If a generalized measurement is performed on a quantum system and we do not know the outcome, are we able to retrodict it with a second measurement? We obtain a necessary and sufficient condition for perfect retrodiction of the outcome of a known generalized measurement, given the final state, for an arbitrary initial state. From this, we deduce that, when the input and output Hilbert spaces have equal (finite) dimension, it is impossible to perfectly retrodict the outcome of any fine-grained measurement [where each positive, operator-valued measure (POVM) element corresponds to a single Kraus operator] for all initial states unless the measurement is unitarily equivalent to a projective measurement. It also enables us to show that every POVM can be realized in such a way that perfect outcome retrodiction is possible for an arbitrary initial state when the number of outcomes does not exceed the output Hilbert space dimension. We then consider the situation where the initial state is not arbitrary, though it may be entangled, and describe the conditions under which unambiguous outcome retrodiction is possible for a fine-grained generalized measurement. We find that this is possible for some state if the Kraus operators are linearly independent. This condition is also necessary when the Kraus operators are nonsingular. From this, we deduce that every trace-preserving quantum operation is associated with a generalized measurement whose outcome is unambiguously retrodictable for some initial state, and also that a set of unitary operators can be unambiguously discriminated iff they are linearly independent. We then examine the issue of unambiguous outcome retrodiction without entanglement. This has important connections with the theory of locally linearly dependent and locally linearly independent operators.
If a generalized measurement is performed on a quantum system and we do not know the outcome, are we able to retrodict it with a second measurement? We obtain a necessary and sufficient condition for perfect retrodiction of the outcome of a known generalized measurement, given the final state, for an arbitrary initial state. From this, we deduce that, when the input and output Hilbert spaces have equal (finite) dimension, it is impossible to perfectly retrodict the outcome of any fine-grained measurement [where each positive, operator-valued measure (POVM) element corresponds to a single Kraus operator] for all initial states unless the measurement is unitarily equivalent to a projective measurement. It also enables us to show that every POVM can be realized in such a way that perfect outcome retrodiction is possible for an arbitrary initial state when the number of outcomes does not exceed the output Hilbert space dimension. We then consider the situation where the initial state is not arbitrary, though it may be entangled, and describe the conditions under which unambiguous outcome retrodiction is possible for a fine-grained generalized measurement. We find that this is possible for some state if the Kraus operators are linearly independent. This condition is also necessary when the Kraus operators are nonsingular. From this, we deduce that every trace-preserving quantum operation is associated with a generalized measurement whose outcome is unambiguously retrodictable for some initial state, and also that a set of unitary operators can be unambiguously discriminated iff they are linearly independent. We then examine the issue of unambiguous outcome retrodiction without entanglement. This has important connections with the theory of locally linearly dependent and locally linearly independent operators.
We study how well we can retrodict results of measurements made on a quantum system if we can make measurements on its final state. We know what measurements were made, but not their results. An initial examination shows that we can gain anywhere from no information to perfect information about the results of previous measurements, depending on the measurements and the initial state of the system. The case of two two-outcome measurements, the second of which is a projective measurement, is examined in some detail. We then look at a model of a qubit interferometer in which measurements are made in order to determine the path the qubit followed. The measurement made on the final state of the qubit depends on the information about previous measurement results that we are trying to determine. One can attempt to find the result of just one of the measurements or all of them or to find a measurement sequence that was not realized. We study all three possibilities.
Can an agent's predictions reveal whether its reasoning is based on a classical or a quantum model? Using known results on matrix factorizations, here we show that quantum Bayesian retrodiction can produce one-step-ahead forecasts incompatible with every classical Bayesian explanation based on learning about a fixed but unknown sampling law from conditionally independent observations. The separation has a sharp threshold in the number of outcomes: for any measurement with at most four outcomes, the quantum predictions admit a classical realization, irrespective of the prior state and the finite Hilbert-space dimension, whereas a five-outcome measurement on a single qubit already violates an explicit classical bound. This bound holds for arbitrary classical latent states, prior probabilities, and sampling laws, and involves only the agent's initial and updated predictions. Thus, in this setting, quantum Bayesian retrodiction is strictly more expressive than classical Bayesian learning, and this difference can be witnessed without observing or specifying the agent's internal description.
Unambiguous measurements play an important role in quantum information, with applications ranging from quantum key distribution to quantum state reconstruction. Recently, such measurements have also been used in quantum algorithms based on Regev's reduction. The key problem for these algorithms is the S-∣LWE> problem in the lattice setting and the Quantum Decoding Problem in the code setting. A key idea for addressing this problem is to use unambiguous measurements to recover k coordinates of a code (or lattice) element x from a quantum state ∣ψx⟩, which corresponds to a noisy word x with errors in quantum superposition. However, a general theoretical framework to analyze this approach has been lacking. In this work, we introduce the notion of fine-grained unambiguous measurements. Given a family of states {∣ψx⟩}x∈{0,1}n, we ask whether there exist measurements that can return, with certainty, k bits of information about x. We study this question in the setting of symmetric states, which naturally arises in the Quantum Decoding Problem. We show that determining the maximal number of parities that a measurement can output can be formulated as a linear program, and we use its dual formulation to derive several upper bounds. In particular, we establish necessary and sufficient conditions for the existence of fine-grained unambiguous measurements and prove impossibility results showing, in particular, that such measurements cannot improve upon the approach of arXiv:2310.20651. Finally, we discuss the implications of these findings for the Quantum Decoding Problem.
Quantum theory of unambiguous measurements In the present paper I formulate a framework that accommodates many unambiguous discrimination problems. I show that the prior information about any type of constituent (state, channel, or observable) allows us to reformulate the discrimination among finite number of alternatives as the discrimination among finite number of average constituents. Using this framework I solve several unambiguous tasks. I present a solution to optimal unambiguous comparison of two ensembles of unknown quantum states. I consider two cases: 1) The two unknown states are arbitrary pure states of qudits. 2) Alternatively, they are coherent states of single-mode optical fields. For this case I propose simple and optimal experimental setup composed of beam-splitters and a photodetector. As a second tasks I consider an unambiguous identification (UI) of coherent states. In this task identical quantum systems are prepared in coherent states and labeled as unknown and reference states, respectively. The promise is that one reference state is the same as the unknown state and the task is to find out unambiguously which one it is. The particular choice of the reference states is unknown to us, and only the probability distribution describing this choice is known. In a general case when multiple copies of unknown and reference states are available I propose a scheme consisting of beamsplitters and photodetectors that is optimal within linear optics. UI can be considered as a search in a quantum database, whose elements are the reference states and the query is represented by the unknown state. This perspective motivated me to show that reference states can be recovered after the measurement and might be used (with reduced success rate) in subsequent UI. Moreover, I analyze the influence of noise in preparation of coherent states on the performance of the proposed setup. Another problem I address is the unambiguous comparison of a pair of unknown qudit unitary channels. I characterize all solutions and identify the optimal ones. I prove that in optimal experiments for comparison of unitary channels the entanglement is necessary. The last task I studied is the unambiguous comparison of unknown non-degenerate projective measurements. I distinguish between measurement devices with apriori labeled and unlabeled outcomes. In both cases only the difference of the measurements can be concluded unambiguously. For the labeled case I derive the optimal strategy if each unknown measurement is used only once. However, if the apparatuses are not labeled, then each measurement device must be used (at least) twice. In particular, for qubit measurement apparatuses with unlabeled outcomes I derive the optimal test state in the two-shots scenario.
To implement a positive-operator-valued measurement (POVM), which is defined on the dS-dimensional Hilbert space of a physical system, one has to extend the Hilbert space to include dA additional dimensions (called the ancilla). This is done via either the tensor product extension (TPE) or the direct sum extension (DSE). The implementation of a POVM utilizes the available resources more efficiently if it requires fewer additional dimensions. To determine how to implement a POVM with the least additional dimensions is, therefore, an important task in quantum information. We have determined the necessary and sufficient (hence minimal) number of the additional dimensions needed to implement the same POVM by the TPE and the DSE, respectively. If the POVM has n elements and ri is the rank of the ith element, then the dimension of the minimal ancilla is ${d}_{A}={\ensuremath{\sum}}_{i=1}^{n}{r}_{i}\ensuremath{-}{d}_{S}$ for the DSE implementation, and this represents a lower bound for the added dimensions in the TPE implementation. In the proof, we explicitly construct the DSE implementation of a general POVM with elements of arbitrary rank. As an example, we determine dA for the unambiguous discrimination of N linearly independent states and provide the full DSE implementation of a state-discriminating POVM for N=2.
In quantum mechanics the statistics of the outcomes of a measuring apparatus is described by a positive operator valued measure (POVM). A quantum channel transforms POVMs into POVMs, generally irreversibly, thus losing some of the information retrieved from the measurement. This poses the problem of which POVMs are “undisturbed,” i.e., they are not irreversibly connected to another POVM. We will call such POVMs clean. In a sense, the clean POVMs would be “perfect,” since they would not have any additional “extrinsical” noise. Quite unexpectedly, it turns out that such a “cleanness” property is largely unrelated to the convex structure of POVMs, and there are clean POVMs that are not extremal and vice versa. In this article we solve the cleannes classification problem for number n of outcomes n⩽d (d dimension of the Hilbert space), and we provide a set of either necessary or sufficient conditions for n>d, along with an iff condition for the case of informationally complete POVMs for n=d2.