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Exploiting the higher-order statistics of random-coefficient pure states for quantum information processing

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Abstract

Quantum mechanics and hence quantum information processing methods widely use two types of states, namely (deterministic-coefficient) pure states and statistical mixtures. Density operators can be associated with them. We here address a third type of states, whose ket coefficients are random variables, as opposed to the deterministic coefficients of usual pure states. We therefore call them Random-Coefficient Pure States, or RCPS. We define physical setups that yield RCPS. We analyze the properties of RCPS and show that they contain much richer information than the density operator and mean of observables that we associate with them , because that operator only exploits the second-order statistics of the random state coefficients, whereas their higher-order statistics contain additional information. That information can be accessed in practice with the multiple-preparation procedure that we propose for RCPS, by using second-order and higher-order statistics of associated random probabilities of measurement outcomes. Exploiting these higher-order statistics yields a very general approach to advanced quantum information processing. We illustrate its relevance with a generic quantum parameter estimation problem related to quantum process tomography , especially considering its blind/unsupervised version. We show that this problem cannot be solved by using only the density operator \( \rho \) of an RCPS and the associated mean value \( Tr( \rho \hat{A} ) \) of the operator \( \hat{A} \) corresponding to the considered physical quantity. We solve it by exploiting a fourth-order statistical parameter of state coefficients, in addition to second-order statistics. Numerical tests validate this result and show that the proposed method yields accurate parameter estimation for the considered number of state preparations.

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Notes

  1. The complete class of moments that may be introduced for a given set of ket coefficients \({\textbf {c}}_{{\textbf {k}}}\) is defined as the expectation of an arbitrary product of factors, where each factor is freely selected to be either a coefficient \({\textbf {c}}_{{\textbf {k}}}\) or its conjugate (and both forms may appear for any given \({\textbf {c}}_{{\textbf {k}}}\)). Here, we only consider a subset of these moments. This is due to the fact that we start from the probabilities \({\textbf {p}}_{{\textbf {k}}}\) (because they are the quantities that we access with measurements), not the coefficients \({\textbf {c}}_{{\textbf {k}}}\), and we then build the quantities (19). If expressing these quantities with respect to ket coefficients, every factor \({\textbf {c}}_{{\textbf {k}}}\) in (19) is constrained to appear together with its conjugate, because (7) shows that \({\textbf {p}}_{{\textbf {k}}}\) is the product of \({\textbf {c}}_{{\textbf {k}}}\) and its conjugate.

  2. Our previous investigations related to BQSS, BQPT and BHPE involve a quantum process. The above-mentioned papers [1,2,3,4] (Section 1.7.2), [6] directly use statistical parameters of the probabilities \({\textbf {p}}_{{\textbf {k}}}\) of measurements performed at the output of that process. In contrast, other investigations, dealing with BQSS, first use the individual values of these classical-form data \({\textbf {p}}_{{\textbf {k}}}\) as the input of a classical processing system, called the separating system. The outputs of that system aim at restoring modulus parameters and combinations of phase parameters of coefficients of several single-qubit quantum states. These parameters thus have some relationships with the quantities \({\textbf {r}}_{{\textbf {k}}}\) and \({\varvec{\phi }_{{\textbf {k}}}}\) in (10). These BQSS methods are based on various statistical parameters of the outputs of the separating system: their generalized moments are used in Section 1.7.3 of [4] and their cumulants in [29], whereas their whole pdf are exploited through their mutual information (see Section 1.5 of [4]), with a connection with the maximum likelihood approach (see Section 1.6 of [4]). All these approaches thus have an indirect link with the higher-order statistics of the \({\textbf {r}}_{{\textbf {k}}}\) and \({\varvec{\phi }_{{\textbf {k}}}}\) parameters, and hence with those of \({\textbf {p}}_{{\textbf {k}}}\).

  3. Although the mean of an observable is here intentionally analyzed without resorting to the content of Sect. 2.3, these two parts of this paper are clearly connected, because (21) is nothing but the quantity \( \textrm{Tr}(\rho \) Â) defined in Sect. 2.3 for an RCPS, and the discussion provided after (21) therefore has connections with the comments we made in Sect. 2.3, mainly about the density operator \(\rho \) of an RCPS and partly about the resulting \( \textrm{Tr}(\rho \) Â).

  4. It should however be noted that using the above function \(g\) has a possibly attractive effect: (23) allows one to access a different linear combination of the probabilities \({\textbf {p}}_{{\textbf {k}}}\) (and cross-terms \( E \{ {{\textbf {c}}_{{\textbf {k}}}} ^* {{\textbf {c}}_{\varvec{\ell }}} \} \)) than (21). Jointly considering (23) for various functions \(g\) and solving the corresponding equations might therefore provide a way to separately estimate the expectation of each probability \({\textbf {p}}_{{\textbf {k}}}\). Anyway, it then remains that: 1) our approach directly based on these probabilities \({\textbf {p}}_{{\textbf {k}}}\) also makes it possible to estimate their expectations and that without having to create and solve the above equations, and 2) the approach based on the mean of observables and of function of observables only accesses these expectations of \({\textbf {p}}_{{\textbf {k}}}\) (and the other second-order parameters \( E \{ {{\textbf {c}}_{{\textbf {k}}}}^*{{\textbf {c}}_{\varvec{\ell }}} \} \) of the ket coefficients), not their other statistics, unlike our approach. Our approach directly based on (all the statistics of) the probabilities \({\textbf {p}}_{{\textbf {k}}}\) therefore remains of much higher interest.

  5. The mean of a function of an observable was not explicitly addressed in Sect. 2.3 and was therefore independently detailed in the present section. However, its connection with Sect. 2.3 may be shown as follows. \( G = g( A) \) is nothing but another observable, with an associated operator Ĝ. Equation (23) defines the mean \( E \{ G \} _{|\varvec{\psi }\rangle } \) of that new observable, that could also be expressed as \( \textrm{Tr}(\rho \) Ĝ) and that therefore has the limitations that we defined for \( E \{ A\} _{|\varvec{\psi }\rangle } = \textrm{Tr}(\rho \) Â) in Sect. 2.3 and at the beginning of the present section, when considering an arbitrary observable \(A\).

  6. See also [12] p. 398 for the other earliest references.

  7. One may expect that higher performance can be obtained by also considering other types of measurements, but this is true for both methods and our goal here is not to derive their ultimate performance depending on the considered measurements but to compare their capabilities for a given, relevant, type of measurements.

  8. Using the variance of \( {\varvec{ p_{0}}} \) instead would be equivalent, as shown by (18).

  9. Classical Blind Source Separation (BSS) methods are sometimes stated to be “semi-blind”, rather than “blind”, because they require some prior knowledge about the source signals to be separated, e.g. these signals may be requested to be statistically independent. That term “semi-blind” is especially used for methods that are more constraining concerning that prior knowledge, e.g. methods that constrain some source moments to be known or to belong to known intervals in addition to requesting source independence. From that point of view, the basic version of the quantum estimation method proposed hereafter might be stated to be “semi-blind” because, in addition to requesting \({{\textbf {r}}}\) and \( \varvec{ \phi } \) to be statistically independent, it uses additional constraints on the marginal statistics of \({{\textbf {r}}}\) and \( \varvec{ \phi } \), as detailed in Sect. 3.3 (in fact, the proposed quantum estimation method does not require one to know all the statistical distributions of \({{\textbf {r}}}\) and \( \varvec{ \phi } \) but only the resulting parameters defined in (40)–(45)). Anyway, it remains that this proposed quantum estimation method does not require the individual values of the input to be known, which is the main feature of blind and associated methods.

  10. In contrast, the non-blind counterpart of the blind method detailed in this paper operates by performing measurements for (copies of) realizations of the input state \(|\varvec{\psi _{in}}\rangle \) of the considered process and then deriving sample statistics for the required statistical parameters of \({{\textbf {r}}}\) and \( \varvec{ \phi } \).

  11. One may also wonder whether continuous RCPS yield different properties than discrete ones.

  12. The terminology in [50,51,52] should be correctly interpreted with respect to the present paper and to other papers from the literature. What is explicitly called a quantum state in these papers [50,51,52] is restricted to a usual von Neumann mixed state, completely described by its density operator. More precisely, [50] states that it considers “ the convex subset of density operators i.e. positive operators with unit trace, also called quantum states ”. Similarly, [51] states that it deals with “density operators (quantum states)” and that “\(\rho \) is a density operator, this is called the quantum characteristic function of the state \(\rho \)”. Finally, [52] states: “A quantum state of the system is given by a density operator \(\rho \)”. These three papers then consider what they call an “ensemble” or “quantum ensemble”, defined by a probability density function or probability measure over a family of such density operators. As compared with the general concept of various types of “random quantum states” considered in the present paper and in papers from the literature cited in the present section, the concept from [50,51,52] that should be considered as what we call an overall type of “random quantum state” is their “ensemble” (of what they call “quantum states”, i.e. density operators).

  13. More precisely, the above sentence in [54], just before its Eq. (19), suggests that the ket coefficients themselves have a Gaussian distribution, which cannot be an accurate model of actual behavior: the modulus of a ket coefficient is upper bounded by one, so that this coefficient cannot have an unbounded Gaussian density . However, in a private communication, the first author of [54], S. Kumar, explained that the above sentence of [54] might be misleading: a normalization is moreover used and the Gaussian distribution in fact applies to the matrix X that appears in Eq. (1) of [54].

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  1. Yannick Deville and Alain Deville contributed equally to this work.

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  1. IRAP (Institut de Recherche en Astrophysique et Planétologie), Université de Toulouse, UPS, CNRS, CNES, OMP, 31400, Toulouse, France

    Yannick Deville

  2. IM2NP UMR 7334, Aix-Marseille Université, CNRS, 13397, Marseille, France

    Alain Deville

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Deville, Y., Deville, A. Exploiting the higher-order statistics of random-coefficient pure states for quantum information processing. Quantum Inf Process 22, 216 (2023). https://doi.org/10.1007/s11128-023-03970-x

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