among the above classes of operations (or instruments), namely, \[\begin{align} \mathrm{LOCC}_\to,\,\mathrm{LOCC}_\leftarrow \subset \mathrm{LOCC}_1\subset \mathrm{LOCC}_2 \subset \ldots \subset \mathrm{LOCC}_\mathbb{N}\subset \mathrm{LOCC}\subset \overline{\mathrm{LOCC}} \subset \mathrm{SEP}\, , \label{eq:infinite95operations} \end{align}\tag{7}\] where \(\overline{\mathrm{LOCC}}\relax\mathop{\mathrm{cl}}_{\mathrm{s}} \left(\mathrm{LOCC}_\mathbb{N}\right)\). Note that \(\mathop{\mathrm{cl}}_{\mathrm{s}} \left(\mathrm{LOCC}_\mathbb{N}\right)=\mathop{\mathrm{cl}}_{\mathrm{s}} \left(\mathrm{LOCC}\right)\) by definition of LOCC. The inclusion relation 7 mirrors the finite-dimensional one in Ref. [7]. One of the results of Ref. [7] is that all inclusions in 7 are strict already in finite dimension. In general systems, we have \[\begin{align} \mathrm{LOCC}_\mathbb{N}\subset \mathrm{SEP}_\mathbb{N}\subset \mathrm{SEP}\, . \label{eq:inclusions95operations95infinite95dim} \end{align}\tag{8}\] Already in finite dimension \(\mathrm{LOCC}_\mathbb{N}\neq\mathrm{SEP}_\mathbb{N}=\mathrm{SEP}\), so that the first inclusion in 8 is in general strict. As for the second, it is also strict in the case of infinite-dimensional systems. A simple example that suffices to show this claim is as follows: consider a “replacer channel” acting on a bipartite system \(AB\) \[\require{physics} \begin{align} \pazocal{E}_{\sigma} (\rho_{AB}) = \sigma_{AB} \Tr\qty[\rho_{AB}]\,, \label{replacer95channel} \end{align}\tag{9}\] where \(\sigma=\sigma_{AB}\) is a separable but not countably separable state. As it turns out, \[\begin{align} \pazocal{E}_{\sigma} \in \mathrm{LOCC}\setminus \mathrm{SEP}_\mathbb{N}\subseteq \mathrm{SEP}\setminus \mathrm{SEP}_\mathbb{N}\, . \label{LOCC95SEPN95gap95replacer95channel} \end{align}\tag{10}\] The fact that \(\pazocal{E}_\sigma \notin \mathrm{SEP}_\mathbb{N}\) can be seen by observing that if \(\rho\) is a product state, the output of any operation in \(\mathrm{SEP}_\mathbb{N}\) acting on \(\rho\) must be a countably separable state, but \(\sigma\) is not countably separable. The fact that \(\pazocal{E}_\sigma \in \mathrm{LOCC}\) is proved as follows. Pick a sequence of countably separable states \(\sigma_\nu\) with \(\lim_{\nu\to\infty} \|\sigma - \sigma_\nu\|_1 = 0\). This is possible because the set of separable states is the closure of that of countably separable states with respect to the trace-norm topology. Then, construct the replacer channels \[\require{physics} \begin{align} \pazocal{E}_\nu :\;\rho \mapsto \sigma_\nu \Tr\qty[\rho]\, . \end{align}\] Note that \(\pazocal{E}_\nu\) is one-way local with respect to any party. Moreover, \(\pazocal{E}_{\nu+1}\circ \pazocal{E}_\nu = \pazocal{E}_{\nu+1}\). Hence, \(\pazocal{E}_{\nu+1}\) is LOCC-linked to \(\pazocal{E}_\nu\) for all \(\nu\). Since \(\pazocal{E}_\nu \mathrel{ \setbox 2=\stackon{\scriptstyle\! \nu\to\infty \!}{\scriptstyle\mathrm{s}} \stackunder[5pt]{ \xrightarrow{\makebox[\dimexpr\wd 2\relax]{\scriptstyle\mathrm{s}}} }{ \scriptstyle\! \nu\to\infty \!\, }} \pazocal{E}_\sigma\), we have \(\pazocal{E}_\sigma\in \mathrm{LOCC}\).

In view of 10 , we see that the strong closure in ?? is needed to include in the definition of SEP some natural channels that we would want to call separable. As it happens for states, it is natural to conjecture that SEP channels that are not in \(\mathrm{SEP}_\mathbb{N}\) still admit a continuous Kraus decomposition with tensor-product Kraus operators. We leave the investigation of this question, which is not essential for our problem, for future work.

2.4 Entanglement cost↩︎

The two fundamental ingredients of entanglement manipulation are entanglement distillation and entanglement dilution, two essentially opposite tasks [8]–[10]. The former consists of transforming many IID copies of a given state \(\rho_{AB}\) into as many ebits as possible while employing only LOCC (or their variants as discussed in Sec. 2.3) and achieving an asymptotically vanishing error. The converse task of entanglement dilution, which will be of interest here, consists of preparing many copies of \(\rho_{AB}\) with LOCC while consuming as few ebits as possible, and again with vanishing error.

More formally, the task of entanglement dilution involves two parties, Alice and Bob, who can perform a restricted family \(\mathcal{O}\) of CPTP maps, assisted by initially shared entanglement in the form of ebits. For the sake of this paper, it suffices to consider the cases \[\begin{align} \mathcal{O}\in \left\{ \mathrm{LOCC}_\to,\, \mathrm{LOCC}_\leftarrow,\, \mathrm{LOCC}_\mathbb{N},\, \mathrm{LOCC},\, \overline{\mathrm{LOCC}},\, \mathrm{SEP}_\mathbb{N},\, \mathrm{SEP}\right\} . \label{eq:classes} \end{align}\tag{11}\] Let \(\require{physics} \rho = \rho_{AB}\in\mathcal{D}\qty(\mathcal{H}_A\otimes\mathcal{H}_B)\) be a target quantum state on a (either finite- or infinite-dimensional) bipartite quantum system \(AB\). The entanglement cost of \(\rho\) under operations in \(\mathcal{O}\) is defined by \[\begin{align} \label{eq:entanglement95cost} E_{c,\, \mathcal{O}} (\rho) \relax\inf\left\{ r >0 :\;\lim_{n\to\infty} \inf_{\pazocal{E}_n \in \mathcal{O}}\frac{1}{2}\left\| \pazocal{E}_n\big(\Phi^{\otimes \floor*{rn}}\big) - \rho^{\otimes n} \right\|_1 = 0\right\} , \end{align}\tag{12}\] where \(\Phi = \ket{\Phi}\!\!\bra{\Phi}\) is the ebit defined by 6 , and for a fixed \(n\), the operation \(\pazocal{E}_n\) acts on a bipartite system composed of \(\floor*{rn}+\floor*{rn}\) qubits and outputs a state in \(A^nB^n\). In 12 , \(r\) should be interpreted as an achievable rate, i.e., as an achievable ebit cost per copy of \(\rho\) to be prepared. If the set on the right-hand side of 12 is empty, then we set \(E_{c,\,\mathcal{O}}(\rho) = +\infty\).

In the definition of entanglement cost 12 , the preparation of \(n\) copies of \(\rho\) can incur a nonzero error, however such error must vanish asymptotically as \(n\to\infty\). Because of the fact that manipulation is not required to be zero-error, the entanglement cost under any class of operations and under its strong closure turns out to be the same. In particular, the cost under \(\mathrm{LOCC}\) and that under \(\mathrm{LOCC}_\mathbb{N}\) coincide. This simple fact is proved below.

Putting together 7 , Remark 3, and Lemma 1, we deduce that \[\label{eq:different95entanglement95costs} E_{c,\, \mathrm{LOCC}_1} (\rho)\geq E_{c,\,\mathrm{LOCC}_2} (\rho)\geq \cdots\geq E_{c,\,\mathrm{LOCC}_\mathbb{N}}(\rho) = E_{c,\,\mathrm{LOCC}}(\rho) \geq E_{c,\, \mathrm{SEP}_\mathbb{N}}(\rho) = E_{c,\, \mathrm{SEP}}(\rho)\tag{16}\] for arbitrary bipartite states \(\rho=\rho_{AB}\). In finite dimension, it is known that all of the above notions of cost, which are a priori different, actually coincide. Our results will show that, perhaps surprisingly, the same happens in infinite-dimensional systems.

2.5 Entanglement of formation↩︎

A foundational result in entanglement theory is the characterization of entanglement cost under finite-dimensional LOCC as the regularization of the entanglement of formation [18], [19], one of the most important entanglement measures. In a finite- or infinite-dimensional quantum system, the entanglement of formation is constructed as the continuous convex-roof extension of the entanglement entropy, i.e. [10], [39], [60] \[\require{physics} \begin{align} E_f\qty(\rho) \relax\inf_{\rho_{AB} = \int \dd\mu(\psi)\, \psi_{AB}} \int \dd\mu(\psi)\, S\big(\psi_A \big)\, , \label{eof} \end{align}\tag{17}\] where the infimum on the right-hand side is taken over all Borel probability measures \(\mu\) on the set of pure states \(\psi = \psi_{AB}=\ket{\psi}\!\!\bra{\psi}_{AB}\) of \(\mathcal{H}_A\otimes \mathcal{H}_B\) having as barycenter \(\rho_{AB}\). Since pure states on \(AB\) have the same local entropy on \(A\) and on \(B\), in 17 , we can equivalently replace \(\require{physics} S\qty(\psi_A)\) with \(\require{physics} S\qty(\psi_B)\). Note that Refs. [39], [60], [61] (see also Ref. [62] for a recent review) have also introduced another possible definition of entanglement of formation, where one replaces the integral in 17 with a discrete, countably infinite sum: \[\require{physics} \begin{align} \widetilde{E}_f\qty(\rho) \relax\inf_{\rho_{AB} = \sum_x p(x)\, \psi_x^{AB}} \sum_x p(x)\, S\big(\psi_x^A \big)\, . \label{EoF95tilde} \end{align}\tag{18}\] Clearly, \[\require{physics} \begin{align} E_f(\rho) \leq \widetilde{E}_f\qty(\rho) \label{EoF95smaller95EoF95tilde} \end{align}\tag{19}\] holds in general, by restricting the measure \(\mu\) in 17 to be a countable convex combination of Dirac measures. If \(AB\) is finite dimensional, it follows from Carathéodory’s theorem [63] that in fact \(E_f = \widetilde{E}_f\). The same is true for infinite-dimensional systems, provided that one considers only states with finite local entropy; that is, as shown in Ref. [61], \[\require{physics} \label{eq:min95S95A95B} E_f(\rho_{AB}) = \widetilde{E}_f(\rho_{AB})\qquad \forall\;\rho_{AB}\in \mathcal{D}\qty(\mathcal{H}_A\otimes \mathcal{H}_B)\quad\text{with}\;\;\min\qty{S\qty(\rho_{A}), S\qty(\rho_{B})}<\infty\, .\tag{20}\] It is presently unknown whether the above two versions of the entanglement of formation, \(E_f\) and \(\widetilde{E}_f\), coincide in general or not [62]. However, the continuous version \(E_f\) has superior proven analytical properties, which is why it may be regarded as the most appropriate definition.² In particular, in Ref. [39] it is proved that:

• \(E_f\) is everywhere lower semi-continuous, meaning that \[\begin{align} \liminf_{n\to\infty} E_f(\rho_n) \geq E_f(\rho) \label{lsc95EoF} \end{align}\tag{21}\] for all converging sequences \((\rho_n)_n\) of states \(\require{physics} \rho_n \in \mathcal{D}\qty(\mathcal{H}_A \otimes \mathcal{H}_B)\) with limit \(\require{physics} \rho\in \mathcal{D}\qty(\mathcal{H}_A \otimes \mathcal{H}_B)\), i.e., such that \(\left\|\rho_n - \rho\right\|_1 \mathrel{ \setbox 2=\stackon{\scriptstyle\! n\rightarrow \infty\!}{\scriptstyle\mathrm{}} \stackunder[5pt]{ \xrightarrow{\makebox[\dimexpr\wd 2\relax]{\scriptstyle\mathrm{}}} }{ \scriptstyle\! n\rightarrow \infty\!\, }} 0\). The same property for \(\widetilde{E}_f\) is known to hold on sufficiently regular sequences but has not been established in general.

• \(E_f\) is monotonic under operations in \(\overline{\mathrm{LOCC}}\), meaning that \[\require{physics} \begin{align} E_f\big(\pazocal{E}(\rho)\big) \leq E_f(\rho)\qquad \forall\;\rho = \rho_{AB} \in \mathcal{D}\qty(\mathcal{H}_A \otimes \mathcal{H}_B)\, , \qquad \forall\;\pazocal{E}\in \overline{\mathrm{LOCC}}\, . \label{monotonicity95locc95EoF} \end{align}\tag{22}\] Note that Ref. [39] only states 22 explicitly for selective local instruments, but by composing them, taking limits, and using the lower semi-continuity 21 , one can establish 22 . In this work, we will extend 22 to all separable operations (see Proposition 15 below).

Additionally, the restriction in 20 immediately implies that the entanglement of formation itself is finite, i.e., \(\require{physics} E_f\qty(\rho)<\infty\), due to the concavity of the entropy. In fact, \(\require{physics} \int \dd\mu(\psi)\, S\big(\psi_A \big) \leq S\left(\int \dd\mu(\psi)\, S\big(\psi_A \big)\right) = S\qty(\rho_{A})\), and analogously for \(B\). In this work, we will always deal with states having finite local entropy, hence the distinction between \(E_f\) and \(\widetilde{E}_f\) is not crucial for us.

Another important property of the entanglement of formation is subadditivity; for every two states \(\rho_{AB}\) and \(\sigma_{A'B'}\), we have that \[E_f\big(\rho_{AB} \otimes \sigma_{A'B'}\big)\leq E_f\big(\rho_{AB}\big) + E_f\big(\sigma_{A'B'}\big),\] where it is understood that on the left-hand side, the entanglement of formation is computed with respect to the bipartition \(AA'\) versus \(BB'\). Note that the additivity \(E_f\big(\rho_{AB} \otimes \sigma_{A'B'}\big)= E_f\big(\rho_{AB}\big) + E_f\big(\sigma_{A'B'}\big)\) does not hold in general [64], [65]. When combined with Fekete’s lemma [66], subadditivity implies that the regularized entanglement of formation exists and can be evaluated as \[\require{physics} \begin{align} E_f^\infty\qty(\rho) \relax\lim_{n\to\infty} \frac{1}{n}\, E_f\big(\rho^{\otimes n}\big) = \inf_{n\in \mathbb{N}_+} \frac{1}{n}\, E_f\big(\rho^{\otimes n}\big) . \label{regularized95eof} \end{align}\tag{23}\] Again, it is understood that the bipartition used on the right-hand sides of 23 is \(A^n\) versus \(B^n\). The main result of Ref. [18] is precisely that \[\require{physics} \begin{align} E_{c,\,\mathrm{LOCC}}(\rho) = E_f^\infty\qty(\rho) \label{Hayden95et95al95result} \end{align}\tag{24}\] for all states \(\rho_{AB}\in \mathcal{D}\big(\mathcal{H}_A\otimes \mathcal{H}_B\big)\) with both \(\dim \mathcal{H}_A < \infty\) and \(\dim \mathcal{H}_B < \infty\), provided that one restricts also the LOCC operations to be finite-dimensional.

In the finite dimension, the entanglement of formation is not only lower semi-continuous but, in fact, asymptotically continuous. Namely, for any two states \(\rho_{AB}\) and \(\omega_{AB}\) of a bipartite system \(AB\) with finite minimum local dimension \(\require{physics} d \relax\min\qty{\dim\mathcal{H}_A,\dim\mathcal{H}_B}\), it holds that [42] \[\require{physics} \begin{align} \left| E_f\qty(\rho) - E_f\qty(\omega) \right| \leq \sqrt{\varepsilon(2-\varepsilon)} \log_2 (d-1) + h_2\qty(\sqrt{\varepsilon(2-\varepsilon)})\, , \label{asymptotic95continuity95EoF} \end{align}\tag{25}\] where \(h_2(x)\relax-x\log_2 x - (1-x) \log_2(1-x)\) is the binary entropy function, and \[\varepsilon\relax\frac{1}{2} \left\|\rho_{AB} - \omega_{AB}\right\|_1\] is the trace distance between \(\rho\) and \(\omega\) (cf.Refs. [40], [41]), assumed to satisfy the inequality \(0\leq \varepsilon\leq 1-\frac{\sqrt{2d-1}}{d}\). The above inequality implies not only the continuity of \(E_f\) but also a special form of uniform continuity that features a linear dependence on \(\log_2 d\) — this is precisely what is commonly referred to as asymptotic continuity.

2.6 What exactly goes wrong in the infinite-dimensional case?↩︎

Before presenting our main result, it is worth discussing why exactly the finite-dimensional proof of 24 in Ref. [18] cannot be directly extended to infinite-dimensional systems. We already gave a broad overview of the problems in Sec. 1.3, but now we can be more technically precise and pinpoint where the issues lie. Essentially, there are four main problems, two in the direct part and two in the converse.

(1) The direct part of the argument in Ref. [18] relies crucially on the existence of an entanglement dilution protocol that prepares any pure state \(\psi_{AB}\) by consuming ebits at a rate equal to \(S(\psi_A)\). The first problem lies precisely in how to construct such a protocol in the infinite-dimensional case. In fact, in order for the cost to be identified with the local quantum entropy, we need to apply some notion of typicality to the locally reduced state \(\psi_A\). How does this work beyond finite dimension? We will solve this issue by resorting to the fact that the notion of weak typicality is still applicable to countably infinite ensembles.

(2) Another more serious problem that arises in attempting to generalize the achievability (direct) part of the argument in Ref. [18] is as follows: given any pure-state decomposition \(\rho_{AB} = \sum_x p(x)\, \psi^{(x)}_{AB}\), instead of \(\rho_{AB}^{\otimes n}\), we aim to prepare its typical part, given, up to normalization, by \(\vphantom{\Big|}\sum_{x^n\in T_{p,n}^\delta} p(x^n)\, \psi^{(x^n)}_{A^nB^n}\), where \(\require{physics} x^n\relax\qty(x_1,\ldots,x_n)\), \(p(x^n) \relax\prod_{i=1}^n p(x_i)\), \(\psi^{(x^n)}_{A^nB^n} \relax\bigotimes_{i=1}^n \psi^{(x_i)}_{A_iB_i}\), and \(T_{p,n}^\delta\) is the strongly typical set [67]. Loosely speaking, \(T_{p,n}^\delta\) comprises all sequences that contain each symbol \(x\in \mathcal{X}\) approximately \(\require{physics} N p\qty(x)\) times, with the approximation being controlled by \(\delta\). Now, the key observation is that in finite dimension, when \(\mathcal{X}\) can be assumed to be finite without loss of generality [63], each of the numbers \(\require{physics} N p\qty(x)\) is either \(0\) (if \(p(x)=0\)) or asymptotically large (if \(p(x)>0\)). In the latter, nontrivial case, we can thus apply the asymptotic entanglement dilution protocol for pure states, with associated cost \(S\big(\psi_x^A\big)\).

If the alphabet $\mathcal{X}$ is infinite, which will be the case
for a generic infinite-dimensional state, then the strongly typical
set [@Ho_2010_unified_typicality] (see also Ref. [@YEUNG-BOOK]) will
include sequences in which some of the symbols, namely those with
very small probabilities, appear a *non-asymptotic number of times*.
We could call this the "rare" part of the state
$\rho_{AB}^{\otimes n}$. What entanglement dilution protocol do we
use for this rare part, given that we cannot rely anymore on the
asymptotic protocol? We will solve this problem by devising a more
"wasteful" protocol that is to be applied specifically to the rare
part of the state. The crux of our argument is to show that the
corresponding waste of resources can be controlled and made
arbitrarily small asymptotically.

(3) At the same time, the converse part of the argument needed to establish 24 also fails in for infinite-dimensional systems. One reason for this failure is because in this case the asymptotic continuity inequality 25 , which plays a key role in proving 24 , trivializes in the limit \(d\to\infty\). And this is for a good reason: the entanglement of formation, like most entropic quantities, ceases to be continuous in the infinite-dimensional case. In practice, the well-known fact that quantum entropy of is discontinuous everywhere if the underlying Hilbert space is infinite dimensional [68] already implies that the entanglement of formation is discontinuous at all pure states. This problem has been identified before in Ref. [43], where it was termed “asymptotic continuity catastrophe.” We will solve it by means of the semi-continuity bound recently established in Ref. [50].

(4) Finally, another property used in the converse part of the finite-dimensional argument in Ref. [18] is the monotonicity of the entanglement of formation under LOCC. It was later shown in Ref. [49] that in finite dimension the monotonicity of the entanglement of formation holds even under separable operations. Indeed, combining this result with the argument in Ref. [18] leads to the conclusion that in finite dimension the entanglement cost under separable operations is given by the same formula as the right-hand side of 24 . It is then natural to wonder whether an analogous formula holds true for infinite-dimensional systems, but this has been challenging to prove due to the lack of analysis of the monotonicity of the entanglement of formation under infinite-dimensional separable operations. With our rigorous formulation of \(\mathrm{SEP}_\mathbb{N}\) and \(\mathrm{SEP}\) in Sec. 2.3, we will solve this problem by showing that we can use an infinite-dimensional version of the Schur–Horn Theorem [69] to extend the result of Ref. [49], first to countably separable and then to all separable operations. This will allow us to establish the converse bound for the entanglement cost under separable operations.

3 Main results↩︎

In this section, we present our main result, that is, the general equality between the entanglement cost and the regularized entanglement of formation, whose validity we extend from finite- to infinite-dimensional systems. In fact, our theorem below shows something even stronger, i.e., that all the different notions of entanglement cost corresponding to the classes of operations in 16 are quantitatively equivalent. Our results can be seen as a complete generalization of the finite-dimensional result in Ref. [18].

The next two sections are devoted to a complete proof of the above Theorem 6. In particular, Sec. 4 deals with the achievability statement, while Sec. 5 contains the converse part.

4 Direct part: Construction of a one-way LOCC protocol for entanglement dilution↩︎

In this section, we construct a family of one-way LOCC protocols for entanglement dilution achieving ?? , proving the direct part of Theorem 6. After a brief introduction on typicality in Sec. 4.1, we show properties of the entanglement cost in Sec. 4.2, which are to be used for our analysis. Then, we give the proof of the direct part in two main steps: first, in Sec. 4.3, we deal with the case of pure states, solving the issue [issue:1] highlighted in Sec. 2.6; second, in Sec. 4.4, we present our general one-way LOCC protocol for entanglement dilution in the case of mixed states, thus addressing the issue [issue:2] in Sec. 2.6 and completing the conceptual developments needed to establish a full achievability proof.

4.1 Notions of typicality↩︎

As in the finite-dimensional case, the crucial notion for constructing an asymptotically optimal entanglement dilution protocol is that of typicality [70], [71], which we summarize here. Let \(\require{physics} \mathcal{X}=\qty{1,2,\ldots}\) denote a finite or countably infinite set, \(X\) a random variable taking on values in \(\mathcal{X}\), and \(p(x)\) the corresponding probability distribution. A sequence of \(n\) independent and identically distributed (IID) copies \(X_i\) of \(X\) will be denoted by \(\require{physics} X^n=\qty(X_1,X_2,\ldots,X_n)\). We write realizations of \(X^n\) as \(\require{physics} x^n=\qty(x_1,x_2,\ldots,x_n)\in \mathcal{X}^n\); the corresponding probability thus is \(\require{physics} p_{n}\qty(x^n)=p\qty(x_1)p\qty(x_2)\cdots p\qty(x_n)\). Define \(\require{physics} q\qty(x|x^n)\) as the empirical distribution of symbols \(x\in \mathcal{X}\) in the string \(x^n\), i.e., \[\require{physics} \begin{align} q\qty(x|x^n) \relax\frac{N\qty(x|x^n)}{n}\, , \end{align}\] where \(\require{physics} N\qty(x|x^n)\) is the number of occurrences of \(x\in\mathcal{X}\) in the sequence \(x^n\). We are now ready to define the main notions of typicality employed here, namely, strong typicality, which can be applied (and will be applied) to finite alphabets only, and weak typicality, which is more generally applicable albeit less powerful. For details, we refer the reader to Refs. [70]–[72]. A succinct discussion can be found in Ref. [73].

If \(\mathcal{X}\) is finite, then for any positive integer \(n\) and any \(\delta>0\), we can define the strongly typical set \(T_\delta^{X^n}\) as the set of sequences \(x^n\in \mathcal{X}^n\) such that, for all \(x\in \mathcal{X}\), if \(p(x)=0\), then \(q(x|x^n) = 0\), and otherwise \(\left|p(x) - q(x|x^n)\right| \leq \delta\). The probability that a sequence is strongly typical approaches one asymptotically, i.e., \[\begin{align} \lim_{n\to\infty} p_n\big(T^{X^n}_\delta\big) = 1 \qquad \forall\;\delta>0\, . \label{unit95prob95strongly95typical95set} \end{align}\tag{26}\]

An alternative notion that can be used for a countably infinite set is that of weak typicality. If \(\mathcal{X}\) is either a finite or countably infinite set, and the random variable \(X\) on \(\mathcal{X}\) has finite entropy \(H(X)\relax-\sum_{x\in\mathcal{X}} p(x)\log_2 p(x)<\infty\), then for any positive integer \(n\) and any \(\delta>0\), we can define the weakly typical set \(W_\delta^{X^n}\) as the set of sequences \(x^n\in \mathcal{X}^n\) such that \[\begin{align} \left|-\frac{1}{n}\, \log_2 p_n(x^n) -H(X)\right|\leq\delta\, . \end{align}\]

Weakly typical sets enjoy a wealth of properties, presented in detail in the aforementioned textbooks. We summarize some of them below:

(a) Asymptotic equipartition property: \[\begin{align} 2^{-n\left( H(X) + \delta \right)} \leq p_n(x^n) \leq 2^{-n\left( H(X) - \delta \right)} \qquad \forall\;x^n\in W^{X^n}_\delta\, . \label{AEP95weakly95typical95set} \end{align}\tag{27}\]

(b) Asymptotically unit probability: \[\begin{align} \lim_{n\to\infty} p_n\big(W^{X^n}_\delta\big) = 1\, , \label{unit95prob95weakly95typical95set} \end{align}\tag{28}\] where \(p_n\big(W^{X^n}_\delta\big) = \sum_{x^n\in W^{X^n}_\delta} p_n(x^n)\).

(c) Exponentially small cardinality: \[\begin{align} \left| W^{X^n}_\delta \right| \leq 2^{n\left(H(X) + \delta\right)} ,\qquad \lim_{n\to\infty} \frac{\left| W^{X^n}_\delta \right|}{2^{n\left(H(X) + \delta\right)}} = 1\, . \label{exp95smaller95weakly95typical95set} \end{align}\tag{29}\]

4.2 Properties of the entanglement cost↩︎

In this section, we show the properties of the entanglement cost relevant to our analysis. In particular, after discussing the additivity of the entanglement cost, we show the convexity of the entanglement cost.

First, we show the additivity of entanglement cost as follows.

An important step in generalizing the finite-dimensional result in Ref. [18] is to realize that the entanglement cost is a convex function of the state. This key fact has been first established in Ref. [74]; however, a closer look at the justification given after Definition 17 on p.14 therein reveals that the argument of Ref. [74] relies on the expression of the entanglement cost as the regularized entanglement formation, the very result that we are trying to generalize. To avoid circular reasoning, we give here a separate simple proof of the convexity of the entanglement cost, which we consider part of the folklore.

4.3 Achievability of entanglement cost for infinite-dimensional pure states↩︎

We are now ready to address the first of the issues listed in Sec. 2.6. To do this, we have to design an entanglement dilution protocol that can prepare any (finite- or infinite-dimensional) pure state \(\psi_{AB}\) with finite local quantum entropy \(S(\psi_A)<\infty\) consuming about \(S(\psi_A)\) ebits per output copy. The main idea of this protocol is the same as in the finite-dimensional case: we simply prepare the locally typical part of the state. The only difference is that we have to use the notion of weak typicality for countably infinite alphabets instead of finite ones. A similar approach has appeared before in Ref. [37].

Before moving on, it is instructive to present a simple corollary of the analysis of entanglement dilution for pure states in combination with the convexity of the entanglement cost. This corollary provides a bound on the entanglement cost for finite-rank infinite-dimensional mixed states, but it still does not suffice to fully characterize the entanglement cost of general infinite-dimensional mixed states.

4.4 Achievability of entanglement cost for infinite-dimensional mixed states↩︎

We now show how to construct a protocol that performs entanglement dilution on an arbitrary mixed state with finite local quantum entropy at a rate equal to the regularized entanglement of formation. We will also see later in Sec. 5 that such a rate is, in fact, optimal. To arrive at the result, we have to solve issue (2) listed in Sec. 2.6. As we have seen with Corollary 8, a mere combination of the result on pure states with the convexity of entanglement cost is insufficient for addressing this issue. One of our main original contributions is precisely an idea to address this issue, which is based on dividing the state into a “common” and a “rare” part, applying the standard entanglement dilution protocol to the former, and a different “wasteful” protocol to the latter. The standard protocol is based on the subroutine presented in Sec. 4.3, while the wasteful protocol will be explained below. See Remark 10 below for more details about how these sub-protocols constitute the overall protocol for the entanglement dilution of infinite-dimensional mixed states.

5 Converse part: Optimality among all entanglement dilution protocols under separable operations↩︎

In this section, we show that no protocol for entanglement dilution, even under separable operations, can achieve a rate that is lower than the regularized entanglement of formation in 23 . In this way, we will establish the converse part of Theorem 6 and, therefore, conclude the proof of our main result. The main original contribution of this section is an argument for the converse part based on two results. One resolves issue (3) presented in Sec. 2.6 by identifying an appropriate replacement for asymptotic continuity in the infinite-dimensional case, i.e., the semi-continuity bound for the entanglement of formation recently introduced in Ref. [50]. The other is the resolution of issue (4) in Sec. 2.6, i.e., the generalization of the proof of the monotonicity of the entanglement of formation to infinite-dimensional separable operations. We present the former in Sec. 5.1 and the latter in Sec. 5.2. Using these results, we finally prove the converse part in Sec. 5.3.

5.1 Semi-continuity of the entanglement of formation↩︎

To discuss the continuity of entropic quantities defined for infinite-dimensional quantum systems, a conventional technique is to impose an energy constraint on the states. To this end, one needs to introduce a Hamiltonian, represented by a (possibly unbounded) self-adjoint operator \(H\) on the underlying Hilbert space \(\mathcal{H}\). For physical reasons, and up to redefining the ground state energy, we will assume that \(H\) is positive semidefinite, meaning that \(\bra{\psi} H \ket{\psi} \geq 0\) for all \(\require{physics} \ket{\psi}\in \mathop{\mathrm{dom}}\qty(H)\), where \(\mathop{\mathrm{dom}}(X)\) denotes the domain of a self-adjoint operator \(X\). The average energy with respect to \(H\) of a state \(\rho\) on \(\mathcal{H}\) with spectral decomposition \(\rho = \sum_{x=0}^\infty p_x \ket{\psi_x}\!\!\bra{\psi_x}\) can then be defined as \[\require{physics} \begin{align} \Tr \rho H \relax\left\{ \begin{array}{ll} \sum_{x=0}^\infty p_x \left\| H^{1/2} \ket{\psi_x} \right\|^2 & \quad \text{if \ket{\psi_x}\in \mathop{\mathrm{dom}}\qty(H^{1/2}) whenever p_x>0,} \\[1.5ex] \infty &\quad \text{otherwise.} \end{array} \right. \end{align}\] Note that the above infinite sum contains only non-negative terms and is therefore always well defined, although possibly infinite. Also, a subtle point that is important to take into consideration is that \(\require{physics} \Tr \rho H\) may be well defined and finite even if \(\require{physics} \mathop{\mathrm{supp}}(\rho)\not\subseteq \mathop{\mathrm{dom}}\qty(H)\) (and thus even if \(\rho H\) is not a trace class operator itself).⁵

With that in mind, we can consider a state \(\rho\) whose average energy with respect to a Hamiltonian \(H\) is bounded by some constant \(E\geq 0\), in the sense that \[\require{physics} \begin{align}\label{eq:energy95constraint} \Tr\qty[\rho H]\leq E. \end{align}\tag{47}\] Intuitively, for such an energy constraint to work as expected, we also need to avoid pathological Hamiltonians \(H\). A mathematically convenient condition to avoid such a pathological behavior is the so-called Gibbs hypothesis, which guarantees that \(H\) is associated with a well-defined thermal state at all temperatures.

For a grounded Hamiltonian that satisfies the Gibbs hypothesis, the Gibbs state \[\require{physics} \begin{align} \gamma_{H,\,\beta} \relax\frac{e^{-\beta H}}{\Tr e^{-\beta H}} \end{align}\] is well defined for all inverse temperatures \(\beta>0\); hence, it also has well-defined energy \[\require{physics} \begin{align} \Tr\qty[\gamma_{H,\,\beta}H]= E\, . \end{align}\] In fact, it is most common to use the above equation to re-parametrize \(\beta\) as a function of \(E\). In this way, we obtain the function \(\beta_H:[0,\infty) \to (0,\infty]\), with \(\beta_H(0) = \infty\) and \(\lim_{E\to\infty} \beta_H(E) = 0\). A key role in the theory is played by the function \[\begin{align} F_H(E) \relax S\big(\gamma_{H,\,\beta_H(E)}\big)\, , \label{FH} \end{align}\tag{49}\] which records the entropy of the Gibbs state of a given energy. If \(H\) is grounded and satisfies the Gibbs hypothesis, \(F_H\) is well-defined for all \(E\geq 0\) and enjoys a wealth of nice properties, some of which have been proved in Ref. [76]. We are only going to need the following.

We will now establish a semi-continuity bound on the entanglement of formation. This is needed to relate the entanglement of formation of the target state and that of the output state of the entanglement dilution protocol, namely, \(\rho^{\otimes n}\) and \(\rho_n \relax\pazocal{E}\big(\Phi^{\otimes \floor*{rn}}\big)\), respectively (cf. 12 ). We are free to study whatever target states we wish, so it makes perfect sense to impose a mild condition on \(\rho\) such as 20 . We cannot impose an energy constraint on \(\rho_n\): in fact, for the converse part, we need to analyze an arbitrary entanglement dilution protocol, which may inject an arbitrarily high amount of energy into the system to output \(\rho_n\). Due to this asymmetry on the constraints between \(\rho^{\otimes n}\) and \(\rho_n\), bounds with energy constraints on both \(\rho\) and \(\rho_n\) are inappropriate for our analysis. By contrast, the characteristic property of the following bound is that it is a one-side continuity bound featuring an energy constraint on \(\rho\) only: as such, it is suitable for our analysis.

We also have the following lemma to convert the constraint in terms of energy into that in terms of entropy. Since our condition on \(\rho_{AB}\) is given in terms of the entropy as 20 , the lemma below will be useful for converting our condition into the energy-constraint condition for the above asymptotic semi-continuity bound.

5.2 Monotonicity of the entanglement of formation under separable operations↩︎

Throughout this section, we extend the result of Ref. [49] on the monotonicity of the entanglement of formation under finite-dimensional separable operations to infinite-dimensional separable operations.

In the finite-dimensional case, the possible ensembles of bipartite pure states produced when separable operations act on a single bipartite pure state are characterized in Ref. [49]. We start by generalizing this finite-dimensional result to the present infinite-dimensional setting. The analysis in Ref. [49] relies on a function defined as the sum of \(N\) smallest eigenvalues of a positive semidefinite operator. In particular, for a \(D\)-dimensional positive semidefinite operator \(M\) with eigenvalues \(\lambda_0 \geq \lambda_1 \geq \cdots \geq \lambda_{D-1} \geq 0\) and spectral decomposition \(M=\sum_{n=0}^{D-1}\lambda_n\ket{n}\!\!\bra{n}\), Ref. [49] defines \[\require{physics} \begin{align} \label{eq:chi95finite} \chi_N\qty(M) &\relax\sum_{n=D-N}^{D-1}\lambda_{n}\, , \end{align}\tag{51}\] with the convention that \(\require{physics} \chi_0\qty(M)\relax 0\). The function \(\chi_N\) played a central role in the finite-dimensional analysis of Ref. [49], but, problematically, \(\chi_N\) by itself is not well defined in the infinite-dimensional case, for the simple reason that there might be no “smallest eigenvalue”. Instead, for any positive semidefinite trace-class operator with spectral decomposition \[\begin{align} T=\sum_{n=0}^{\infty}\lambda_n\ket{n}\!\!\bra{n}\, ,\quad \lambda_0\geq \lambda_1\geq \lambda_2 \geq\cdots \end{align}\] and any finite \(N>0\), we here use the \(N\) largest eigenvalues \(\lambda_0,\ldots,\lambda_{N-1}\) to introduce \[\require{physics} \begin{align} \label{eq:chi} \widetilde{\chi}_N\qty(T)&\relax\Tr T -\sum_{n=0}^{N-1}\lambda_n =\sum_{n=N}^{\infty}\lambda_n\, , \end{align}\tag{52}\] and \(\require{physics} \widetilde{\chi}_0\qty(T)\relax\Tr T\) for \(N=0\). For a \(D\)-dimensional positive semidefinite operator \(M\) with \(\lambda_0\geq\lambda_1\geq\cdots\geq\lambda_{D-1}\geq 0\), we can define \(\widetilde{\chi}_N(M)\) by taking formally \(\lambda_{D}=\lambda_{D+1}=\cdots=0\). With our function \(\widetilde{\chi}_N\) at hand, we are now ready to generalize of the key technical result [49] to our infinite-dimensional setting.

In our proof of Lemma 8, an infinite-dimensional version of the Schur–Horn Theorem [69] plays an essential role. In both the finite- and the infinite-dimensional setting, this latter result characterizes the relationship between the diagonal and the spectrum of positive semidefinite compact operators in terms of majorization.

Using Lemma 9, we can now prove Lemma 8.

Using the above lemma, we can generalize Ref. [49] so as to obtain the following result.

Using the above technical lemmas, we can characterize the probabilistic pure state transformations under infinite-dimensional countably separable operations in terms of a majorization condition, thereby generalizing the finite-dimensional result of Ref. [49].

To say something about the monotonicity of the entanglement of formation, we need to work with the entropy rather than with the function \(\widetilde{\chi}_N\) for representing majorization. To translate the relations given by Proposition 13 into statements involving entropy, we here develop a strategy that, to the best of our knowledge, has not been worked out in the literature so far. The linchpin of it is the following integral representation of the quantum entropy in finite- or infinite-dimensional systems, which we believe is of independent interest. It was inspired by recent work done in this direction — but in finite dimension — in Refs. [78]–[80].

Note that the function \(\mathbb{N}\ni N \mapsto \widetilde{\chi}_N(\rho) + N\mu\) admits a minimum for all values of \(\mu > 0\), simply because \(\lim_{N\to\infty} \left\{ \widetilde{\chi}_N(\rho) + N\mu \right\} = +\infty\). The proof of Lemma 11 is relegated to Appendix 8. We use the above integral formula to deduce the following corollary.

Combining the above result with Proposition 13, we obtain the following key statement. Note that if the target ensemble is composed of a single pure state only, then the deduction of the monotonicity of entanglement entropy from the majorization relation would be an immediate corollary of Ref. [81]. The main point, of course, is that we consider arbitrary pure-state ensembles as targets. We also note that infinite-dimensional bipartite pure state transformations under LOCC and stochastic LOCC (SLOCC) have been studied in Refs. [82]–[84], but the techniques found there are not directly applicable here since our focus is on a different family of operations, i.e., separable operations.

5.3 Proof of the converse part↩︎

We are finally ready to state and prove the converse part of our main result, Theorem 6.

6 Conclusion↩︎

Infinite-dimensional systems, whether natural or engineered, are common in physics. We commonly manipulate entanglement in such infinite-dimensional physical systems in experiments. However, most of the significant results on asymptotic entanglement transformation established in the history of entanglement theory are simply lost as soon as one transitions to infinite-dimensional systems, due to the unavailability of key mathematical techniques that the finite-dimensional analyses rely on.

Here we have solved the fundamental problem of characterizing an operational measure of entanglement, the entanglement cost [8]–[10], [18], [19], in the general infinite-dimensional setting. To wit, we have proved that the entanglement cost of any infinite-dimensional state \(\rho_{AB}\) with finite quantum entropy on either \(A\) or \(B\) is given by the regularized entanglement of formation. This extends the foundational finite-dimensional result of Refs. [18], [19] to the case of infinite-dimensional quantum systems. We believe that the restriction to states with finite local entropy is physically justified, as states with finite local energy always obey it, for any choice of Hamiltonian that satisfies the Gibbs hypothesis.

The techniques used in the finite-dimensional analysis in Ref. [18], such as strong typicality, monotonicity, and asymptotic continuity, are no longer directly applicable if we allow manipulation of infinite-dimensional systems. Nevertheless, we constructed an optimal protocol for entanglement dilution of infinite-dimensional mixed states by employing a known notion of typicality for countably infinite alphabets and a new “wasteful” protocol that deals specifically with the “rare” part of the state. Remarkably, the protocol is implementable with one-way LOCC and a finite amount of classical communication assisted by a finite number of ebits, yet it still approximately outputs copies of the infinite-dimensional target state.

We also proved the converse bound, which implies that this protocol is optimal even among all protocols that employ infinite-dimensional separable operations. To establish the converse, we have introduced techniques for proving the monotonicity of the entanglement of formation under countably separable operations, generalizing the results on finite-dimensional LOCC and SEP presented in Refs. [46]–[49]. Among these techniques there is a new integral representation for the quantum entropy in infinite-dimensional systems, which was inspired by recent works on integral representations of quantum \(f\)-divergences. The other key piece of techniques we have used is the recent semi-continuity bound of the entanglement of formation for infinite-dimensional states in Ref. [50], which is used in place of the conventional asymptotic continuity, appropriate only for finite-dimensional states [40]–[42].

A fundamental open question that we leave for future investigation is whether other operational measures of more general quantum resources on infinite-dimensional systems [44], [86]–[88] can also enjoy a characterization in terms of entropic quantities, in a similar way to what we did here. Also, from a more practical perspective, it would be interesting to consider infinite-dimensional multipartite entanglement manipulation on quantum networks [89]–[91].

Acknowledgments↩︎

L.L.thanks Maksim E.Shirokov and Mark M.Wilde for useful comments on the first version of this manuscript, and in particular the former for providing an insightful argument to complete the last part of the proof of Proposition 15. H.Y.acknowledges JST PRESTO Grant Number JPMJPR201A, JPMJPR23FC, JSPS KAKENHI Grant Number JP23K19970, and MEXT Quantum Leap Flagship Program (MEXT QLEAP) JPMXS0118069605, JPMXS0120351339. K.K.was supported by a Mike and Ophelia Lazaridis Fellowship, a Funai Overseas Scholarship, and a Perimeter Residency Doctoral Award. P.H.acknowledges support from ARO (award W911NF2120214), DOE (Q-NEXT), CIFAR and the Simons Foundation. LL acknowledges financial support from the European Union under the European Research Council (ERC Grant Agreement No. 101165230) and from MIUR (Ministero dell’Istruzione, dell’Università e della Ricerca) through the project ‘Dipartimenti di Eccellenza 2023–2027’ of the ‘Classe di Scienze’ department at the Scuola Normale Superiore.

Part of this work was carried out during the workshop “Quantum resources: from mathematical foundations to operational characterisation” held in Singapore in December 2022. Other key discussions happened at the conference “It from Qubit”, held at the Perimeter Institute for Theoretical Physics in July–August 2023.

Data availability statement. No data sets were generated or analyzed during this study.

Competing interests. The authors declare no competing interests.

7 Proof of lemma on Hamiltonian associated with a state↩︎

In this appendix, we present a simplified and fully self-contained proof of the very handy Lemma 7, derived by us before we became aware of Ref. [76], which features a slightly more general statement.

We are now ready to provide a full proof of Lemma 7.

8 Proof of the integral representation of the entropy↩︎

We give a proof of Lemma 11 on the integral representation of the entropy.

Let \(\rho\in \mathcal{D}(\mathcal{H})\) be a quantum state over a (possibly infinite-dimensional) separable Hilbert space \(\mathcal{H}\). Then, its quantum entropy \(\require{physics} S(\rho) = -\Tr \rho \log_2\rho\) defined by 4 , be it finite or infinite, admits the integral representation \[\require{physics} \begin{align} S(\rho) = \frac{1}{\ln 2} \left(\int_0^1 \frac{\dd \mu}{\mu}\, \min_{N\in \mathbb{N}} \big\{ \widetilde{\chi}_N(\rho) + N\mu\big\} - 1\right) , \label{integral95representation95entropy95appendix} \end{align}\tag{71}\] where \(N\) runs on the non-negative integers, and \(\widetilde{\chi}_N\) is defined by 52 .

References↩︎