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arXiv:quant-ph/0102064 · Physical Review Letters

Statistical Distinguishability between Unitary Operations

Antonio Acín

Published Oct 4, 2001• 288 citations
arXivHTMLPDF

Abstract

The problem of distinguishing two unitary transformations, or quantum gates, is analyzed and a function reflecting their statistical distinguishability is found. Given two unitary operations, U1 and U2, it is proved that there always exists a finite number N such that U(x in circle N)(1) and U(x in circle N)(2) are perfectly distinguishable, although they were not in the single-copy case. This result can be extended to any finite set of unitary transformations. Finally, a fidelity for one-qubit gates, which satisfies many useful properties from the point of view of quantum information theory, is presented.

DOI
10.1103/physrevlett.87.177901

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The problem of distinguishing two unitary transformations, or quantum gates, is analyzed and a function reflecting their statistical distinguishability is found. Given two unitary operations, U1 and U2, it is proved that there always exists a finite number N such that U(x in circle N)(1) and U(x in circle N)(2) are perfectly distinguishable, although they were not in the single-copy case. This result can be extended to any finite set of unitary transformations. Finally, a fidelity for one-qubit gates, which satisfies many useful properties from the point of view of quantum information theory, is presented.
Distinguishability is a fundamental and operational measure generally connected to information applications. In quantum information theory, from the postulates of quantum mechanics it often has an intrinsic limitation, which then dictates and also characterises capabilities of related information tasks. In this work, we consider discrimination between bipartite two-qubit unitary transformations by local operations and classical communication (LOCC) and its relations to entangling capabilities of given unitaries. We show that a pair of entangling unitaries which do not contain local parts, if they are perfectly distinguishable by global operations, can also be perfectly distinguishable by LOCC. There also exist non-entangling unitaries, e.g. local unitaries, that are perfectly discriminated by global operations but not by LOCC. The results show that capabilities of LOCC are strictly restricted than global operations in distinguishing bipartite unitaries for a finite number of repetitions, contrast to discrimination of a pair of bipartite states and also to asymptotic discrimination of unitaries.
An essential element of classical computation is the ``if-then'' construct, that accepts a control bit and an arbitrary gate, and provides conditional execution of the gate depending on the value of the controlling bit. On the other hand, quantum theory prevents the existence of an analogous universal construct accepting a control qubit and an arbitrary quantum gate as its input. Nevertheless, there are controllable sets of quantum gates for which such a construct exists. Here we provide a necessary and sufficient condition for a set of unitary transformations to be controllable, and we give a complete characterization of controllable sets in the two-dimensional case. This result reveals an interesting connection between the problem of controllability and the problem of extracting information from an unknown quantum gate while using it.
The problem studied in this paper is, if the number of queries to unitary operations is fixed, say kkk, then when do local operations and classical communication (LOCC) suffice for optimally distinguishing bipartite unitary operations? We consider the above problem for two-qubit unitary operations in the case of k=1k=1k=1, showing that for two two-qubit entangling unitary operations without local parties, LOCC achieves the same distinguishability as the global operations. Specifically, we obtain the following: (i) if such two unitary operations are perfectly distinguishable by global operations, then they are perfectly distinguishable by LOCC also, and (ii) if they are not perfectly distinguishable by global operations, then LOCC can achieve the same optimal discrimination probability as the global operations.
We study optimal perfect distinguishability between a unitary and a general quantum operation. In 2-dimensional case we provide a simple sufficient and necessary condition for sequential perfect distinguishability and an analytical formula of optimal query time. We extend the sequential condition to general d-dimensional case. Meanwhile, we provide an upper bound and a lower bound for optimal sequential query time. In the process a new iterative method is given, the most notable innovation of which is its independence to auxiliary systems or entanglement. Following the idea, we further obtain an upper bound and a lower bound of (entanglement-assisted) q-maximal fidelities between a unitary and a quantum operation. Thus by the recursion in [1] an upper bound and a lower bound for optimal general perfect discrimination are achieved. Finally our lower bound result can be extended to the case of arbitrary two quantum operations.
We prove a general limitation in quantum information that unifies the impossibility principles such as no-cloning and no-anticloning. Further, we show that for an unknown qubit one cannot design a universal Hadamard gate for creating equal superposition of the original and its complement state. Surprisingly, we find that Hadamard transformations exist for an unknown qubit chosen either from the polar or equatorial great circles. Also, we show that for an unknown qubit one cannot design a universal unitary gate for creating unequal superpositions of the original and its complement state. We discuss why it is impossible to design a controlled-NOT gate for two unknown qubits and discuss the implications of these limitations. 03.67.Hk, 03.65.Ta
We study the problem of information masking through nonzero linear operators that distribute information encoded in single qubits to the correlations between two qubits. It is shown that a nonzero linear operator cannot mask any nonzero measure set of qubit states. We prove that the maximal maskable set of states on the Bloch sphere with respect to any masker is the ones on a spherical circle. Any states on a spherical circle on the Bloch sphere are maskable, which also proves the conjecture on maskable qubit states given by Modi et al. [Phys. Rev. Lett. 120, 230501 (2018)]. we provide explicitly operational unitary maskers for all maskable sets. As applications, different protocols for secret sharing are introduced.