In the paper is discussed complete probabilistic description of quantum systems with application to multiqubit quantum computations. In simplest case it is a set of probabilities of transitions to some fixed set of states. The probabilities in the set may be represented linearly via coefficients of density matrix and it is very similar with description using mixed states, but also may give some alternative view on specific properties of quantum circuits due to possibility of direct comparison with classical statistical paradigm.
In this work, we provide a complete analysis to minimum-error discrimination of mixed four qubit states with arbitrary prior probabilities. For the complete anaysis, the most important work to do is to find the necessary and sufficient conditions for the existence of null measurement operator. From the geometric structure of qubit states, we obtain the analytic condition for deciding the existence of a null operator in minimum-error measurement for mixed four qubit states, which also gives the necessary and sufficient conditions for every optimal POVM to have non-zero elements. Using the condi…
The question of the generation of random mixed states is discussed, aiming for the computation of probabilistic characteristics of composite finite dimensional quantum systems. Particularly, we consider the generation of the random Hilbert-Schmidt and Bures ensembles of qubit and qutrit pairs and compute the corresponding probabilities to find a separable state among the states of a fixed rank.
The problems of Hadamard quantum coin flipping in n-trials and related generalized Fibonacci sequences of numbers were introduced in [1]. It was shown that for an arbitrary number of repeated consecutive states, probabilities are determined by Fibonacci numbers for duplicated states, Tribonacci numbers for triplicated states and N-Bonacci numbers for arbitrary N-plicated states. In the present paper we generalize these results for direct product of multiple qubit states and arbitrary position of repeated states. The calculations are based on structure of Fibonacci trees in space of qubit state…
For three given probability distributions describing positions of three classical coins the quantum density matrix of spin-$1/2$ state is constructed and its matrix elements are associated with triada of Malevich's squares. The superposition principle of spin-$1/2$ states is presented in the form of nonlinear addition rule for these classical coin probabilities. We illustrate the formulas by the statement"God does not play dice - God plays coins".
Bures/statistical distinguishability probabilities of triseparable and biseparable Eggeling-Werner States
在以前的一些研究中，我们已经研究了使用Bures（最小单调）度量的体积元素-同样，统计可扩展性（SD）度量的四分之一-作为n x n密度矩阵的（n^2-1）维凸集上的自然度量。这使得我们在n = 4和6的情况下估计了量子位-量子位和量子位-四元组对可分离的先验（Bures/SD）概率。在这里，我们将这项工作从这样的二分系统扩展到最近由Eggeling和Werner（Phys. Rev. A 63 [2001]，042324）构造的具有U × U × U对称性的三分“实验室”量子系统。我们推导出相关的SD度量张量的三个量子比特和三个qutrit的情况下，然后使用Monte Carlo方法获得各种相关的Bures/SD概率的估计。
We analyze optimal schemes and also propose some practical schemes for the nonlocal conversion from two shared Bell pairs to four-qubit entangled states in optical quantum networks. In the analysis, we consider two-qubit operations as nonlocal operations and minimize the number of access to ancillary qubits as possible. First, we consider two-qubit unitary operations without using ancillary qubits and derive a necessary and sufficient condition for convertible states. Second, we consider nonlocal optical systems composed of passive linear optics and postselection. For the passive linear optica…
We derive a general expression for standard tensor norm of $N$-body correlation tensors for $N$-qubit complete graph states. With the help of this expression, we formulate a separability criterion that identifies non-$k$-separability of a class of $N$-qubit complete graph states, including GHZ state. We illustrate the performance of our criterion by considering the $N$-qubit complete graph states added with colored product noise. We also demonstrate that few local measurement settings are sufficient to evaluate our criterion.
Qubit-Qubit and Qubit-Qutrit Separability Functions and Probabilities
We list in increasing order -- 1/3, 3/8, 2/5, 135 pi/1024, 16/(3 pi^2), 3 pi/16, 5/8, 105 pi/512, 2 - 435 pi/1024, 11/16, 1 -- a number of exact two-qubit Hilbert-Schmidt (HS) separability probabilities, we are able to compute. Each probability corresponds to a specific scenario -- a class of 4 x 4 density matrices (rho) with an m-subset (m<6) of its six off-diagonal pairs of symmetrically located entries set to zero. Initially, we consider only scenarios in which the (6-m) non-nullified pairs are real, but then permit them to be complex (as well as quaternionic) in nature also. The general an…
We consider tensor-network stabilizer codes and show that their tensor-network decoder has the property that independent logical qubits can be decoded in parallel. As long as the error rate is below threshold, we show that this parallel decoder is essentially optimal. As an application, we verify this for the max-rate holographic Steane (heptagon) code. For holographic codes this tensor-network decoder was shown to be efficient with complexity polynomial in n, the number of physical qubits. Here we show that, by using the parallel decoding scheme, the complexity is also linear in k, the number…
The problem of Hadamard quantum coin measurement in $n$ trials, with arbitrary number of repeated consecutive last states is formulated in terms of Fibonacci sequences for duplicated states, Tribonacci numbers for triplicated states and $N$-Bonacci numbers for arbitrary $N$-plicated states. The probability formulas for arbitrary position of repeated states are derived in terms of Lucas and Fibonacci numbers. For generic qubit coin, the formulas are expressed by Fibonacci and more general, $N$-Bonacci polynomials in qubit probabilities. The generating function for probabilities, the Golden Rati…
We develop a unitary dependence theory to characterize the behaviors of quantum circuits and states in terms of how quantum gates manipulate qubits and determine their measurement probabilities. A qubit has dependence on a 1-qubit unitary gate if its measurement probabilities depend on the parameters of the gate. A 1-qubit unitary creates such a dependence onto the target qubit, and a CNOT gate copies all the dependences from the control qubit to the target qubit. The complete dependence picture of the output state details the connections qubits may have when being manipulated or measured. Com…
We show that the tensor rank of tensor product of two three-qubit W states is not less than eight. Combining this result with the recent result of M. Christandl, A. K. Jensen, and J. Zuiddam that the tensor rank of tensor product of two three-qubit W states is at most eight, we deduce that the tensor rank of tensor product of two three-qubit W states is eight. We also construct the upper bound of the tensor rank of tensor product of many three-qubit W states.
We investigate the tunable quantum entanglement and the probabilities of excitations in a system of three qubits in a non-stationary cavity due to the dynamical Lamb effect, caused by non-adiabatic fast change of the boundary conditions of the cavity. The transition amplitudes and the probabilities of excitation of qubits due to the dynamical Lamb effect have been evaluated. The conditional concurrence and the conditional residual tangle for each fixed amount of created photons are introduced and calculated as measures of the pairwise or three-way dynamical quantum entanglement of the qubits. …
The CANDECOMP/PARAFAC (CP) decomposition is a generalization of the spectral decomposition of matrices to higher-order tensors. In this paper we use the CP decomposition to study unitary equivalence of higher order tensors and construct several invariants of local unitary equivalence for general higher order tensors. Based on this new method, we study the coefficient tensors of $3$-qubit states and obtain a necessary and sufficient criterion for local unitary equivalence of general tripartite states in terms of the CP decomposition. We also generalize this method to obtain some invariants of l…
Quantum process tomography (QPT), where a quantum channel is reconstructed through the analysis of repeated quantum measurements, is an important tool for validating the operation of a quantum processor. We detail the combined use of an existing QPT approach based on tensor networks (TNs) and unsupervised learning with TN contraction path finding algorithms in order to use TNs of arbitrary topologies for reconstruction. Experiments were conducted on the 7-qubit IBM Quantum Falcon Processor ibmq_casablanca, where we demonstrate this technique by matching the topology of the tensor networks used…
Fundamental questions in chemistry and physics may never be answered due to the exponential complexity of the underlying quantum phenomena. A desire to overcome this challenge has sparked a new industry of quantum technologies with the promise that engineered quantum systems can address these hard problems. A key step towards demonstrating such a system will be performing a computation beyond the capabilities of any classical computer, achieving so-called quantum supremacy. Here, using 9 superconducting qubits, we demonstrate an immediate path towards quantum supremacy. By individually tuning …
We express the matrix elements of the density matrix of the qutrit state in terms of probabilities associated with artificial qubit states. We show that the quantum statistics of qubit states and observables is formally equivalent to the statistics of classical systems with three random vector variables and three classical probability distributions obeying special constrains found in this study. The Bloch spheres geometry of qubit states is mapped onto triangle geometry of qubits. We investigate the triada of Malevich's squares describing the qubit states in quantum suprematism picture and the…
In [Science 340, 1205, 7 June (2013)], via polytopes Michael Walter et al. proposed a sufficient condition detecting the genuinely entangled pure states. In this paper, we indicate that generally, the coefficient vector of a pure product state of $n$ qubits cannot be decomposed into a tensor product of two vectors, and show that a pure state of $n$ qubits is a product state if and only if there exists a permutation of qubits such that under the permutation, its coefficient vector arranged in ascending lexicographical order can be decomposed into a tensor product of two vectors. The contraposit…
We consider the problem of minimum-error quantum state discrimination for single-qubit mixed states. We present a method which uses the Helstrom conditions constructively and analytically; this algebraic approach is complementary to existing geometric methods, and solves the problem for any number of arbitrary signal states with arbitrary prior probabilities.
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 …
Great advances have been achieved in studying characteristics of entanglement for fundamentals of quantum mechanics and quantum information processing. However, even for N-qubit systems, the problem of entanglement criterion has not been well solved. In this Letter, using the method of state decomposition and high order singular value decomposition (HOSVD), we propose a necessary and sufficient entanglement criterion for general N-qubit systems. As an example, we apply our method to study the multi-qubit W state with white noise. We not only obtain the separability critical point, which is tig…
We analyze positivity of a tensor product of two linear qubit maps, $Φ_1 \otimes Φ_2$. Positivity of maps $Φ_1$ and $Φ_2$ is a necessary but not a sufficient condition for positivity of $Φ_1 \otimes Φ_2$. We find a non-trivial sufficient condition for positivity of the tensor product map beyond the cases when both $Φ_1$ and $Φ_2$ are completely positive or completely co-positive. We find necessary and (separately) sufficient conditions for $n$-tensor-stable positive qubit maps, i.e. such qubit maps $Φ$ that $Φ^{\otimes n}$ is positive. Particular cases of 2- and 3-tensor-stable positive qubit …
We detect a certain pattern of behavior of separability probabilities $p(r_A,r_B)$ for two-qubit systems endowed with Hilbert-Schmidt, and more generally, random induced measures, where $r_A$ and $r_B$ are the Bloch radii ($0 \leq r_A,r_B \leq 1$) of the qubit reduced states ($A,B$). We observe a relative repulsion of radii effect, that is $p(r_A,r_A) < p(r_A,1-r_A)$, except for rather narrow "crossover" intervals $[\tilde{r}_A,\frac{1}{2}]$. Among the seven specific cases we study are, firstly, the "toy" seven-dimensional $X$-states model and, then, the fifteen-dimensional two-qubit states ob…
Milz and Strunz ({\it J. Phys. A}: {\bf{48}} [2015] 035306) recently studied the probabilities that two-qubit and qubit-qutrit states, randomly generated with respect to Hilbert-Schmidt (Euclidean/flat) measure, are separable. They concluded that in both cases, the separability probabilities (apparently exactly $\frac{8}{33}$ in the two-qubit scenario) hold {\it constant} over the Bloch radii ($r$) of the single-qubit subsystems, jumping to 1 at the pure state boundaries ($r=1$). Here, firstly, we present evidence that in the qubit-qutrit case, the separability probability is uniformly distrib…
We treat 3-qubits states with maximally disordered subsystems, by using Hilbert-Schmidt decompositions.By using unfolding methods, the tensors are converted into matrices and by applying singular values decompositions to these matrices the number 27 of the Hilbert Schmidt parameters, is reduced to 9 and under the condition that the sum of absolute values of these parameters is not larger than 1, we conclude that the density matrix is fully separable . In another method we divide the 27 Hilbert Schmidt parameters into 9 triads where for each triad we calculate the Frobenius norm of 3 Hilbert-Sc…
Within context of quantum logic, it is possible to assign dispersion-free probabilities to experimental propositions pertaining to qubits. This makes qubits distinct from the rest of quantum systems since the latter do not admit probabilities having only values 0 and 1. The present paper shows that erasing qubit discrimination leads to a model of computation which permits execution of many primitive operations in a massive parallel way. In the paper, it is demonstrated that such a model (that can be called a quantum parallel random-access machine, QPRAM) is quantum mechanically plausible.
We consider the problem of discriminating qubit states that are sent over a quantum channel and derive a necessary and sufficient condition for an optimal measurement to be preserved by the channel. We apply the result to the characterization of optimal measurement preserving (OMP) channels for a given qubit ensemble, e.g., a set of two states or a set of multiple qubit states with equal a priori probabilities. Conversely, we also characterize qubit ensembles for which a given channel is OMP, such as unitary and depolarization channels. Finally, we show how the sets of OMP channels for a given…
