In this paper, we demonstrate that higher order singular value decomposition (HOSVD) can be used to identify special states in three qubits by local unitary (LU) operations. Since the matrix unfoldings of three qubits are related to their reduced density matrices, HOSVD simultaneously diagonalizes the one-body reduced density matrices of three qubits. From the all-orthogonality conditions of HOSVD, we computed the special states of three qubits. Furthermore, we showed that it is possible to construct a polytope that encapsulates all the special states of three qubits by LU operations with HOSVD.
We propose channel matrices by using unfolding matrices from their reduced density matrices. These channel matrices can be a criterion for a channel whether the channel can teleport or not any qubit state. We consider a special case, teleportation of the arbitrary two-qubit state by using the four-qubit channel. The four-qubit channel can only teleport if the rank of the related channel matrix is four.
Explicitly separable density matrices are constructed for all separable two-qubits states based on Hilbert-Schmidt (HS) decompositions. For density matrices which include only two-qubits correlations the number of HS parameters is reduced to 3 by using local rotations, and for two-qubits states which include single qubit measurements, the number of parameters is reduced to 4 by local Lorentz transformations. For both cases we related the absolute values of the HS parameters to probabilities, and the outer products of various Pauli matrices were transformed to pure states density matrices produ…
The generalized n-qubit Greenberger-Horne-Zeilinger (GHZ) states and their local unitary equivalents are the only states of n qubits that are not uniquely determined among pure states by their reduced density matrices of n-1 qubits. Thus, among pure states, the generalized GHZ states are the only ones containing information at the n-party level. We point out a connection between local unitary stabilizer subgroups and the property of being determined by reduced density matrices.
We construct $\ell $-spin-flipping matrices from the coefficient matrices of pure states of $n$ qubits and show that the $\ell $-spin-flipping matrices are congruent and unitary congruent whenever two pure states of $n$ qubits are SLOCC and LU equivalent, respectively. The congruence implies the invariance of ranks of the $\ell $-spin-flipping matrices under SLOCC and then permits a reduction of SLOCC classification of n qubits to calculation of ranks of the $\ell $-spin-flipping matrices. The unitary congruence implies the invariance of singular values of the $\ell $-spin-flipping matrices un…
Hilbert-Schmidt (HS) decompositions and Frobenius norms are used to analyze biseparability of 3-qubit systems, with particular emphasis on density matrices with maximally disordered subsystems (MDS) and on the W state mixed with white noise. The biseparable form of a MDS density matrix is obtained by using the Bell states of a 2-qubit subsystem, multiplied by density matrices of the third qubit, which include the relevant HS parameters. Using our methods a sufficient condition and explicit biseparability of the W state mixed with white noise are given. They are compared with the sufficient con…
We show that higher order inter-group covariances involving even number of qubits are necessarily positive semidefinite for N qubit separable states, which are completely symmetric under permutations of the qubits. This identification leads to a family of sufficient conditions of inseparability based on the negativity of 2kth order inter-group covariance matrices of symmetric N-qubit systems. These conditions have a simple structure and detect entanglement in all even partitions of the symmetric multiqubit system. The observables involved are feasible experimental quantities and do not demand …
Explicit separability of general two qubits density matrices is related to Lorentz transformations. We use the 4-dimensional form R(u,v=0,1,2,3) of the Hilbert-Schmidt (HS) decomposition of the density matrix. For the generic case in which Lorentz transformations diagonalize R(u,v=0,1,2,3) (into s(0),s(1),s(2),s(3)) we give relations between the R parameters and the s parameters. In particular we consider two cases: a) Two qubits density matrices with one pair of linear terms in the HS decomposition. b) Two qubits density matrices with two or three symmetric pairs of linear terms. Some of the …
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…
We find an algebraic formula for the N-partite concurrence of N qubits in an X-matrix. X- matricies are density matrices whose only non-zero elements are diagonal or anti-diagonal when written in an orthonormal basis. We use our formula to study the dynamics of the N-partite entanglement of N remote qubits in generalized N-party Greenberger-Horne-Zeilinger (GHZ) states. We study the case when each qubit interacts with a partner harmonic oscillator. It is shown that only one type of GHZ state is prone to entanglement sudden death; for the rest, N-partite entanglement dies out momentarily. Algeb…
The ability to simulate one Hamiltonian with another is an important primitive in quantum information processing. In this paper, a simulation method for arbitrary $σ_z \otimes σ_z$ interaction based on Hadamard matrices (quant-ph/9904100) is generalized for any pairwise interaction. We describe two applications of the generalized framework. First, we obtain a class of protocols for selecting an arbitrary interaction term in an n-qubit Hamiltonian. This class includes the scheme given in quant-ph/0106064v2. Second, we obtain a class of protocols for inverting an arbitrary, possibly unknown n-qu…
Random numbers are critical for any cryptographic application. However, the data that is flowing through the internet is not secure because of entropy deprived pseudo random number generators and unencrypted IoTs. In this work, we address the issue of lesser entropy of several data formats. Specifically, we use the large information space associated with the n-qubit permutation matrices to expand the entropy of any data without increasing the size of the data. We take English text with the entropy in the range 4 - 5 bits per byte. We manipulate the data using a set of n-qubit (n $\leq$ 10) per…
Explicit separable density matrices, for mixed two qubits states, are derived by the use of Hilbert Schmidt decompositions and Peres Horodecki criterion. A strongly separable two qubits mixed state is defined by multiplications of two density matrices given with pure states while weakly separable two qubits state is defined by multiplications of two density matrices which includes non-pure states. We find the sufficient and necessary condition for separability of two-qubits density matrices and show that under this condition the two-qubit density matrices are strongly separable.
We pursue a number of analytical directions, motivated to some extent initially by the possibility of developing a methodology for formally proving or disproving a certain conjecture of quantum-theoretical relevance (quant-ph/0308037). It asserts that the 15-dimensional volume occupied by the separable two-qubit density matrices is (\sqrt{2}-1)/3, as measured in terms of the statistical distinguishability metric (four times the Bures or minimal monotone metric). Somewhat disappointingly, however, the several various analyses that we report, though we hope of independent/autonomous interest, ap…
This paper is a continuation of our previous paper [8], in which we have studied the dynamics of quantum correlations of two qubits embedded each into its own disordered multiconnected environment. We modeled the environment by random matrices of large size allowing for a possibility to describe meso- and even nanoenvironments. In this paper we also study the dynamics of quantum correlations of two qubits but embedded into a common environment which we also model by random matrices of large size. We obtain the large size limit of the reduced density matrix of two qubits. We then use an analog …
We establish a novel procedure to analyze the entanglement properties of extremal density matrices depending on the parameters of a finite dimensional Hamiltonian. It was applied to a general 2-qubit Hamiltonian which could exhibit Kramers degeneracy. This is done through the extremal density matrix formalism, which allows to extend the conventional variational principle to mixed states. By applying the positive partial transpose criterion in terms of the Correlation and Schlienz-Mahler matrices on the extremal density matrices, we demonstrate that it is possible to reach both pure and mixed e…
Hilbert-Schmidt (HS) decompositions are employed for analyzing systems of n-qubits, and a qubit with a qudit. Negative eigenvalues, obtained by partial-transpose (PT) plus local unitary transformations (PTU) for one qubit from the whole system, are used for indicating inseparability. A sufficient criterion for full separability of the n-qubits and qubit-qudit systems is given. We use the singular value decomposition (SVD) for improving the criterion for full separability. General properties of entanglement and separability are analyzed for a system of a qubit and a qudit and n-qubits systems, …
New inequalities for tomographic probability distributions and density matrices of qutrit states are obtained by means of generalization of qubit portrait method. The approach based on the qudit portrait method to get new entropic inequalities is proposed. It can be applied to the case of arbutrary nonnegative hermitian matrices including the density matrices of multipartite qudit states.
We construct coefficient matrices of size 2^l by 2^{n-l} associated with pure n-qubit states and prove the invariance of the ranks of the coefficient matrices under stochastic local operations and classical communication (SLOCC). The ranks give rise to a simple way of partitioning pure n-qubit states into inequivalent families and distinguishing degenerate families from one another under SLOCC. Moreover, the classification scheme via the ranks of coefficient matrices can be combined with other schemes to build a more refined classification scheme. To exemplify we classify the nine families of …
We propose a sufficient criterion ${S}=λ_1+λ_2-(λ_1-λ_2)^2<0$ to detect Einstein-Podolsky-Rosen steering for arbitrary two-qubit density matrix $ρ_{AB}$. Here $λ_1,λ_2$ are respectively the minimal and the second minimal eigenvalues of $ρ^{T_B}_{AB}$, which is the partial transpose of $ρ_{AB}$. By investigating several typical two-qubit states such as the isotropic state, Bell-diagonal state, maximally entangled mixed state, etc., we show this criterion works efficiently and can make reasonable predictions for steerability. We also present a mixed state of which steerability always exists, and…
We study the Chern-Simons approach to the topological quantum computing. We use quantum $\mathcal{R}$-matrices as universal quantum gates and study the approximations of some one-qubit operations. We make some modifications to the known Solovay-Kitaev algorithm suitable for our particular problem.
All mixed states of two qubits can be brought into normal form by the action of SLOCC operations of the kind $ρ'=(A\otimes B)ρ(A\otimes B)^\dagger$. These normal forms can be obtained by considering a Lorentz singular value decomposition on a real parameterization of the density matrix. We show that the Lorentz singular values are variationally defined and give rise to entanglement monotones, with as a special case the concurrence. Next a necessary and sufficient criterion is conjectured for a mixed state to be convertible into another specific one with a non-zero probability. Finally the form…
Quantum entanglement is an enigmatic and powerful property that has attracted much attention due to its usefulness in new ways of communications, like quantum teleportation and quantum key distribution. Much effort has been done to quantify entanglement. Indeed, there exist some well-established separability criterion and analytical formulas for the entanglement of bipartite systems. In some of these, the crucial elements are the eigenvalues of the partial transpose of the density matrix. In this paper, we show that one can also have information about the entanglement of bipartite state, in C2…
We derive necessary and sufficient conditions for local unitary (LU) operators to leave invariant the set of 1-qubit reduced density matrices of a multi-qubit state. LU operators with this property are tensor products of {\it cyclic local} operators, and form a subgroup, the centralizer subgroup of the set of reduced states, of the Lie group $SU(2)^{\otimes n}$. The dimension of this subgroup depends on the type of reduced density matrices. It is maximum when all reduced states are maximally mixed and it is minimum when none of them is maximally mixed. For any given multi-qubit state, pure or …
In this work, we present a method to exponentiate non-sparse indefinite low-rank matrices on a quantum computer. Given an operation for accessing the elements of the matrix, our method allows singular values and associated singular vectors to be found quantum mechanically in a time exponentially faster in the dimension of the matrix than known classical algorithms. The method extends to non-Hermitian and non-square matrices via embedding matrices. In the context of the generic singular value decomposition of a matrix, we discuss the Procrustes problem of finding a closest isometry to a given m…
We investigate the proportional relationships for spectrums and for SJNFs (Standard Jordan Normal Forms) of the matrices constructed from coefficient matrices of two SLOCC (stochastic local operations and classical communication) equivalent states of $n$ qubits. Invoking the proportional relationships for spectrums and for SJNFs, pure states of $n$ ($\geq 4$) qubits are partitioned into 12 groups and 34 families under SLOCC, respectively. Specially, it is true for four qubits.
By focusing on the X-matrix part of a density matrix of two qubits we provide an algebraic lower bound for the concurrence. The lower bound is generalized for cases beyond two qubits and can serve as a sufficient condition for non-separability for bipartite density matrices of arbitrary dimension. Experimentally, our lower bound can be used to confirm non-separability without performing a complete state tomography.
The nonnegativity of the determinant of the partial transpose of a two-qubit (4 x 4) density matrix is both a necessary and sufficient condition for its separability. While the determinant is restricted to the interval [0,1/256], the determinant of the partial transpose can range over [-1/16,1/256], with negative values corresponding to entangled states. We report here the exact values of the first nine moments of the probability distribution of the partial transpose over this interval, with respect to the Hilbert-Schmidt (metric volume element) measure on the nine-dimensional convex set of re…
By using the quantum Yang-Baxterization approach to the three different Hamiltonians, we investigate the behavior of the quantum Fisher information (QFI) under the actions of these Hamiltonians on the different two-qubit input states and by estimating the meaningful parameter $φ$. We address the overall estimation properties by evaluating the QFI for the whole system undergone different unitary evolution. The results show that the behavior of the QFI depends on the choice of the initial states. Choosing the optimal input states can improve the precision of quantum parameter estimation. On the …
We derive necessary and sufficient conditions for the LU-equivalence of two general (pure or mixed) $n$-qubit states as well as we determine the local unitary operators connecting them. Almost all relevant information is contained in the $1$-qubit reduced matrices of the multiqubit states under investigation Our technique relies on identifying {\it ab initio} all local symmetries and the corresponding local cyclic unitary operators. To derive the above conditions we use the reference forms of the multiqubit states whose definition requires the diagonalization of the 1-qubit reduced matrices. B…
Hypothesis testing is a fundamental issue in statistical inference and has been a crucial element in the development of information sciences. The Chernoff bound gives the minimal Bayesian error probability when discriminating two hypotheses given a large number of observations. Recently the combined work of Audenaert et al. [Phys. Rev. Lett. 98, 160501] and Nussbaum and Szkola [quant-ph/0607216] has proved the quantum analog of this bound, which applies when the hypotheses correspond to two quantum states. Based on the quantum Chernoff bound, we define a physically meaningful distinguishabilit…
Suppose we have an n-qubit system, and we are given a collection of local density matrices rho_1,...,rho_m, where each rho_i describes some subset of the qubits. We say that rho_1,...,rho_m are "consistent" if there exists a global state sigma (on all n qubits) whose reduced density matrices match rho_1,...,rho_m. We prove the following result: if rho_1,...,rho_m are consistent with some state sigma > 0, then they are also consistent with a state sigma' of the form sigma' = (1/Z) exp(M_1+...+M_m), where each M_i is a Hermitian matrix acting on the same qubits as rho_i, and Z is a normalizing f…
A general scheme is presented to decompose a $d$-by-$d$ unitary matrix as the product of two-level unitary matrices with additional structure and prescribed determinants. In particular, the decomposition can be done by using two-level matrices in $d-1$ classes, where each class is isomorphic to the group of $2\times 2$ unitary matrices. The proposed scheme is easy to apply, and useful in treating problems with the additional structural restrictions. A Matlab program is written to implement the scheme, and the result is used to deduce the fact that every quantum gate acting on $n$-qubit registe…
Two-qubit X-matrices have been the subject of considerable recent attention, as they lend themselves more readily to analytical investigations than two-qubit density matrices of arbitrary nature. Here, we maximally exploit this relative ease of analysis to formally derive an exhaustive collection of results pertaining to the separability probabilities of generalized two-qubit X-matrices endowed with Hilbert-Schmidt and, more broadly, induced measures. Further, the analytical results obtained exhibit interesting parallels to corresponding earlier (but, contrastingly, not yet fully rigorous) res…
In the Bloch sphere picture, one finds the coefficients for expanding a single-qubit density operator in terms of the identity and Pauli matrices. A generalization to $n$ qubits via tensor products represents a density operator by a real vector of length $4^n$, conceptually similar to a statevector. Here, we study this approach for the purpose of quantum circuit simulation, including noise processes. The tensor structure leads to computationally efficient algorithms for applying circuit gates and performing few-qubit quantum operations. In view of variational circuit optimization, we study ``b…
We theoretically study operations with a four-level superconducting circuit as a two-qubit system. Using a mapping on a two-qubit system, we show how to implement iSWAP gates and Hadamard gates through pulses on transitions between particular pairs of energy levels. Our approach allows one to prepare pure two-qubit entangled states with desired form of reduced density matrices of the same purity and, in particular, arbitrary identical reduced states of qubits. We propose using schemes for the Hadamard gate and two-qubit entangled states with identical reduced density matrices in order to verif…
We consider the problems of maximizing the entanglement negativity of X-form qubit-qutrit density matrices with (i) a fixed spectrum and (ii) a fixed purity. In the first case, the problem is solved in full generality whereas, in the latter, partial solutions are obtained by imposing extra spectral constraints such as rank-deficiency and degeneracy, which enable a semidefinite programming treatment for the optimization problem at hand. Despite the technically-motivated assumptions, we provide strong numerical evidence that three-fold degenerate X states of purity $P$ reach the highest entangle…
Using techniques from symplectic geometry, we prove that a pure state of three qubits is up to local unitaries uniquely determined by its one-particle reduced density matrices exactly when their ordered spectra belong to the boundary of the, so called, Kirwan polytope. Otherwise, the states with given reduced density matrices are parameterized, up to local unitary equivalence, by two real variables. Given inevitable experimental imprecisions, this means that already for three qubits a pure quantum state can never be reconstructed from single-particle tomography. We moreover show that knowledge…
We study 12 parameter families of two qubit density matrices, arising from a special class of two-fermion systems with four single particle states or alternatively from a four-qubit state with amplitudes arranged in an antisymmetric matrix. We calculate the Wooters concurrences and the negativities in a closed form and study their behavior. We use these results to show that the relevant entanglement measures satisfy the generalized Coffman-Kundu-Wootters formula of distributed entanglement. An explicit formula for the residual tangle is also given. The geometry of such density matrices is elab…
We derive the spectral density of the equiprobable mixture of two random density matrices of a two-level quantum system. We also work out the spectral density of mixture under the so-called quantum addition rule. We use the spectral densities to calculate the average entropy of mixtures of random density matrices, and show that the average entropy of the arithmetic-mean-state of $n$ qubit density matrices randomly chosen from the Hilbert-Schmidt ensemble is never decreasing with the number $n$. We also get the exact value of the average squared fidelity. Some conjectures and open problems rela…
Recently a framework for assisted quantum error correction was proposed in which a specific type of error is allowed to occur on auxiliary qubits, which is in contrast to standard entanglement assistance that requires noiseless auxiliary qubits. However, while the framework maintains the ability to import any binary or quaternary linear code without sacrificing active error correction power, it requires the code designer to turn a parity-check matrix of the underlying classical code into an equivalent one in standard form. This means that classical coding theoretic techniques that require pari…
Oscillating neutrino beams exhibit quantum coherence over distances of thousands of kilometers. Their unambiguously quantum nature suggests an appealing test system for direct quantum simulation. Such techniques may enable presently analytically intractable calculations involving multi-neutrino entanglements, such as collective neutrino oscillations in supernovae, but only once oscillation phenomenology is properly re-expressed in the language of quantum circuits. Here we resolve outstanding conceptual issues regarding encoding of arbitrarily mixed neutrino flavor states in the Hilbert space o…
In singular value decomposition (SVD) of a complex matrix A, the singular vectors or the eigenvectors of AA† and A†A are unique up to complex phase factors. Thus, the two unitary matrices in SVD are unique up to diagonal matrices of phase factors, the phase-factor matrices. Also, the product of these two phase-factor matrices, or the product of phase factors of the corresponding singular vectors with the same singular value, is unique. In the Schmidt decomposition, a phase-factor matrix is a phase rotation operator acting on a subsystem alone. We summarize here three simple steps to consistent…
In connection with optimal state determination for two qubits, the question was raised about the maximum number of pairwise complementary reductions. The main result of the paper tells that the maximum number is 4, that is, if A1, A2,... Ak are pairwise complementary (or quasi-orthogonal) subalgebras of the algebra of all 4x4$ matrices and they are isomorphic to the algebra of all 2x2 matrices, then k is at most 4. In the way to this result, contributions are made to the understanding of the structure of complementary reductions.
The entanglement characteristics of two qubits are encoded in the invariants of the adjoint action of SU(2) x SU(2) group on the space of density matrices defined as the space of positive semi-definite Hermitian matrices. The corresponding ring of polynomial invariants is studied. The special integrity basis for this ring is described and constraints on its elements due to the positive semi-definiteness of density matrices are given explicitly in the form of polynomial inequalities. The suggested basis is characterized by the property that only a minimal number of invariants, namely two primar…
