The superposition principle lies at the heart of many nonclassical properties of quantum mechanics. Motivated by this, we introduce a rigorous resource theory framework for the quantification of superposition of a finite number of linear independent states. This theory is a generalization of resource theories of coherence. We determine the general structure of operations which do not create superposition, find a fundamental connection to unambiguous state discrimination, and propose several quantitative superposition measures. Using this theory, we show that trace decreasing operations can be completed for free which, when specialized to the theory of coherence, resolves an outstanding open question and is used to address the free probabilistic transformation between pure states. Finally, we prove that linearly independent superposition is a necessary and sufficient condition for the faithful creation of entanglement in discrete settings, establishing a strong structural connection between our theory of superposition and entanglement theory.
The superposition principle lies at the heart of many nonclassical properties of quantum mechanics. Motivated by this, we introduce a rigorous resource theory framework for the quantification of superposition of a finite number of linear independent states. This theory is a generalization of resource theories of coherence. We determine the general structure of operations which do not create superposition, find a fundamental connection to unambiguous state discrimination, and propose several quantitative superposition measures. Using this theory, we show that trace decreasing operations can be completed for free which, when specialized to the theory of coherence, resolves an outstanding open question and is used to address the free probabilistic transformation between pure states. Finally, we prove that linearly independent superposition is a necessary and sufficient condition for the faithful creation of entanglement in discrete settings, establishing a strong structural connection between our theory of superposition and entanglement theory.
A combination of a finite number of linear independent states forms superposition in a way that cannot be conceived classically. Here, using the tools of resource theory of superposition, we give the conditions for a class of superposition state transformations. These conditions strictly depend on the scalar products of the basis states and reduce to the well-known majorization condition for quantum coherence in the limit of orthonormal basis. To further superposition-free transformations of d-dimensional systems, we provide superposition-free operators for a deterministic transformation of superposition states. The linear independence of a finite number of basis states requires a relation between the scalar products of these states. With this information in hand, we determine the maximal superposition states which are valid over a certain range of scalar products. Notably, we show that, for $d\ensuremath{\ge}3$, scalar products of the pure superposition-free states have a greater place in seeking maximally resourceful states. Various explicit examples illustrate our findings.
Abstract The resource theory of superposition extends resource theories of coherence, offering a rigorous framework to quantify the superposition within a finite set of linearly independent states. Based on this theory, we propose and analyze several legitimate superposition measures, including the superposition of formation (SOF) and the logarithmic measure, demonstrating that both satisfy the required conditions for a valid superposition measure. We then investigate the intriguing properties of various superposition measures. We find that the relative entropy of superposition cannot be expressed as a closed form, and it is superadditive under tensor products and monotone under partial trace. Furthermore, we show that the SOF also exhibits superadditive properties. It is shown that the upper bound of the relative entropy of superposition measure is determined by the SOF. In contrast, we find that the logarithmic superposition measure exhibits subadditive properties, and its regularized version also exist. Additionally, we discuss the rank-measure of superposition and show that it satisfies additivity. Finally, we investigate the general quantum resource measure via the standard convex roof construction in quantum resource theory, and some properties of a general resource measure are also discussed.
One of the fundamental features of quantum mechanics is the superposition principle, a manifestation of which is embodied in quantum coherence. Coherence of a quantum state is invariably defined with respect to a preferred set of pointer states, and there exist quantum coherence measures with respect to deterministically as well as probabilistically distinguishable sets of quantum state vectors. Here we study the resource theory of quantum coherence with respect to an arbitrary set of quantum state vectors, that may not even be probabilistically distinguishable. Geometrically, a probabilistically indistinguishable set of quantum state vectors forms a linearly dependent set. In quantum optics, the coherent states form an ``overcomplete basis'' of linearly dependent states and are useful in dealing with states that can be prepared in optical systems. Also, the resource theory of magic can be looked upon as a resource theory of quantum coherence with respect to a set of basis vectors that are probabilistically indistinguishable. These motivate us to consider a resource theory of coherence with respect to probabilistically indistinguishable pointers. We find the free states of the resource theory, and analyze the corresponding free operations, obtaining a necessary condition for an arbitrary quantum operation to be free. We identify a class of measures of the quantum coherence and, in particular, establish the monotonicity property of the measures. We find a connection of an arbitrary set of quantum state vectors with positive operator-valued measurements with respect to the resource theory being considered, which paves the way for an alternate definition of the free states. We subsequently examine the wave-particle duality in a double-slit setup in which superposition of probabilistically indistinguishable quantum state vectors is possible. Specifically, we report a complementary relation between quantum coherence and path distinguishability in such a setup.
Any quantum resource theory is based on free states and free operations, i.e., states and operations which can be created and performed at no cost. In the resource theory of coherence free states are diagonal in some fixed basis, and free operations are those which cannot create coherence for some particular experimental realization. Recently, some problems of this approach have been discussed, and new sets of operations have been proposed to resolve these problems. As an example for such a problem, a quantum operation can be incoherent in one particular experimental realization, but it still can create large amount of coherence in another. This problem is resolved by the framework of genuine quantum coherence. Here, we investigate this concept further and study other more general sets of operations which cannot create coherence in any experimental realization. We analyze in detail the mathematical structure of this class of maps and use this to study possible state transformations under these operations. We show that deterministic manipulation is severely limited, even in the asymptotic settings. In particular, this framework does not have a unique golden unit, i.e., there is no single state from which all other states can be created deterministically with the free operations. On the other hand, we show that these theories allow for a richer structure in what comes to stochastic manipulation of states. In particular, we determine the optimal probability of conversion among pure states with genuinely incoherent operations. Still, the aforementioned limitations suggest that any reasonably powerful resource theory of coherence must contain free operations which can potentially create coherence in some experimental realization.
Quantum physics currently provides our deepest description of nature. To understand precisely how it differs from classical physics is not only of foundational interest, but also important to advance technology: the advent of quantum information has led to the insight that certain quantum phenomena, e.g., entanglement and coherence, are resources which enable us to solve practical tasks that are unsolvable within the realm of classical physics. But how can we quantify quantum resources and how do we decide which resource is relevant for a specific operational advantage in quantum cryptography, quantum computation, or quantum metrology? How can we employ these resources optimally? Addressing these questions is the motivation for the development of quantum resource theories, a mathematically rigorous framework that allows us to investigate a specific quantum phenomenon as a resource. Formally, such theories emerge from restrictions which are imposed in addition to the laws of quantum mechanics. A prominent example is the restriction to local operations and classical communication, which singles out entanglement as a resource. Typically, such theories describe the value of quantum states, i.e., static resources. An important question in such theories concerns their structure as determined by the restriction under consideration, which is analyzed for coherence theory on qubits in this thesis. Only recently, so-called dynamical resource theories that quantify the value of operations were proposed, which are advantageous for several reasons: they include resource theories of states as special cases, allow us to quantify additional quantum properties such as the detection or usage of static resources, which will be a main topic of this thesis, and are more natural from an operational perspective. But are different types of dynamical resources present in quantum operations related? For the case of dynamical coherence and entanglement, it will be shown that there is not only a quantitative, but also a qualitative connection.
Quantum mechanics exhibits a wide range of nonclassical features, of which entanglement in multipartite systems takes a central place. In several specific settings, it is well known that nonclassicality (e.g., squeezing, spin squeezing, coherence) can be converted into entanglement. In this work, we present a general framework, based on superposition, for structurally connecting and converting nonclassicality to entanglement. In addition to capturing the previously known results, this framework also allows us to uncover new entanglement convertibility theorems in two broad scenarios, one which is discrete and one which is continuous. In the discrete setting, the classical states can be any finite linearly independent set. For the continuous setting, the pertinent classical states are "symmetric coherent states," connected with symmetric representations of the group SU(K). These results generalize and link convertibility properties from the resource theory of coherence, spin coherent states, and optical coherent states, while also revealing important connections between local and nonlocal pictures of nonclassicality.