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We introduce an axiomatic approach for channel divergences and channel relative entropies that is based on three information-theoretic axioms of monotonicity under superchannels, i.e., generalized data processing inequality, additivity under tensor products, and normalization, similar to the approach given for the state domain in Gour and Tomamiche...
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Citations
... We add that the optimal upper bound q of {N (|x⟩⟨x|)} x∈[m] can be computed directly from Lemma 26 as follows: where κ is an integer 1 ⩽ κ ⩽ k, and the inequality follows from Lemma 33. Following the same line of reasoning as the proof of Theorem 4 in the supplemental material section of [29] shows that ...
Quantum channels represent a broad spectrum of operations crucial to quantum information theory, encompassing everything from the transmission of quantum information to the manipulation of various resources. In the domain of states, the concept of majorization serves as a fundamental tool for comparing the uncertainty inherent in both classical and quantum systems. This paper establishes a rigorous framework for assessing the uncertainty in both classical and quantum channels. By employing a specific class of superchannels, we introduce and elucidate three distinct approaches to channel majorization: constructive, axiomatic, and operational. Intriguingly, these methodologies converge to a consistent ordering. This convergence not only provides a robust basis for defining entropy functions for channels but also clarifies the interpretation of entropy in this broader context. Most notably, our findings reveal that any viable entropy function for quantum channels must assume negative values, thereby challenging traditional notions of entropy.
... In some situations it is important to consider pairs of quantum states and evaluate how different they are from each other. To this end, various quantifiers have been defined, such as the trace distance, the fidelity [38] or quantum divergences [17,14]. These quantifiers all show that, whenever the same channel is applied to each element of a pair of quantum states, in general our ability to distinguish the resulting states is decreased. ...
... A resource theory of quantum distinguishability is given as follows [17,14]. Resources : ((ρ, σ ), H) are pairs of quantum states, that is, ρ, σ ∈ L(H) where H is a finite-dimensional Hilbert space. ...
... operations , non-Markovian quantum process [25][26][27][28][29][30][31][32][33][34][35][36][37], and dynamical resource theory [38][39][40][41][42][43][44][45][46][47][48][49][50][51][52]. They can also be interpreted as quantum functional programming [53,54]. ...
Isometry operations encode the quantum information of the input system to a larger output system, while the corresponding decoding operation would be an inverse operation of the encoding isometry operation. Given an encoding operation as a black box from a d -dimensional system to a D -dimensional system, we propose a universal protocol for isometry inversion that constructs a decoder from multiple calls of the encoding operation. This is a probabilistic but exact protocol whose success probability is independent of D . For a qubit ( d = 2 ) encoded in n qubits, our protocol achieves an exponential improvement over any tomography-based or unitary-embedding method, which cannot avoid D -dependence. We present a quantum operation that converts multiple parallel calls of any given isometry operation to random parallelized unitary operations, each of dimension d . Applied to our setup, it universally compresses the encoded quantum information to a D -independent space, while keeping the initial quantum information intact. This compressing operation is combined with a unitary inversion protocol to complete the isometry inversion. We also discover a fundamental difference between our isometry inversion protocol and the known unitary inversion protocols by analyzing isometry complex conjugation and isometry transposition. General protocols including indefinite causal order are searched using semidefinite programming for any improvement in the success probability over the parallel protocols. We find a sequential "success-or-draw" protocol of universal isometry inversion for d = 2 and D = 3 , thus whose success probability exponentially improves over parallel protocols in the number of calls of the input isometry operation for the said case.
... In some situations it is important to consider pairs of quantum states and evaluate how different they are from each other. To this end, various quantifiers have been defined, such as the trace distance, the fidelity [37] or quantum divergences [16,13]. These quantifiers all show that, whenever the same channel is applied to each element of a pair of quantum states, in general our ability to distinguish the resulting states is decreased. ...
... A resource theory of quantum distinguishability is given as follows [16,13]. Resources : ((ρ, σ ), H) are pairs of quantum states, that is, ρ, σ ∈ L(H) where H is a finite-dimensional Hilbert space. ...
In this paper we generalize the framework proposed by Gour and Tomamichel regarding extensions of monotones for resource theories. A monotone for a resource theory assigns a real number to each resource in the theory signifying the utility or the value of the resource. Gour and Tomamichel studied the problem of extending monotones using set-theoretical framework when a resource theory embeds fully and faithfully into the larger theory. One can generalize the problem of computing monotone extensions to scenarios when there exists a functorial transformation of one resource theory to another instead of just a full and faithful inclusion. In this article, we show that (point-wise) Kan extensions provide a precise categorical framework to describe and compute such extensions of monotones. To set up monontone extensions using Kan extensions, we introduce partitioned categories (pCat) as a framework for resource theories and pCat functors to formalize relationship between resource theories. We describe monotones as pCat functors into $([0,\infty], \leq)$, and describe extending monotones along any pCat functor using Kan extensions. We show how our framework works by applying it to extend entanglement monotones for bipartite pure states to bipartite mixed states, to extend classical divergences to the quantum setting, and to extend a non-uniformity monotone from classical probabilistic theory to quantum theory.
... where D(· ·) is the quantum relative entropy [67,78,79] defined as ...
Quantum measurement is a class of quantum channels that sends quantum states to classical states. We set up resource theories of quantum coherence and quantum entanglement for quantum measurements and find relations between them. For this, we conceive a relative entropy type quantity to account for the quantum resources of quantum measurements. The quantum coherence of a quantum measurement can be converted into the entanglement in a bipartite quantum measurement through coherence-non generating transformations. Conversely, a quantum entanglement monotone of quantum measurements induces a quantum coherence monotone of quantum measurement. Our results confirm that the intuition on the link between quantum coherence and quantum entanglement is valid even for quantum measurements which do not generate any quantum resource.
... Universal transformations of quantum states have played an essential role in the fundamental understanding of quantum information theory and its applications [1]. Recently, higher-order quantum transformations, namely, universal transformations of quantum operations given as black boxes, have been studied in the contexts of processing unitary operations , non-Markovian quantum process [24][25][26][27][28][29][30][31][32][33][34][35][36], and dynamical resource theory [37][38][39][40][41][42][43][44][45][46][47][48]. They can also be interpreted as quantum functional programming [49,50]. ...
This work considers transformations of encoding operations of quantum information that are given as isometry operations. We propose a universal protocol for isometry inversion that constructs a decoder from multiple calls of an encoding black box transforming a $d$-dimensional system to a $D$-dimensional system. This protocol is probabilistic but exact, and uses the black boxes in parallel. Our protocol can significantly outperform other protocols based on embedding isometry inversion in $D$-dimensional unitary inversion or quantum process tomography. This is because the success probability of our protocol is independent of $D$. The advantage of the performance of this protocol can be understood as being originated in the impossibility of exact isometry complex conjugation for $D \geq 2d$. This means that isometry inversion cannot be achieved by concatenating isometry complex conjugation and isometry transposition. We also perform semidefinite programming to obtain the optimal success probability of isometry inversion, isometry complex conjugation, and isometry transposition in parallel, sequential, and general protocols including indefinite causal order to investigate the performance improvement beyond the parallel protocols. We numerically find a sequential ``success-or-draw'' protocol for isometry inversion for $d=2$ and $D=3$. This implies that there exists a sequential protocol whose success probability is exponentially improved over parallel protocols in the number of calls of the input isometry operation.
... |t k | max Proof. We first refer to previous literature on extensions of channel divergences [24,25]. From this, we define the minimal extension D and maximal extension D of any channel divergence D which reduces to the Kullback-Leibler divergence D on states as: where H S is the Shannon entropy. ...
Given entropy's central role in multiple areas of physics and science, one important task is to develop a systematic and unifying approach to defining entropy. Games of chance become a natural candidate for characterising the uncertainty of a physical system, as a system's performance in gambling games depends solely on the uncertainty of its output. In this work, we construct families of games which induce pre-orders corresponding to majorization, conditional majorization, and channel majorization. Finally, we provide operational interpretations for all pre-orders, show the relevance of these results to dynamical resource theories, and find the only asymptotically continuous classical dynamic entropy.
Quantum measurement is a class of quantum channels that sends quantum states to classical states. We set up resource theories of quantum coherence and quantum entanglement for quantum measurements and find relations between them. For this, we conceive a relative entropy-type quantity to account for the quantum resources of quantum measurements. The quantum coherence of a quantum measurement can be converted into the entanglement in a bipartite quantum measurement through coherence nongenerating transformations. Conversely, a quantum entanglement monotone of quantum measurements induces a quantum coherence monotone of quantum measurements. Our results confirm that the understanding on the link between quantum coherence and quantum entanglement is valid even for quantum measurements which do not generate any quantum resource.
Quantifying coherence has received significant attention in recent years. Many research is focus on the resource theory of static coherence. In this paper, we introduce the coherence weight of quantum channel to study the quantum resource theory of dynamical coherence. The free channel is classical channel and the free super-channel are the maximally incoherent superchannels (MISC) and incoherent superchannels (ISC). We first study some basic properties (such as convexity and monotonicity) of coherence weight of quantum channel. By introducing entanglement-assisted super-subchannel exclusion task, we show that the advantage of coherent channels in such task. Finally, we compute the coherence weight of quantum channel for some channels.
The crucial role of channels in physics and information theory motivates the task of characterizing the entropy, or uncertainty, of a channel. Games of chance become a natural candidate for this task, as a system's performance in a gambling game depends solely on the uncertainty of its output. In this work, we construct families of games that induce preorders corresponding to majorization, conditional majorization, and channel majorization. Finally, we provide operational interpretations for all preorders, show the relevance of these results to dynamical resource theories, and find the only asymptotically continuous classical channel entropy.
