Keiji Matsumoto, Fumitaka Yura
We study the entanglement cost of the states in the contragredient 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 $\C^d \otimes (\C^d)^{d-2}$. Combined with the arguments in \cite{Matsumoto02}, additivity of channel capacity of some quantum channels is also shown.
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Quantum entanglement is the quantum information processing resource. Thus it is of importance to understand how much of entanglement particular quantum states have, and what kinds of laws entanglement and also transformation between entanglement states subject to. Therefore, it is essentialy important to use proper measures of entanglement which have nice properties. One of the major candidates of such measures is "entanglement of formation", and whether this measurement is additive or not is an important open problem. We aim at certain states so-called "antisymmetric states" for which the add…
We study the additivity property of three multipartite entanglement measures, i.e. the geometric measure of entanglement (GM), the relative entropy of entanglement and the logarithmic global robustness. First, we show the additivity of GM of multipartite states with real and non-negative entries in the computational basis. Many states of experimental and theoretical interests have this property, e.g. Bell diagonal states, maximally correlated generalized Bell diagonal states, generalized Dicke states, the Smolin state, and the generalization of Dür's multipartite bound entangled states. We als…
We investigate multiparty communication scenarios where information is sent from several sender to several receivers. We establish a relation between the quantum capacity of multiparty communication channels and their distillability properties which enables us to show that the quantum capacity of such channels is not additive.
Super-additivity of quantum capacity in simple channels
本文研究了“generalized platypus channel”的信道容量，证明其私有和经典容量均为1，并证实其与qudit erasure channels及multilevel amplitude damping channels结合时仍具量子容量超加性。
We show that for the tensor product of an entanglement-breaking quantum channel with an arbitrary quantum channel, both the minimum entropy of an output of the channel and the Holevo-Schumacher-Westmoreland capacity are additive. In addition, for the tensor product of two arbitrary quantum channels, we give a bound involving entanglement of formation for the amount of subadditivity (for minimum entropy output) or superadditivity (for channel capacity) that can occur.
The purpose of these notes is to discuss the relation between the additivity questions regarding the quantities (Holevo) capacity of a quantum channel T and entanglement of formation of a given bipartite state. In particular, using the Stinespring dilation theorem, we give a formula for the channel capacity involving entanglement of formation. This can be used to show that additivity of the latter for some states can be inferred from the additivity of capacity for certain channels. We demonstrate this connection for a family of group--covariant channels, allowing us to calculate the entangleme…
An example is given of a qubit quantum channel which requires four inputs to maximize the Holevo capacity. The example is one of a family of channels which are related to 3-state channels. The capacity of the product channel is studied and numerical evidence presented which strongly suggests additivity. The numerical evidence also supports a conjecture about the concavity of output entropy as a function of entanglement parameters. However, an example is presented which shows that for some channels this conjecture does not hold for all input states. A numerical algorithm for finding the capacit…
The additivity of both the entanglement of formation and the classical channel capacity is known to be a consequence of the strong superadditivity conjecture. We show that, conversely, the strong superadditivity conjecture follows from the additivity of the entanglement of formation; this means that the two conjectures are equivalent and that the additivity of the classical channel capacity is a consequence of them.
Additivity of the Holevo capacity is proved for product channels, under the condition that one of the channels is in a certain class of unital qubit channels, with the other completely arbitrary. This qubit class includes the depolarizing channel. As a byproduct this proves that the Holevo bound is the ultimate information capacity of such qubit channels (assuming no prior entanglement between sender and receiver). Additivity of minimal entropy and multiplicativity of p-norms are also proved under the same assumptions.
An elementary introduction into algebraic approach to unified quantum information theory and operational approach to quantum entanglement as generalized encoding is given. After introducing compound quantum state and two types of informational divergences, namely, Araki-Umegaki (a-type) and of Belavkin-Staszewski (b-type) quantum relative entropic information, this paper treats two types of quantum mutual information via entanglement and defines two types of corresponding quantum channel capacities as the supremum via the generalized encodings. It proves the additivity property of quantum chan…
We consider two apparently separated problems: in the first part of the paper we study the concept of a scalable (approximate) programmable quantum gate (SPQG). These are special (approximate) programmable quantum gates, with nice properties that could have implications on the theory of universal computation. Unfortunately, as we prove, such objects do not exist in the domain of usual quantum theory. In the second part the problem of noisy dense coding (and generalizations) is addressed. We observe that the additivity problem for the classical capacity obtained is of apparently greater general…
Sufficient conditions for additivity of the zero-error classical capacity of quantum channels
研究量子信道零误差经典容量可加性，通过建立独立数乘法性的充分条件及研究非交换图块形式，给出满足条件的量子信道例子，推导相关条件，为零误差量子通信基本限制提供新见解。
We consider the additivity of the minimal output entropy and the classical information capacity of a class of quantum channels. For this class of channels the norm of the output is maximized for the output being a normalized projection. We prove the additivity of the minimal output Renyi entropies with entropic parameters contained in [0, 2], generalizing an argument by Alicki and Fannes, and present a number of examples in detail. In order to relate these results to the classical information capacity, we introduce a weak form of covariance of a channel. We then identify several instances of w…
We prove additivity of the minimum output entropy and the Holevo capacity for rotationally invariant quantum channels acting on spin-1/2 and spin-1 systems. The physical significance of these channels and their relations to other known channels is also discussed.
Recently Shor proved equivalence of several open (sub)additivity problems related to the Holevo capacity and the entanglement of formation [15]. In our previous note [6] equivalence of these to the additivity of the Holevo capacity for channels with arbitrary linear constraints was shown. This note is the development of the previous one in the direction of channels with general constraints. Introducing input constraints provides greater flexibility in the treatment of the additivity conjecture. The Holevo capacity of arbitrarily constrained channel is considered and the characteristic property…
We present a constructive example of violation of additivity of minimum output Rényi entropy for each p>2. The example is provided by antisymmetric subspace of a suitable dimension. We discuss possibility of extension of the result to go beyond p>2 and obtain additivity for p=0 for a class of entanglement breaking channels.
We prove the strong super-additivity of the entanglement of formation for stabilizer pure states, and the set of mixed states which minimize their average entropy of entanglement as a mixture of stabilizer pure states sharing the same stabilizer group up to phases. The implications of the result on the additivity of the Holevo capacity of a quantum channel transmitting stabilizer states with Pauli noise is discussed.
The design of error-correcting codes used in modern communications relies on information theory to quantify the capacity of a noisy channel to send information [1]. This capacity can be expressed using the mutual information between input and output for a single use of the channel: although correlations between subsequent input bits are used to correct errors, they cannot increase the capacity. For quantum channels, it has been an open question whether entangled input states can increase the capacity to send classical information [2]. The additivity conjecture [3,4] states that entanglement do…
It is known that the additivity conjecture of Holevo capacity, output minimum entoropy, and the entanglement of formation (EoF), are equivalent with each other. Among them, the output minimum entropy is simplest, and hence many researchers are focusing on this quantity. Here, we suggest yet another entanglement measure, whose strong superadditivity and additivity are equivalent to the additivity of the quantities mentioned above. This quantity is as simple as the output minimum entropy, and in existing proofs of additivity conjecture of the output minimum entropy for the specific examples, the…
We study the additivity problems for the classical capacity of quantum channels, the minimal output entropy and its convex closure. We show for each of them that additivity for arbitrary pairs of channels holds iff it holds for arbitrary equal pairs, which in turn can be taken to be unital. In a similar sense, weak additivity is shown to imply strong additivity for any convex entanglement monotone. The implications are obtained by considering direct sums of channels (or states) for which we show how to obtain several information theoretic quantities from their values on the summands. This prov…
A conjecture arising naturally in the investigation of additivity of classical information capacity of quantum channels states that the maximal purity of outputs from a quantum channel, as measured by the p-norm, should be multiplicative with respect to the tensor product of channels. We disprove this conjecture for p>4.79. The same example (with p=infinity) also disproves a conjecture for the multiplicativity of the injective norm of Hilbert space tensor products.
The notion of the Holevo capacity for arbitrarily constrained infinite dimensional quantum channels is introduced. It is shown that despite nonexistence of an optimal ensemble in this case it is possible to define the notion of the output optimal average state for such a channel. The characterization of the output optimal average state and a "minimax" expression for the Holevo capacity are obtained. This makes it possible to prove equivalence of several additivity properties for infinite dimensional quantum channels. The notion of the $χ$-function for an infinite dimensional channel is conside…
The information carrying capacity of the d-dimensional depolarizing channel is computed. It is shown that this capacity can be achieved by encoding messages as products of pure states belonging to an orthonormal basis of the state space, and using measurements which are products of projections onto this same orthonormal basis. In other words, neither entangled signal states nor entangled measurements give any advantage for information capacity. The result follows from an additivity theorem for the capacity of the product of the depolarizing channel with an arbitrary channel. We establish the A…
The capacity of quantum channel with product input states was formulated by the quantum coding theorem. However, whether entangled input states can enhance the quantum channel is still open. It turns out that this problem is reduced to aspecial case of the more general problem whether the capacity of product quantum channel exhibits additivity. In the present study, we apply one of the quantum Arimoto-Blahut type algorithms to the latter problem. The results suggest that the additivity of product quantum channel capacity always holds and that entangled input states cannot enhance the quantum c…
We prove multiplicativity of maximal output $p$ norm of classical noise channels and thermal noise channels of arbitrary modes for all $p>1$ under the assumption that the input signal states are Gaussian states. As a direct consequence, we also show the additivity of the minimal output entropy and that of the energy-constrained Holevo capacity for those Gaussian channels under Gaussian inputs. To the best of our knowledge, newly discovered majorization relation on symplectic eigenvalues, which is also of independent interest, plays a central role in the proof.
Quantum information theory determines the maximum rates at which information can be transmitted through physical systems described by quantum mechanics. Here we consider the communication protocol known as quantum reading. Quantum reading is a protocol for retrieving the information stored in a digital memory by using a quantum probe, e.g., shining quantum states of light to read an optical memory. In a variety of situations using a quantum probe enhances the performances of the reading protocol in terms of fidelity, data density and energy dissipation. Here we review and characterize the quan…
The property of the optimal signal ensembles of entanglement assisted channel capacity is studied. A relationship between entanglement assisted channel capacity and one-shot capacity of unassisted channel is obtained. The data processing inequalities, convexity and additivity of the entanglement assisted channel capacity are reformulated by simple methods.
When can noiseless quantum information be sent across noisy quantum devices? And at what maximum rate? These questions lie at the heart of quantum technology, but remain unanswered because of non-additivity -- a fundamental synergy which allows quantum devices (aka quantum channels) to send more information than expected. Previously, non-additivity was known to occur in very noisy channels with coherent information much smaller than that of a perfect channel; but, our work shows non-additivity in a simple low-noise channel. Our results extend even further. We prove a general theorem concerning…
New constructive counterexamples to additivity of minimum output Rényi p-entropy of quantum channels
该研究受Grudka等人相关成果启发，构造了几类新的量子信道反例，证明其最小输出Rényi p熵在p>2时不满足可加性，这类信道具有反对称子空间的扩张与子空间特征。
The additivity problem asks if the use of entanglement can boost the information-carrying capacity of a given channel beyond what is achievable by coding with simple product states only. This has recently been shown not to be the case for phase-insensitive one-mode Gaussian channels, but remains unresolved in general. Here we consider two general classes of bosonic noise channels, which include phase-insensitive Gaussian channels as special cases: these are beamsplitters with general, potentially non-Gaussian environment states and classical noise channels with general probabilistic noise. We …
Machine learning techniques are increasingly being used in fundamental research to solve various challenging problems. Here we explore one such technique to address an important problem in quantum communication scenario. While transferring quantum information through a noisy quantum channel, the utility of the channel is characterized by its quantum capacity. Quantum channels, however, display an intriguing property called super-additivity of coherent information. This makes the calculation of quantum capacity a hard computational problem involving optimization over an exponentially increasing…
Generic nonadditivity of quantum capacity in simple channels
该研究针对量子信道超加性示例稀缺的问题，提出platypus信道族，发现其与不同信道联合时的超加性效应，验证了超加性在大容量信道间的普遍性，拓展了量子信息理论认知。
We study quantum channels that are close to another channel with weakly additive Holevo information and derive upper bounds on their classical capacity. Examples of channels with weakly additive Holevo information are entanglement-breaking channels, unital qubit channels, and Hadamard channels. Related to the method of approximate degradability, we define approximation parameters for each class above that measure how close an arbitrary channel is to satisfying the respective property. This gives us upper bounds on the classical capacity in terms of functions of the approximation parameters, as…
Communication rates over quantum channels can be boosted by entanglement, via superadditivity phenomena or entanglement assistance. Superadditivity refers to the capacity improvement from entangling inputs across multiple channel uses. Nevertheless, when unlimited entanglement assistance is available, the entanglement between channel uses becomes unnecessary -- the entanglement-assisted (EA) capacity of a single-sender and single-receiver channel is additive. We generalize the additivity of EA capacity to general multiple-access channels (MACs) for the total communication rate. Furthermore, fo…
We study the power of quantum channels with little or no capacity for private communication. Because privacy is a necessary condition for quantum communication, one might expect that such channels would be of little use for transmitting quantum states. Nevertheless, we find strong evidence that there are pairs of such channels that, when used together, can transmit far more quantum information than the sum of their individual private capacities. Because quantum transmissions are necessarily private, this would imply a large violation of additivity for the private capacity. Specifically, we pre…
In this article, we investigate the additivity phenomenon in the dynamic capacity of a quantum channel for trading classical communication, quantum communication and entanglement. Understanding such additivity property is important if we want to optimally use a quantum channel for general communication purpose. However, in a lot of cases, the channel one will be using only has an additive single or double resource capacity, and it is largely unknown if this could lead to an superadditive double or triple resource capacity. For example, if a channel has an additive classical and quantum capacit…
A class of problems in quantum information theory, having an elementary formulation but still resisting solution, concerns the additivity properties of various quantities characterizing quantum channels, notably the "classical capacity", and the "maximal output purity". All known results, including extensive numerical work, are consistent with the conjecture that these quantities are indeed additive (resp. multiplicative) with respect to tensor products of channels. A proof of this conjecture would have important consequences in quantum information theory. In particular, according to this conj…
We give a direct proof of the additivity of the minimum output entropy of a particular quantum channel which breaks the multiplicativity conjecture. This yields additivity of the classical capacity of this channel, a result obtained by a different method by Matsumoto and Yura [quant-ph/0306009]. Our proof relies heavily upon certain concavity properties of the output entropy which are of independent interest.
Due to Csiszar and Koerner, the private capacity of classical wiretap channels has a single-letter characterization in terms of the private information. For quantum wiretap channels, however, it is known that regularization of the private information is necessary to reach the capacity. Here, we study hybrid classical-quantum wiretap channels in order to resolve to what extent quantum effects are needed to witness non-additivity phenomena in quantum Shannon theory. For wiretap channels with quantum inputs but classical outputs, we prove that the characterization of the capacity in terms of the …
We reduce the number of open additivity problems in quantum information theory by showing that four of them are equivalent. We show that the conjectures of additivity of the minimum output entropy of a quantum channel, additivity of the Holevo expression for the classical capacity of a quantum channel, additivity of the entanglement of formation, and strong superadditivity of the entanglement of formation, are either all true or all false.
Additivity of quantum capacities in simple non-degradable quantum channels
本文提出两类非退化量子信道，其相干信息保持可加性，从而简化量子容量计算，并分析结构约束对强、弱可加性的影响。
We study subadditivity of the minimum output entropy ($H_{min}$) of quantum multiple access channels (MACs). We provide an example of violation of the additivity theorem for $H_{min}$ known in classical information theory. Our result is based on a fundamental property of $MACs$ i.e. independence of each sender. The channels used in the example can be constructed explicitly. On the basis of subadditivity of $H_{min}$ we also provide an example of extremal superadditivity (super activation) of the classical capacity region of MACs.
We study classical capacities of quantum multi-access channels in geometric terms revealing breaking of additivity of Holevo-like capacity. This effect is purely quantum since, as one points out, any classical multi-access channels have their regions additive. The observed non-additivity in quantum version presented here seems to be the first effect of this type with no additional resources like side classical or quantum information (or entanglement) involved. The simplicity of quantum channels involved resembles butterfly effect in case of classical channel with two senders and two receivers.
Entanglement indcued non--additivity of classical communication capacity in networks consisting of quantum channels is considered. Communication lattices consisiting of butterfly-type entanglement breaking channels augmented, with some probability, by identity channels are analyzed. The capacity superadditivity in the network is manifested in directed correlated bond percolation which we consider in two flavours: simply directed and randomly oriented. The obtained percolation properties show that high capacity information transfer sets in much faster in the regime of superadditive communicatio…
We introduce an infinite sequence of quantum channels for which the Holevo capacity is additive. The channel series is closely related to the quantum channels arising from universal quantum cloning machines. The additivity proof is motivated by a special property the studied channels enjoy: the property of conjugate degradability. As a consequence of the announced proof, we also provide an easy way of proving the additivity of the Holevo capacity for the original Unruh channel for which the quantum capacity is already known. Consequently, we present not only an infinite series of finite-dimens…
Counterexamples to additivity of minimum output $p$-Rényi entropy of quantum channels for all $p\ge 0$
研究量子信道最小输出 \(p -\) 雷尼熵可加性问题，通过结合两种随机投影构造，证明对于 \(p > 3/4\) 或 \(0\leq p < 1/4\) 存在反例，缩小未解决区间并改进输出维度阈值。
Allowing leakage can increase quantum transmission
研究发现允许量子信息泄漏到环境可反直觉地提升信道容量，源于双字母级超可加性，并可能提高量子密钥分发速率。
Information theory establishes the fundamental limits on data transmission, storage, and processing. Quantum information theory unites information theoretic ideas with an accurate quantum-mechanical description of reality to give a more accurate and complete theory with new and more powerful possibilities for information processing. The goal of both classical and quantum information theory is to quantify the optimal rates of interconversion of different resources. These rates are usually characterized in terms of entropies. However, nonadditivity of many entropic formulas often makes finding a…
We prove the equivalence of an arbitrary single-mode Gaussian quantum channel and a newly defined fiducial channel preceded by a phase shift and followed by a Gaussian unitary operation. This equivalence implies that the energy-constrained classical capacity of any single-mode Gaussian channel can be calculated based on this fiducial channel, which is furthermore simply realizable with a beam splitter, two identical single-mode squeezers, and a two-mode squeezer. In a large domain of parameters, we also provide an analytical expression for the Gaussian classical capacity, exploiting its additi…
The additivity of minimal output entropy is proved for the Weyl channel obtained by the deformation of a q-c Weyl channel. The classical capacity of channel is calculated.
We prove additivity violation of minimum output entropy of quantum channels by straightforward application of ε-net argument and Lévy's lemma. The additivity conjecture was disproved initially by Hastings. Later, a proof via asymptotic geometric analysis was presented by Aubrun, Szarek and Werner, which uses Dudley's bound on Gaussian process (or Dvoretzky's theorem with Schechtman's improvement). In this paper, we develop another proof along Dvoretzky's theorem in Milman's view showing additivity violation in broader regimes than the existing proofs. Importantly, Dvoretzky's theorem works wel…
From Permutation Symmetry to Communication Bounds and Additivity
本文证明完全置换不变性限制量子信道通信优势，使渐近优化简化为单次使用，并建立强逆定理与输出熵界，揭示可加性结果的统一结构。
It is conjectured that the Holevo capacity of a product channel Ω\otimes Φis achieved when product states are used as input. Amosov, Holevo and Werner have also conjectured that the maximal p-norm of a product channel is achieved with product input states. In this paper we establish both of these conjectures in the case that Ωis arbitrary and Φis a CQ or QC channel (as defined by Holevo). We also establish the Amosov, Holevo and Werner conjecture when Ωis arbitrary and either Φis a qubit channel and p=2, or Φis a unital qubit channel and p is integer. Our proofs involve a new conjecture for th…
Recently there has been considerable activity on the subject of additivity of various quantum channel capacities. Here, we construct a family of channels with sharply bounded classical, hence private capacity. On the other hand, their quantum capacity when combined with a zero private (and zero quantum) capacity erasure channel, becomes larger than the previous classical capacity. As a consequence, we can conclude for the first time that the classical private capacity is non-additive. In fact, in our construction even the quantum capacity of the tensor product of two channels can be greater th…
Additivity of quantum relative entropies as a single-copy criterion
研究量子信息中正则化是否必要的问题，提出单副本优化器满足特定性质时Umegaki相对熵等量可加，适用于假设检验和量子资源理论。
The additivity of the minimal output entropy and that of the $χ$-capacity are known to be equivalent for finite-dimensional irreducibly covariant channels. In this paper we formulate a list of conditions allowing to establish similar equivalence for infinite-dimensional covariant channels with constrained input. This is then applied to Bosonic Gaussian channels with quadratic input constraint to extend the classical capacity results of the recent paper \cite{ghg} to the case where the complex structures associated with the channel and with the constraint operator need not commute. In particula…
We develop an approximation approach to infinite dimensional quantum channels based on detailed investigation of the continuity properties of entropic characteristics of quantum channels and operations (trace-nonincreasing completely positive maps) as functions of a pair ``channel, input state''. The obtained results are then applied to the following problems: continuity of the $χ$-capacity as function of a channel; strong additivity of the $χ$-capacity for infinite dimensional channels; the analytical expression for the convex closure of the output entropy of arbitrary quantum channel.
We consider generalizations of depolarizing channels to maps in which the identity channel is replaced by a convex combinations of unitary conjugations. We show that one can construct unital channels of this type for which the input which achieves maximal output purity is unique. We give conditions under which multiplicativity of the maximal p-norm and additivity of the minimal output entropy. We also show that the Holevo capacity need not equal log d - the minimal entropy as one might expect for a convex combination of unitary conjugations. Conversely, we give examples for which this conditio…
We make a number of simplifications in Gour and Friedland's proof of local additivity of minimum output entropy of a quantum channel. We follow them in reframing the question as one about entanglement entropy of bipartite states associated with a $d_B \times d_E $ matrix. We use a different approach to reduce the general case to that of a square positive definite matrix. We use the integral representation of the log to obtain expressions for the first and second derivatives of the entropy, and then exploit the modular operator and functional calculus to streamline the proof following their und…
We give a capacity formula for the classical information transmission over a noisy quantum channel, with separable encoding by the sender and limited resources provided by the receiver's pre-shared ancilla. Instead of a pure state, we consider the signal-ancilla pair in a mixed state, purified by a "witness". Thus, the signal-witness correlation limits the resource available from the signal-ancilla correlation. Our formula characterizes the utility of different forms of resources, including noisy or limited entanglement assistance, for classical communication. With separable encoding, the send…
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