Floyd-Hoare logic is a foundation of axiomatic semantics of classical programs, and it provides effective proof techniques for reasoning about correctness of classical programs. To offer similar techniques for quantum program verification and to build a logical foundation of programming methodology for quantum computers, we develop a full-fledged Floyd-Hoare logic for both partial and total correctness of quantum programs. It is proved that this logic is (relatively) complete by exploiting the power of weakest preconditions and weakest liberal preconditions for quantum programs.

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Abstract-The efficiency of parameter estimation of quantum channels is studied in this paper. We introduce the concept of programmable parameters to the theory of estimation. It is found that programmable parameters obey the standard quantum limit strictly; hence no speedup is possible in its estimation. We also construct a class of non-unitary quantum channels whose parameter can be estimated in a way that the standard quantum limit is broken. The study of estimation of general quantum channels also enables an investigation of the effect of noises on quantum estimation.

We provide a feasible necessary and sufficient condition for when an unknown quantum operation (quantum device) secretly selected from a set of known quantum operations can be identified perfectly within a finite number of queries, and thus complete the characterization of the perfect distinguishability of quantum operations. We further design an optimal protocol which can achieve the perfect discrimination between two quantum operations by a minimal number of queries. Interestingly, we find that an optimal perfect discrimination between two isometries is always achievable without auxiliary systems or entanglement.

We explicitly exhibit a set of four ququad-ququad orthogonal maximally entangled states that cannot be perfectly distinguished by means of local operations and classical communication. Before our work, it was unknown whether there is a set of d locally indistinguishable d⊗d orthogonal maximally entangled states for some positive integer d. We further show that a 2⊗2 maximally entangled state can be used to locally distinguish this set of states without being consumed, thus demonstrate a novel phenomenon of entanglement discrimination catalysis. Based on this set of states, we construct a new set K consisting of four locally indistinguishable states such that K(⊗m) (with 4(m) members) is locally distinguishable for some m greater than one. As an immediate application, we construct a noisy quantum channel with one sender and two receivers whose local zero-error classical capacity can achieve the full dimension of the input space but only with a multi-shot protocol.

We show that a unitary operation (quantum circuit) secretely chosen from a finite set of unitary operations can be determined with certainty by sequentially applying only a finite amount of runs of the unknown circuit. No entanglement or joint quantum operations is required in our scheme. We further show that our scheme is optimal in the sense that the number of the runs is minimal when discriminating only two unitary operations.PACS numbers: 03.65. Ta, 03.65.Ud, Entanglement is a valuable physical resource for accomplishing many useful quantum computing and quantum information processing tasks [1]. For certain tasks such as superdense coding [2] and quantum teleportation [3], it has been demonstrated that entanglement is an indispensable ingredient. For many other tasks entanglement is also used to enhance the efficiency [4,5,6,7]. One important instance among these tasks is the discrimination of unitary operations. Although two nonorthogonal quantum states cannot be discriminated with certainty whenever only finitely many number of copies are available [8], a perfect discrimination between two different unitary can always be achieved by taking a suitable entangled state as input and then applying only a finite number of runs of the unknown unitary operation [5,6]. It is widely believed that this remarkable effect is essentially due to the use of quantum entanglement. As entanglement is a kind of nonlocal correlation existing between different quantum systems, creation of entanglement needs to perform joint quantum operations on two or more systems. These joint operations are generally difficult and expensive. Consequently, it is of great importance to consume as small amount of entanglement as possible in accomplishing a given task. This motivates us to ask: "What kind of tasks can be achieved without entanglement?" Some pioneering works have been devoted to a good understanding of the exact role of quantum entanglement in the context of quantum computing. It has been shown that for certain problems, including DeutschJoza's problem [9], Simon's problem [10], and quantum search problem [11], quantum computing devices may still have advantages over than any known classical computing devices even without the presence of entanglement [12,13,14,15]. It was also argued that it may be the interference and the orthogonality but not the entanglement which are responsible for the power of quantum computing [13].In this letter we contribute a new instance of this kind of problems in the context of quantum information by reporting a somewhat counterintuitive result: Entanglement is not necessary for perfect discrimination between unitary operations. We achieve this goal by explicitly constructing a simple scheme where no entanglement is needed to discriminate any two given unitary operations with certainty.The basic idea behind our scheme can be best understood in the following scenario. Suppose we are given an unknown quantum circuit which is secretely chosen from two alternatives: U or V . Here both U and V are unitary ...

We prove that the states secretly chosen from a mixed state set can be perfectly discriminated if and only if they are orthogonal. The sufficient and necessary condition under which nonorthogonal mixed quantum states can be unambiguously discriminated is also presented. Furthermore, we derive a series of lower bounds on the inconclusive probability of unambiguous discrimination of states from a mixed state set with a priori probabilities.

Although security of quantum cryptography is provable based on principles of quantum mechanics, it can be compromised by flaws in the design of quantum protocols. So, it is indispensable to develop techniques for verifying and debugging quantum cryptographic systems. Model-checking has proved to be effective in the verification of classical cryptographic protocols, but an essential difficulty arises when it is applied to quantum systems: the state space of a quantum system is always a continuum even when its dimension is finite. To overcome this difficulty, we introduce a novel notion of quantum Markov chain, especially suited for modelling quantum cryptographic protocols, in which quantum effects are encoded as super-operators labelling transitions, leaving the location information (nodes) being classical. Then we define a quantum extension of probabilistic computation tree logic (PCTL) and develop a model-checking algorithm for quantum Markov chains.