Abstract In this work, we investigate the question of which objects can be discriminated by totally “interaction-free” measurements. To this end, we interpret the Elitzur–Vaidman bomb-tester experiment as a quantum channel discrimination problem and generalize the notion of “interaction-free” measurement to arbitrary quantum channels. Our main result is a necessary and sufficient criterion for when it is possible or impossible to discriminate quantum channels in an “interaction-free” manner (i.e., such that the discrimination error probability and the “interaction” probability can be made arbitrarily small). For the case where our condition holds, we devise an explicit protocol with the property that both probabilities approach zero with an increasing number of channel uses, N. More specifically, the “interaction” probability in our protocol decays as $$\frac{1}{N}$$ 1 N and we show that this rate is the optimal achievable one. Furthermore, our protocol only needs at most one ancillary qubit and might thus be implementable in near-term experiments. For the case where our condition does not hold, we prove an inequality that quantifies the trade-off between the error probability and the “interaction” probability.
Abstract In this work, we investigate the question of which objects can be discriminated by totally “interaction-free” measurements. To this end, we interpret the Elitzur–Vaidman bomb-tester experiment as a quantum channel discrimination problem and generalize the notion of “interaction-free” measurement to arbitrary quantum channels. Our main result is a necessary and sufficient criterion for when it is possible or impossible to discriminate quantum channels in an “interaction-free” manner (i.e., such that the discrimination error probability and the “interaction” probability can be made arbitrarily small). For the case where our condition holds, we devise an explicit protocol with the property that both probabilities approach zero with an increasing number of channel uses, N. More specifically, the “interaction” probability in our protocol decays as N1 1 N and we show that this rate is the optimal achievable one. Furthermore, our protocol only needs at most one ancillary qubit and might thus be implementable in near-term experiments. For the case where our condition does not hold, we prove an inequality that quantifies the trade-off between the error probability and the “interaction” probability.
Abstract In this paper, we mainly study the indistinguishability of classically simulable measurements (measurements that can be effectively simulated on classical computers) in the tasks of quantum channel discrimination. First, we investigate the limitations of classically simulable measurements under unambiguous and perfect discrimination strategies. For unambiguous discrimination, we establish a conditional limitation result for channel-induced output-state discrimination scenarios, while for perfect discrimination we derive a sufficient condition under which classically simulable measurements fail to perfectly distinguish quantum channels. When considering the minimum-error discrimination strategy, we formulate the corresponding bilinear optimization problem and discuss its formal Lagrange dual formulation. Subsequently, we give the lower bound of the minimum-error probability obtained by using classically simulable measurements in the minimum-error discrimination task under the adaptive protocol. Furthermore, we give two measures of the nonstabilizerness (magic) of measurements. We prove that one measure we proposed can be used as an exact quantifier for the advantage in some tasks of quantum state discrimination. Finally, we investigate the relationship between classically simulable measurements and the data hiding ratio.
Interaction-free measurement is a quantum process where, in the ideal situation, an object can be detected as if no interaction took place with the probing photon. Here we show that the problem of interaction-free measurement can be regarded as a problem of quantum-channel discrimination. In particular, we look for the optimal photonic states that can minimize the detection error and the photon loss in detecting the presence or absence of the object, which is taken to be semitransparent, and the number of the interrogation cycle is assumed to be finite. Furthermore, we also investigated the possibility of minimizing the detection error through the use of entangled photons, which is essentially a setting of quantum illumination. However, our results indicate that entanglement does not exhibit a clear advantage; the same performance can be achieved with unentangled photonic states.
Quantum hypothesis testing is an important tool for quantum information processing.Two main strategies have been widely adopted: in a minimum error discrimination strategy, the average error probability is minimized; while in an unambiguous discrimination strategy, an inconclusive decision (abstention) is allowed to vanish any possibility of errors when a conclusive result is obtained.In both scenarios, the testing between quantum states is relatively well understood, for example, the ultimate limits of the performance are established decades ago; however, the testing between quantum channels is less understood.Although the ultimate limit of minimum error discrimination between channels has been explored recently, the corresponding limit of unambiguous discrimination is unknown.In this paper, we formulate an approximate unambiguous discrimination scenario, and derive the ultimate limits of the performance for both states and channels.In particular, in the channel case, our lower bound of the inconclusive probability holds for arbitrary adaptive sensing protocols.For the special class of "teleportation-covariant" channels, the lower bound is achievable with maximum entangled inputs and no adaptive strategy is necessary.
Interaction-free measurements (IFMs) demonstrate that the existence of an interaction channel may be inferred without the interaction itself occurring. In the standard Elitzur–Vaidman bomb tester, the interaction capability remains independently verifiable after completion of the protocol because the bomb survives and may subsequently be tested.This conceptual note considers an idealized modification in which the interaction capability exists only during the photon's passage through the interferometer and becomes permanently operationally uncertifiable before the detector outcome is available. The quantum evolution is assumed to remain identical to that of the standard interaction-free measurement protocol; only the possibility of independent post-experimental verification is removed.The paper formulates the operational question of whether interaction-free measurement depends solely on the existence of an interaction channel during the measurement interval, or whether post-experimental operational certifiability is an implicit assumption of the standard protocol. No interpretation-dependent conclusions are drawn.
In this work, we study the problem of single-shot discrimination of von Neumann measurements, which we associate with measure-and-prepare channels. There are two possible approaches to this problem. The first one is simple and does not utilize entanglement. We focus only on the discrimination of classical probability distributions, which are outputs of the channels. We find necessary and sufficient criterion for perfect discrimination in this case. A more advanced approach requires the usage of entanglement. We quantify the distance between two measurements in terms of the diamond norm (called sometimes the completely bounded trace norm). We provide an exact expression for the optimal probability of correct distinction and relate it to the discrimination of unitary channels. We also state a necessary and sufficient condition for perfect discrimination and a semidefinite program which checks this condition. Our main result, however, is a cone program which calculates the distance between the measurements and hence provides an upper bound on the probability of their correct distinction. As a by-product, the program finds a strategy (input state) which achieves this bound. Finally, we provide a full description for the cases of Fourier matrices and mirror isometries.
Single-shot quantum channel discrimination is the fundamental task of determining, given only a single use, which of two known quantum channels is acting on a system. In this thesis we investigate the well-known phenomenon that entanglement to an auxiliary system can provide an advantage in this task. In particular, we consider the questions: (1) How much entanglement is in general necessary to achieve an optimal discrimination strategy? (2) What is the maximal advantage provided by entanglement? \n \nGiven a linear map Ψ : L(Cn) → L(Cm), its multiplicity maps are defined as the family of linear maps Ψ ⊗ 1L(Ck) : L(Cn ⊗ Ck) → L(Cm ⊗ Ck), where 1L(Ck) is the identity on L(Ck). Due to the Holevo-Helstrom theorem, the optimal performance using an auxiliary system of dimension k is quantified in terms of the norm of Ψ⊗1L(Ck), where Ψ is a linear map that depends on the parameters of the discrimination problem. Hence, the advantage provided by entanglement is represented in the growth of the norm of Ψ ⊗ 1L(Ck ) with k, a classic phenomenon in the theory of operator algebras. \n \nWe formalize question (1) by investigating, relative to the input and output dimensions of the channels to be discriminated, how large of an auxiliary system is necessary to achieve an optimal strategy. Mathematically, this is connected to when the norm of Ψ ⊗ 1L(Ck) stops growing with k. It is well- known that an auxiliary system of dimension equal to the input is always sufficient to achieve an optimal strategy, and that this is sometimes necessary when the output dimension is at least as large as the input. We prove that, even when the output dimension is arbitrarily small compared to the input, it is still sometimes necessary to use an auxiliary system as large as the input to achieve an optimal strategy. \n \nFor question (2), we investigate, with respect to a fixed input dimension, how large the gap between the optimal performances with and without entanglement can be. Mathematically, this is quantified by the rate of growth of the norm of Ψ ⊗ 1L(Ck ) in k. It is known that matrix transposition has the fastest possible growth, and we prove that it is essentially the unique linear map with this property. We use this to prove that a discrimination problem defined in terms of the Werner-Holevo channels is essentially the unique game satisfying a norm relation that states that the game can be won with certainty using entanglement, but is hard to win without entanglement. \n \nAlong the way, we prove characterizations of the structure of maximal entanglement, as measured according to the entanglement negativity, as well as relative to a large class of entanglement measures. We also give various characterizations of complete trace-norm isometries, and reversible quantum channels.