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Can Two Differently Prepared Mixed Quantum-Ensembles be Discriminated Via Measurement Variance ?

Chandan Kumar

Published Feb 1, 2017• 0 citations
OpenAlex

Abstract

Alice prepares two large qubit-ensembles E1 and E2 in the following states: She individually prepares each qubit of E1 in |0> or |1>, the eigenstates of Pauli-z operator Z, depending on the outcome of an unbiased coin toss. Similarly, she individually prepares each qubit of E2 in |+> or |-> the eigenstates of Pauli-x operator X. Bob, who is aware of the above states preparation procedures, but knows neither which of the two is E1 nor Alice's outcomes of coin tosses, needs to discriminate between the two maximally mixed ensembles. Here we argue that Bob can partially purify the mixed states (E1, E2), using the information supplied by central limit theorem. We will show that, subsequently Bob can discriminate between ensembles E1 and E2 by individually rotating each qubit state about the x-axis on Bloch sphere by a random angle, and then projectively measuring Z. By these operations, the variance of sample mean of Z measurement outcomes corresponding to the ensemble E1 gets reduced. On the other hand, qubit states in E2 are invariant under the x-rotations and therefore the variance remains unaltered. Thus Bob can discriminate between the two maximally mixed ensembles. We analyse the above problem both analytically as well as numerically, and show that the latter supports the former.

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Alice prepares two large qubit-ensembles E1 and E2 in the following states: She individually prepares each qubit of E1 in |0> or |1>, the eigenstates of Pauli-z operator Z, depending on the outcome of an unbiased coin toss. Similarly, she individually prepares each qubit of E2 in |+> or |-> the eigenstates of Pauli-x operator X. Bob, who is aware of the above states preparation procedures, but knows neither which of the two is E1 nor Alice's outcomes of coin tosses, needs to discriminate between the two maximally mixed ensembles. Here we argue that Bob can partially purify the mixed states (E1, E2), using the information supplied by central limit theorem. We will show that, subsequently Bob can discriminate between ensembles E1 and E2 by individually rotating each qubit state about the x-axis on Bloch sphere by a random angle, and then projectively measuring Z. By these operations, the variance of sample mean of Z measurement outcomes corresponding to the ensemble E1 gets reduced. On the other hand, qubit states in E2 are invariant under the x-rotations and therefore the variance remains unaltered. Thus Bob can discriminate between the two maximally mixed ensembles. We analyse the above problem both analytically as well as numerically, and show that the latter supports the former.
Abstract Given an ensemble of qubits, which we are told consists of a mixture of two pure states, one with probability η 0 and one with probability η 1 , we want to find a POVM that will discriminate between the two states by measuring the qubits. We do not know the states, and for any given qubit, we do not know which of the two states it is in. This can be viewed as learning a POVM from quantum data. Once found, the POVM can be used to separate the remaining qubits in the ensemble into two groups, corresponding to the two states present in the ensemble. In order to find the POVM, we need more information about the possible states. We examine several cases. First, we suppose that we know that the Bloch vectors of the states lie in the x – z plane and their a priori probabilities are equal. We next keep the restriction to the x – z plane, but allow the a priori probabilities to be different. Finally, we consider the case in which the Bloch vectors of the states have the same z component.
The discrimination of quantum measurements is an important subject of quantum information processes. In this paper we present a novel protocol for local quantum measurement discrimination with multi-qubit entanglement systems. It is shown that, in the case of Bob did not know Alice's result of measurement, if both two observers (Alice and Bob) agreed in advance that Alice should measure her qubits before an appointed time t, the local discrimination of two different kinds of measurement can be completed via numerous eight-qubit GHZ entangled states and selective projective measurement.
Abstract A central tenet of quantum theory is that ensembles sharing the same density matrix are indistinguishable under measurements on individual systems. We show that this indistinguishability can break down when additional structure is present across multiple copies. We investigate ensembles of qubits prepared in either the computational (ZZZ) or Hadamard (XXX) basis under a same-basis constraint. All qubits within a batch are prepared in the same basis, but with randomly distributed outcomes such that the reduced state of each individual qubit is maximally mixed. We introduce an experimentally realizable protocol that exploits multi-copy statistical structure to distinguish these ensembles. The graph-state protocol identifies the two basis through stabilizer measurements and higher-order statistical features. Our protocol has been implemented on the \texttt{ibm\_fez} superconducting quantum processor and demonstrates robust performance in the presence of realistic noise, achieving near-unity classification accuracy as the number of samples increases. While single-copy measurements are uninformative, aggregating outcomes across multiple instances produces a clear and scalable separation between basis families. Our result identifies a regime in which quantum information is not accessible at the level of individual systems but emerges through structured multi-copy measurements. This highlights the role of higher-order statistical correlations in quantum information processing and provides a practical route for exploiting such structure on near-term quantum devices.
We investigate sampling procedures that certify that an arbitrary quantum state on nnn subsystems is close to an ideal mixed state φ⊗nφ^{\otimes n}φ⊗n for a given reference state φφφ, up to errors on a few positions. This task makes no sense classically: it would correspond to certifying that a given bitstring was generated according to some desired probability distribution. However, in the quantum case, this is possible if one has access to a prover who can supply a purification of the mixed state. In this work, we introduce the concept of mixed-state certification, and we show that a natural sampling protocol offers secure certification in the presence of a possibly dishonest prover: if the verifier accepts then he can be almost certain that the state in question has been correctly prepared, up to a small number of errors. We then apply this result to two-party quantum coin-tossing. Given that strong coin tossing is impossible, it is natural to ask "how close can we get". This question has been well studied and is nowadays well understood from the perspective of the bias of individual coin tosses. We approach and answer this question from a different---and somewhat orthogonal---perspective, where we do not look at individual coin tosses but at the global entropy instead. We show how two distrusting parties can produce a common high-entropy source, where the entropy is an arbitrarily small fraction below the maximum (except with negligible probability).
We discuss sequential unambiguous state-discrimination measurements performed on the same qubit. Alice prepares a qubit in one of two possible states. The qubit is first sent to Bob, who measures it, and then on to Charlie, who also measures it. The object in both cases is to determine which state Alice sent. In an unambiguous state discrimination measurement, we never make a mistake, i.e., misidentify the state, but the measurement may fail, in which case we gain no information about which state was sent. We find that there is a nonzero probability for both Bob and Charlie to identify the state, and we maximize this probability. The probability that Charlie's measurement succeeds depends on how much information about the state Alice sent is left in the qubit after Bob's measurement, and this information can be quantified by the overlap between the two possible states in which Bob's measurement leaves the qubit. This Letter is a first step toward developing a theory of nondestructive sequential quantum measurements, which could be useful in quantum communication schemes.
We consider two different optimized measurement strategies for the discrimination of nonorthogonal quantum states. The first is ambiguous discrimination with a minimum probability of inferring an erroneous result, and the second is unambiguous, i.e., error-free, discrimination with a minimum probability of getting an inconclusive outcome, where the measurement fails to give a definite answer. For distinguishing between two mixed quantum states, we investigate the relation between the minimum-error probability achievable in ambiguous discrimination, and the minimum failure probability that can be reached in unambiguous discrimination of the same two states. The latter turns out to be at least twice as large as the former for any two given states. As an example, we treat the case where the state of the quantum system is known to be, with arbitrary prior probability, either a given pure state, or a uniform statistical mixture of any number of mutually orthogonal states. For this case we derive an analytical result for the minimum probability of error and perform a quantitative comparison with the minimum failure probability.