Quantum circuit model is the most popular paradigm for implementing complex quantum computation. Based on Cartan decomposition, we show that $2(N-1)$ generalized controlled-$X$ (GCX) gates, $6$ single-qubit rotations about the $y$- and $z$-axes, and $N+5$ single-partite $y$- and $z$-rotation-types which are defined in this paper are sufficient to simulate a controlled-unitary gate $\mathcal{U}_{cu(2\otimes N)}$ with $A$ controlling on $\mathbb{C}^2\otimes \mathbb{C}^N$. In the scenario of the unitary gate $\mathcal{U}_{cd(M\otimes N)}$ with $M\geq3$ that is locally equivalent to a diagonal unitary on $\mathbb{C}^M\otimes \mathbb{C}^N$, $2M(N-1)$ GCX gates and $2M(N-1)+10$ single-partite $y$- and $z$-rotation-types are required to simulate it. The quantum circuit for implementing $\mathcal{U}_{cu(2\otimes N)}$ and $\mathcal{U}_{cd(M\otimes N)}$ are presented. Furthermore, we find $\mathcal{U}_{cu(2\otimes2)}$ with $A$ controlling has Schmidt rank two, and in other cases the diagonalized form of the target unitaries can be expanded in terms of specific simple types of product unitary operators.
The Schmidt number is a fundamental parameter characterizing the properties of quantum states, and the local projections are a fundamental operation in quantum physics. We investigate the relation between the Schmidt numbers of bipartite states and their projected states. We show that there exist bipartite positive-partial-transpose (PPT) entangled states of any given Schmidt number. We further construct the notion of joint Schmidt number for multipartite states, and its relation with the Schmidt number of bipartite reduced density operators.
Our previous work about algebraic-geometric invariants of the mixed states are extended and a stronger separability criterion is given. We also show that the Schmidt number of pure states in bipartite quantum systems, a classical concept, is actually an algebraic-geometric invariant.
We prove a lower bound for Schmidt numbers of bipartite mixed states. This lower bound can be applied easily to low rank bipartite mixed states. From this lower bound it is known that generic low rank bipartite mixed states have relatively high Schmidt numbers and thus entangled. We can compute Schmidt numbers exactly for some mixed states by this lower bound as shown in Examples. This lower bound can also be used effectively to determine that some mixed states cannot be convertible to other mixed states by local operations and classical communication. The results in this paper are proved by O…
It is shown that for each mixed state there exists a Schmidt (super state vector) decomposition in terms of Hermitian operators. Its utilization for finding all twins is illustrated in full detail in the case of the two spin-one-half-particle states with maximally disordered subsystems (mixtures of Bell states).
It is well known that the Schmidt decomposition exists for all pure states of a two-party quantum system. We demonstrate that there are two ways to obtain an analogous decomposition for arbitrary rank-1 operators acting on states of a bipartite finite-dimensional Hilbert space. These methods amount to joint Schmidt-type decompositions of two pure states where the two sets of coefficients and local bases depend on the properties of either state, however, at the expense of the local bases not all being orthonormal and in one case the complex-valuedness of the coefficients. With these results we …
Operator entanglement of two-qubit joint unitary operations is revisited. Schmidt number is an important attribute of a two-qubit unitary operation, and may have connection with the entanglement measure of the unitary operator. We found the entanglement measure of two-qubit unitary operators is classified by the Schmidt number of the unitary operators. The exact relation between the operator entanglement and the parameters of the unitary operator is clarified too.
We consider the Schmidt decomposition of a bipartite density operator induced by the Hilbert-Schmidt scalar product, and we study the relation between the Schmidt coefficients and entanglement. First, we define the Schmidt equivalence classes of bipartite states. Each class consists of all the density operators (in a given bipartite Hilbert space) sharing the same set of Schmidt coefficients. Next, we review the role played by the Schmidt coefficients with respect to the separability criterion known as the `realignment' or `computable cross norm' criterion; in particular, we highlight the fact…
We consider a bipartite mixed state of the form, $ρ=\sum_{α, β=1}^{l}a_{αβ} | ψ_α> < ψ_ β}| $, where $| ψ_α>$ are normalized bipartite state vectors, and matrix $(a_{αβ})$ is positive semidefinite. We provide a necessary and sufficient condition for the state $ρ$ taking the form of maximally correlated states by a local unitary transformation. More precisely, we give a criterion for simultaneous Schmidt decomposability of $| ψ_α>$ for $α=1,2,..., l$. Using this criterion, we can judge completely whether or not the state $ρ$ is equivalent to the maximally correlated state, in which the distilla…
We study the problem of transforming a tripartite pure state to a bipartite one using stochastic local operations and classical communication (SLOCC). It is known that the tripartite-to-bipartite SLOCC convertibility is characterized by the maximal Schmidt rank of the given tripartite state, i.e. the largest Schmidt rank over those bipartite states lying in the support of the reduced density operator. In this paper, we further study this problem and exhibit novel results in both multi-copy and asymptotic settings. In the multi-copy regime, we observe that the maximal Schmidt rank is strictly s…
Implementing nonlocal unitary operators is an important and hard question in quantum computing and cryptography. We show that any bipartite nonlocal unitary operator of Schmidt rank three on the $(d_A \times d_B)$-dimensional system is locally equivalent to a controlled unitary when $d_A$ is at most three. This operator can be locally implemented assisted by a maximally entangled state of Schmidt rank $r=\min\{d_A^2,d_B\}$. We further show that stochastic-equivalent nonlocal unitary operators are indeed locally equivalent, and propose a sufficient condition on which nonlocal and controlled uni…
We prove, using a new method based on map-state duality, lower bounds on entanglement resources needed to deterministically implement a bipartite unitary using separable (SEP) operations, which include LOCC (local operations and classical communication) as a particular case. It is known that the Schmidt rank of an entangled pure state resource cannot be less than the Schmidt rank of the unitary. We prove that if these ranks are equal the resource must be uniformly (maximally) entangled: equal nonzero Schmidt coefficients. Higher rank resources can have less entanglement: we have found numerica…
We investigate the differential geometry of bipartite quantum states. In particular the manifold structures of pure bipartite states are studied in detail. The manifolds with respect to all normalized pure states of arbitrarily given Schmidt ranks or Schmidt coefficients are explicitly presented. The dimensions of the related manifolds are calculated.
We prove that every unitary acting on any multipartite system and having operator Schmidt rank equal to 2 can be diagonalized by local unitaries. This then implies that every such multipartite unitary is locally equivalent to a controlled unitary with every party but one controlling a set of unitaries on the last party. We also prove that any bipartite unitary of Schmidt rank 2 is locally equivalent to a controlled unitary where either party can be chosen as the control, and at least one party can control with two terms, which implies that each such unitary can be implemented using local opera…
It is known from Schmidt decomposition that Schmidt number of nonlocal two-qubit quantum gates is 2 or 4. We identify conditions on geometrical points of a gate to have Schmidt number 2. A simple analysis reveals that Schmidt number 2 corresponds to controlled unitary gates with CNOT being the only perfect entangler. Further, it is shown that Schmidt strength and entangling power are maximum only for CNOT in the controlled unitary family.
It is known that any bipartite unitary operator of Schmidt rank three is equivalent to a controlled unitary under local unitaries. We propose a standard form of such operators. Using the form we improve the upper bound for the entanglement cost to implement such operators under local operations and classical communications (LOCC), and provide a corresponding protocol. A part of our protocol is based on a recursive-control protocol which is helpful for implementing other unitary operators. We show that any bipartite permutation unitary of Schmidt rank three can be implemented using LOCC and two…
We review some results on the equivalence of quantum states under local unitary transformations (LUT). In particular, the classification of two-qubit Schmidt correlated (SC) states under LUT is investigated. By presenting the standard form of quantum states under LUT, the sufficient and necessary conditions of whether two different SC states are local unitary equivalent are provided. The correlations of SC states are also discussed.
In this letter we have established the physical character of pure bipartite states with the same amount of entanglement in the same Schmidt rank that either they are local unitarily connected or they are incomparable. There exist infinite number of deterministically locally inequivalent classes of pure bipartite states in the same Schmidt rank (starting from three) having same amount of entanglement. Further, if there exists incomparable states with same entanglement in higher Schmidt ranks (greater than three), then they should differ in at least three Schmidt coefficients.
Constructing four six-dimensional mutually unbiased bases (MUBs) is an open problem in quantum physics and measurement. We investigate the existence of four MUBs including the identity, and a complex Hadamard matrix (CHM) of Schmidt rank three. The CHM is equivalent to a controlled unitary operation on the qubit-qutrit system via local unitary transformation $I_2\otimes V$ and $I_2\otimes W$. We show that $V$ and $W$ have no zero entry, and apply it to exclude examples as members of MUBs. We further show that the maximum of entangling power of controlled unitary operation is $\log_2 3$ ebits. …
Unitary operations are physically implementable. We further the understanding of such operations by studying the possible forms of nonlocal unitary operators, which are bipartite or multipartite unitary operators that are not tensor product operators. They are of broad relevance in quantum information processing. We prove that any nonlocal unitary operator of Schmidt rank three on a $d_A\times d_B$ bipartite system is locally equivalent to a controlled unitary. This operator can be locally implemented assisted by a maximally entangled state of Schmidt rank $\min\{d_A^2,d_B\}$ when $d_A\le d_B$…
We study Schmidt rank for a vector (i.e., a pure state) and Schmidt number for a mixed state which are entanglement measures. We show that if a subspace of a certain bipartite system contains no vector of Schmidt rank $\leqslant k$, then any state supported on that space has Schmidt number at least $k+1$. A construction of subspace of $\mathbb{C}^m \otimes \mathbb{C}^n$ of maximal dimension, which does not contain any vector of Schmidt rank less than $3$, is given here.
We describe a decomposition of the Lie group of unitary evolutions for a bipartite quantum system of arbitrary dimensions. The decomposition is based on a recursive procedure which systematically uses the Cartan classification of the symmetric spaces of the Lie group SO(n). The resulting factorization of unitary evolutions clearly displays the local and entangling character of each factor.
In this note we generalize Nielsen's marjoization criterion for the convertibility of bipartite pure states [Phys. Rev. Lett \textbf{83}, 436(1999)] to a special class of multipartite pure states which have generalized Schmidt decompositions.
The nonlocal properties of arbitrary dimensional bipartite quantum systems are investigated. A complete set of invariants under local unitary transformations is presented. These invariants give rise to both sufficient and necessary conditions for the equivalence of quantum states under local unitary transformations: two density matrices are locally equivalent if and only if all these invariants have equal values.
The equivalence of arbitrary dimensional bipartite states under local unitary transformations (LUT) is studied. A set of invariants and ancillary invariants under LUT is presented. We show that two states are equivalent under LUT if and only if they have the same values for all of these invariants.
The Schmidt number is an entanglement measure whose logarithm quantifies the zero-error entanglement cost of generating a given quantum state using local operations and classical communication (LOCC). %However, the Schmidt number is a notoriously difficult quantity to compute, and its relationship to other entanglement measures is largely unknown. In this paper we show that the Schmidt number is highly non-multiplicative in the sense that for any integer $n$, there exists states whose Schmidt number remains constant when taking $n$ copies of the given state. These states also provide a rare in…
We show that any unitary operator on the $d_A\times d_B$ system ($d_A\ge 2$) can be decomposed into the product of at most $4d_A-5$ controlled unitary operators. The number can be reduced to $2d_A-1$ when $d_A$ is a power of two. We also prove that three controlled unitaries can implement a bipartite complex permutation operator, and discuss the connection to an analogous result on classical reversible circuits. We further show that any $n$-partite unitary on the space $\mathbb{C}^{d_1}\otimes...\otimes\mathbb{C}^{d_n}$ is the product of at most $[2\prod^{n-1}_{j=1}(2d_j-2)-1]$ controlled unit…
Local distinguishability of bipartite unitary operations has recently received much attention. A nontrivial and interesting question concerning this subject is whether there is a sequential scheme for locally discriminating between two bipartite unitary operations, because a sequential scheme usually represents the most economic strategy for discrimination. An affirmative answer to this question was given in the literature, however with two limitations: (i) the unitary operations to be discriminated were limited to act on $d\otimes d$, i.e., a two-qudit system, and (ii) the inverses of the uni…
A quantum state's entanglement across a bipartite cut can be quantified with entanglement entropy or, more generally, Schmidt norms. Using only Schmidt decompositions, we present a simple iterative algorithm to maximize Schmidt norms. Depending on the choice of norm, the optimizing states maximize or minimize entanglement, possibly across several bipartite cuts at the same time and possibly only among states in a specified subspace. Recognizing that convergence but not success is certain, we use the algorithm to explore topics ranging from fermionic reduced density matrices and varieties of pu…
This paper considers the control landscape of quantum transitions in multi-qubit systems driven by unitary transformations with single-qubit interaction terms. The two-qubit case is fully analyzed to reveal the features of the landscape including the nature of the absolute maximum and minimum, the saddle points and the absence of traps. The results permit calculating the Schmidt state starting from an arbitrary two-qubit state following the local gradient flow. The analysis of multi-qubit systems is more challenging, but the generalized Schmidt states may also be located by following the local…
Inspired by the `computable cross norm' or `realignment' criterion, we propose a new point of view about the characterization of the states of bipartite quantum systems. We consider a Schmidt decomposition of a bipartite density operator. The corresponding Schmidt coefficients, or the associated symmetric polynomials, are regarded as quantities that can be used to characterize bipartite quantum states. In particular, starting from the realignment criterion, a family of necessary conditions for the separability of bipartite quantum states is derived. We conjecture that these conditions, which a…
For any bipartite quantum system the Schmidt decomposition allows us to express the state vector in terms of a single sum instead of double sums. We show the existence of the Schmidt decomposition for tripartite system under certain condition. If the partial inner product of a basis (belonging to a Hilbert space of smaller dimension) with the state of the composite system gives a disentangled basis, then the Schmidt decomposition for a tripartite system exists. In this case the reduced density matrix of each of the subsystem has equal spectrum in the Schmidt basis.
We investigate quantum properties of light in optical parametric oscillators (OPOs) based on four-wave mixing gain in media with third-order non-linearities. In spite of other competing $χ^{(3)}$ effects, such as phase modulation, bipartite and tripartite entanglement are predicted above threshold. These findings are relevant for recent implementations of CMOS-compatible on-chip OPOs.
The Schmidt coefficients capture all entanglement properties of a pure bipartite state and therefore determine its usefulness for quantum information processing. While the quantification of the corresponding properties in mixed states is important both from a theoretical and a practical point of view, it is considerably more difficult, and methods beyond estimates for the concurrence are elusive. In particular this holds for a quantitative assessment of the most valuable resource, the maximum possible Schmidt number of an arbitrary mixed state. We derive a framework for lower bounding the appr…
Generalized Schmidt decomposition of pure three-qubit states has four positive and one complex coefficients. In contrast to the bipartite case, they are not arbitrary and the largest Schmidt coefficient restricts severely other coefficients. We derive a nonstrict inequality between three-qubit Schmidt coefficients, where the largest coefficient defines the least upper bound for the three nondiagonal coefficients or, equivalently, the three nondiagonal coefficients together define the greatest lower bound for the largest coefficient. In addition, we show the existence of another inequality whic…
In this brief report, we consider the equivalence between two sets of $m+1$ bipartite quantum states under local unitary transformations. For pure states, this problem corresponds to the matrix algebra question of whether two degree $m$ matrix polynomials are unitarily equivalent; i.e. $UA_iV^\dagger=B_i$ for $0\leq i\leq m$ where $U$ and $V$ are unitary and $(A_i, B_i)$ are arbitrary pairs of rectangular matrices. We present a randomized polynomial-time algorithm that solves this problem with an arbitrarily high success probability and outputs transforming matrices $U$ and $V$.
In this work we estimate the transverse Schmidt number for the bipartite bright squeezed vacuum state by means of second-order intensity correlation function measurement. Assuming that the number of modes is equal in both beams we determine the Schmidt number considering only one of the subsystems. The obtained results demonstrate that this approach is equally efficient over the whole propagation of the state from the near field to the far field regions of its emitter.
Any bipartite nonlocal unitary operation can be carried out by teleporting a quantum state from one party to the other, performing the unitary gate locally, and teleporting a state back again. This paper investigates unitaries which can be carried out using less prior entanglement and classical communication than are needed for teleportation. Large families of such unitaries are constructed using (projective) representations of finite groups. Among the tools employed are: a diagrammatic approach for representing entangled states, a theorem on the necessary absence of information at certain tim…
We propose a method for obtaining the Schmidt decomposition of bipartite systems with continuous variables. It approximates the modes to the prescribed accuracy by well known orthogonal functions. We give some criteria for the control of errors. We illustrate the method comparing its results with the already published analysis for entanglement of biphotons. The agreement is excellent.
The problem behind this paper is, if the number of queries to unitary operations is fixed, say $k$, then when do local operations and classical communication (LOCC) suffice for optimally distinguishing bipartite unitary operations? We consider the above problem for two-qubit unitary operations in the case of $k=1$, showing that for two two-qubit entangling unitary operations without local parties, LOCC achieves the same distinguishability as the global operations. Specifically, we obtain: (i) if such two unitary operations are perfectly distinguishable by global operations, then they are perfe…
Entanglement measures quantify the amount of quantum entanglement that is contained in quantum states. Typically, different entanglement measures do not have to be partially ordered. The presence of a definite partial order between two entanglement measures for all quantum states, however, allows for meaningful conceptualization of sensitivity to entanglement, which will be greater for the entanglement measure that produces the larger numerical values. Here, we have investigated the partial order between the normalized versions of four entanglement measures based on Schmidt decomposition of bi…
We consider experimental routes to determine the nonclassical degree of states of a field mode. We adopt a distance-type criterium based on the Hilbert-Schmidt metric to quantify the nonclassicality. The concept of nonclassical degree is extended for states of bipartite systems, allowing us to discuss a possible connection between nonclassicality and entanglement measures.
We compare three different characterizations of the globalness of bipartite unitary operations, namely, delocalization power, entanglement cost, and entangling power, to investigate global properties of unitary operations. We show that the globalness of the same unitary operation depends on whether input states are given by unknown states representing pieces of quantum information, or a set of known states for the characterization. We extend our analysis on the delocalization power in two ways. First, we show that the delocalization power differs whether the global operation is applied on one …
The general class of Gaussian Schmidt-number witness operators for bipartite systems is studied. It is shown that any member of this class is reducible to a convex combination of two types of Gaussian operators using local operations and classical communications. This gives rise to a simple operational method, which is solely based on measurable covariance matrices of quantum states. Our method bridges the gap between theory and experiment of entanglement quantification. In particular, we certify lower bounds of the Schmidt number of squeezed thermal and phase-randomized squeezed vacuum states…
In this short note we show two completely opposite methods of constructing entangled states. Given a bipartite state $γ\in M_k\otimes M_k$, define $γ_S=(Id+F)γ(Id+F)$, $γ_A=(Id-F)γ(Id-F)$, where $F\in M_k\otimes M_k$ is the flip operator. In the first method, entanglement is a consequence of the inequality $\text{rank}(γ_S)<\sqrt{\text{rank}(γ_A)}$. In the second method, there is no correlation between $γ_S$ and $γ_A$. These two methods show how diverse is quantum entanglement. We prove that any bipartite state $γ\in M_k\otimes M_k$ satisfies $\displaystyle SN(γ)\geq\max \left\{ \frac{\text{ra…
