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Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum

Authors:
  • Serán BioScience

Abstract

We have measured probability distributions of quadrature-field amplitude for both vacuum and quadrature-squeezed states of a mode of the electromagnetic field. From these measurements we demonstrate the technique of optical homodyne tomography to determine the Wigner distribution and the density matrix of the mode. This provides a complete quantum mechanical characterization of the measured mode.
... Naively, one may attempt to perform tomography of the state of the system. This amounts to making a sufficiently large number of different measurements on the system so that the density operator describing the state can be inferred [1][2][3][4][5]. ...
... Consider the case where the system consists of n qubits, corresponding to the total dimension of D = 2 n . In this case, shadow tomography can be performed by making (identical or not identical) generalised measurements {E (1) , E (2) , . . . , E (n) } on each of the qubit, each described by a collection of N i effects, ...
... Theoretically, this corresponds to a measurement of a generalised measurement E tot on the whole system with each effect labelled by a string of outcomes {k (1) , k (2) , . . . , k (n) }, ...
Preprint
Advances in quantum technology require scalable techniques to efficiently extract information from a quantum system, such as expectation values of observables or its entropy. Traditional tomography is limited to a handful of qubits and shadow tomography has been suggested as a scalable replacement for larger systems. Shadow tomography is conventionally analysed based on outcomes of ideal projective measurements on the system upon application of randomised unitaries. Here, we suggest that shadow tomography can be much more straightforwardly formulated for generalised measurements, or positive operator valued measures. Based on the idea of the least-square estimator, shadow tomography with generalised measurements is both more general and simpler than the traditional formulation with randomisation of unitaries. In particular, this formulation allows us to analyse theoretical aspects of shadow tomography in detail. For example, we provide a detailed study of the implication of symmetries in shadow tomography. Shadow tomography with generalised measurements is also indispensable in realistic implementation of quantum mechanical measurements, when noise is unavoidable. Moreover, we also demonstrate how the optimisation of measurements for shadow tomography tailored toward a particular set of observables can be carried out.
... In quantum information, Quantum Tomography refers to a set of methods for inferring the description of quantum systems [15]. Quantum State Tomography [16] and Quantum Process Tomography [17] refer to methods that aim to provide descriptions of quantum states and of quantum evolution processes, respectively. Hence, the natural approach to characterize errors in a quantum network is to use quantum tomography methods. ...
... with probabilities α(s) given by (17). By comparing (16) and (27), there is no gain in using GHZ states. This is intuitively understood by considering the fact that only n − 1 bits of the GHZ state are used to parameterize the necessary information, which is the same amount of bits used in the initial case. ...
Preprint
The fragile nature of quantum information makes it practically impossible to completely isolate a quantum state from noise under quantum channel transmissions. Quantum networks are complex systems formed by the interconnection of quantum processing devices through quantum channels. In this context, characterizing how channels introduce noise in transmitted quantum states is of paramount importance. Precise descriptions of the error distributions introduced by non-unitary quantum channels can inform quantum error correction protocols to tailor operations for the particular error model. In addition, characterizing such errors by monitoring the network with end-to-end measurements enables end-nodes to infer the status of network links. In this work, we address the end-to-end characterization of quantum channels in a quantum network by introducing the problem of Quantum Network Tomography. The solution for this problem is an estimator for the probabilities that define a Kraus decomposition for all quantum channels in the network, using measurements performed exclusively in the end-nodes. We study this problem in detail for the case of arbitrary star quantum networks with quantum channels described by a single Pauli operator, like bit-flip quantum channels. We provide solutions for such networks with polynomial sample complexity. Our solutions provide evidence that pre-shared entanglement brings advantages for estimation in terms of the identifiability of parameters.
... Measuring the density matrix of quantum states is therefore a fundamental task in a wide variety of quantum applications. The standard method for measuring the density matrix of quantum states is the quantum state tomography (QST) technique [1][2][3][4], which can be used to reconstruct the entire density matrix. As the dimension of the investigated quantum system's Hilbert space grows, the number of measurements for QST and the complexity of the reconstruction algorithm also increases, and the QST method becomes increasingly inefficient. ...
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A direct measurement protocol allows reconstructing specific elements of the density matrix of a quantum state without using quantum state tomography. Until now, most direct measurement protocols have relied on ancillary pointers, which add complexity and are not always easy to implement experimentally. In this paper, we experimentally validate a direct measurement protocol requiring no ancillary pointers that calculates the value of a specific element of a quantum state's density matrix using only six projection measurements. We prepare path-encoded arbitrary four-dimensional quantum states on a silicon-based quantum photonic chip and measure the density-matrix elements of the four-dimensional quantum states using this protocol, with experimental results deviating from ideal values by an average of only 0.010 ± 0.004. This approach has the potential to be applied to quantum information applications where only partial information about the quantum state needs to be extracted, for example, problems such as entanglement witnessing, fidelity estimation of quantum systems, and quantum coherence estimation.
... Thus, in the case where quantum uncertainty relations are expressed in terms of classical trajectories, the uncertainty relations provide the inequality conditions for the trajectories as, for the parametric oscillator with a specific time dependence of the frequency, they can give a particular condition for the solutions expressed in terms of the functions more complicated than in the case of free motion or standard vibrations with constant frequency. For optical tomogram [13,24] w α (X|θ, t) of the parametric oscillator states under consideration, we have an explicit expression; e.g., for the coherent state |α, t , the optical tomogram is the normal probability distribution of the quadrature, ...
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The states of quantum oscillator with time-dependent frequency are described by the tomographic probability distributions. The integrals of motion, being linear in the position and momentum operators, are used to construct the Gaussian squeezed and correlated states of the oscillator associated with normal probability distributions of the quadrature determining the density matrices of the states. The even and odd coherent states of the oscillator and their symplectic tomograms are given in terms of probability distributions. Considering free particle as a partial case of the oscillator with zero frequency, we find tomograms of even and odd coherent states of free particle in the probability representation.
... The main idea behind tomographic methods is to reconstruct the classical and/or quantum phase space probability distributions via direct measurements. Even though those methods were initially formulated for somewhat restricted cases [106,107,108,109], recent studies seem to be promising in terms of providing wider application areas [110,111,112,54]. We suggest that tomography techniques, combined with the construction presented here, can in principle be used to analyse systems within a time dependent thermodynamic setting. ...
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... In general, the detection of complexly structured quantum light is experimentally challenging. In particular, phasesensitive measurements typically require a well-defined, ex-ternal reference phase, such as provided by the local oscillator in balanced homodyne detection [26]. For polarization measurements, interference properties of the two polarisation components suffice to characterize the quantum state [16,17]. ...
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... This has been a typical problem and has been investigated by many authors. In quantum information experiments, quantum tomography has been also discussed [14][15][16][17]. As far as the author knows, these studies do not refer to the comparison of several models in terms of information quantity. ...
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