Lu-Ming Duan, Guang-Can Guo
We show that, there are physical means for cloning two non-orthogonal pure states which are secretly chosen from a certain set $% \$={ | Ψ_0 > , | Ψ_1 > }$. The states are cloned through a unitary evolution together with a measurement. The cloning efficiency can not attain 100%. With some negative measurement results, the cloning fails.
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The no-cloning principle tells us that non-orthogonal quantum states cannot be cloned, but it does not tell us that orthogonal states can always be cloned. We suggest a situation where the cloning transformations are restricted, leading to a novel type of no-cloning principle. In the case of a composite system made of two subsystems: if the subsystems are only available one after the other then there are various cases when orthogonal states cannot be cloned. Surprising examples are given, which give a radically better insight regarding the basic concepts of quantum cryptography.
All existing quantum cryptosystems use non-orthogonal states as the carriers of information. Non-orthogonal states cannot be cloned (duplicated) by an eavesdropper. In result, any eavesdropping attempt must introduce errors in the transmission, and therefore, can be detected by the legal users of the communication channel. Orthogonal states are not used in quantum cryptography, since they can be faithfully cloned without altering the transmitted data. In this Letter we present a cryptographic scheme based on orthogonal states, which also assures the detection of any eavesdropper.
Since, in general, non-orthogonal states cannot be cloned, any eavesdropping attempt in a Quantum Communication scheme using non-orthogonal states as carriers of information introduces some errors in the transmission, leading to the possibility of detecting the spy. Usually, orthogonal states are not used in Quantum Cryptography schemes since they can be faithfully cloned without altering the transmitted data. Nevertheless, L. Goldberg and L. Vaidman [\prl 75 (1995) 1239] proposed a protocol in which, even if the data exchange is realized using two orthogonal states, any attempt to eavesdrop i…
A fundamental question in quantum mechanics is, whether it is possible to replicate an arbitrary unknown quantum state. Then famous quantum no-cloning theorem [Nature 299, 802 (1982)] says no to the question. But it leaves open the following question: If the state is not arbitrary, but secretly chosen from a certain set $\$={ | Ψ_1> ,| Ψ_2> ,... ,| Ψ_n> } $, whether is the cloning possible? This question is of great practical significance because of its applications in quantum information theory. If the states $| Ψ_1>, | Ψ_2>,...$ and $| Ψ_n> $ are linearly-dependent, similar to the proof of t…
We discuss (im)possibility of the exact cloning of orthogonal but genuinely entangled three qubit states aided with entangled ancila under local operation and classical communication. Whereas any two orthogonal GHZ states taken from the canonical GHZ basis, can be cloned with the help of a known GHZ state, surprisingly we find that no two W states can be cloned by using any known three qubit (possibly entangled) state as blank copy.
Orthogonal pure states can be cloned as well as deleted. However if there is an initial disorder in the system, that is for orthogonal mixed states, one cannot perform deletion. And cloning, in such cases, necessarily produces an irreversibility, in the form of leakage of information into the environment.
Encrypted Qubits can be Cloned
该研究提出通过加密克隆技术实现量子态的加密复制，利用幺正变换生成多个加密克隆并解密，为量子存储提供了新的可能性。
In conventional quantum mechanics, quantum no-deleting and no-cloning theorems indicate that two different and nonorthogonal states cannot be perfectly and deterministically deleted and cloned, respectively. Here, we investigate the quantum deleting and cloning in a pseudo-unitary system. We first present a pseudo-Hermitian Hamiltonian with real eigenvalues in a two-qubit system. By using the pseudo-unitary operators generated from this pseudo-Hermitian Hamiltonian, we show that it is possible to delete and clone a class of two different and nonorthogonal states, and it can be generalized to a…
We examine the perfect cloning of non-local, orthogonal states with only local operations and classical communication. We provide a complete characterisation of the states that can be cloned under these restrictions, and their relation to distinguishability. We also consider the case of catalytic cloning, which we show provides no enhancement to the set of clonable states.
The correspondence between local unitary equivalence of bipartite quantum states and simultaneous orthogonal equivalence is thoroughly investigated and strengthened. It is proved that local unitary equivalence can be studied through simultaneous similarity under projective orthogonal transformations, and four parametrization independent algorithms are proposed to judge when two density matrices on $\mathbb C^{d_1}\otimes \mathbb C^{d_2}$ are locally unitary equivalent in connection with trace identities, Weierstrass pencils, Albert determinants and Smith normal forms.
It is well known that classical information can be cloned, but non-orthogonal quantum states cannot be cloned, and non-commuting quantum states cannot be broadcast. We conceive a scenario in which the object we want to broadcast is the statistical distinguishability, as quantified by quantum Fisher information, about a signal parameter encoded in quantum states. We show that quantum Fisher information cannot be cloned, whilst it might be broadcast even when the input states are non-commuting. This situation interpolates between cloning of classical information and no-broadcasting of quantum in…
We consider non-universal cloning maps, namely cloning transformations which are covariant under a proper subgroup G of the universal unitary group U(d), where d is the dimension of the Hilbert space H of the system to be cloned. We give a general method for optimizing cloning for any cost-function. Examples of applications are given for the phase-covariant cloning (cloning of equatorial qubits) and for the Weyl-Heisenberg group (cloning of "continuous variables").
It is well known that (non-orthogonal) pure states cannot be cloned so one may ask: how much or what kind of additional (quantum) information is needed to supplement one copy of a quantum state in order to be able to produce two copies of that state by a physical operation? For classical information, no supplementary information is required. However for pure quantum (non-orthogonal) states, we show that the supplementary information must always be as large as it can possibly be i.e. the clone must be able to be generated from the additional information alone, independently of the first (given)…
Learning Orthogonal Random Unitary Channels with Contracted Quantum Approaches and Simplex Optimization
提出一种利用 Pauli 学习和收缩量子学习过程在正交酉基上学习随机酉信道的方法，通过多目标极小化支持局部等价信道学习，并在不同噪声程度下验证了其可扩展性。
Local unitary equivalence of absolutely maximally entangled states constructed from orthogonal arrays
本研究利用不变量分析由正交阵列构造的绝对最大纠缠态的局部酉等价性，证明存在无限多个局部酉不等价的三方和五方AME态。
In this paper, we study the superposition invariance of unitary operators and maximally entangled state respectively. Furthermore, we discuss the set of orthogonal maximally entangled states. We find that orthogonal basis of maximally entangled states can be divided into k subspaces. It is shown that some entanglement properties of superposed state in every subspace are invariant.
Cloning Encrypted Quantum States in Arbitrary Dimensions
本文扩展了加密克隆协议，证明其适用于更高阶量子系统，并提出新的单位ary加密算子以满足量子门要求。
We construct a new class of multipartite states possessing orthogonal symmetry. This new class defines a convex hull of multipartite states which are invariant under the action of local unitary operations introduced in our previous paper "On multipartite invariant states I. Unitary symmetry". We study basic properties of multipartite symmetric states: separability criteria and multi-PPT conditions.
The study of quantum systems evolving from initial states to distinguishable, orthogonal final states is important for information processing applications such as quantum computing and quantum metrology. However, for most unitary evolutions and initial states the system does not evolve to an orthogonal quantum state. Here, we ask what proportion of quantum states evolves to nearly orthogonal systems as a function of the dimensionality of the Hilbert space of the system, and numerically study the evolution of quantum states in low-dimensional Hilbert spaces. We find that, as well as the speed o…
No-cloning theorem says that there is no unitary operation that makes perfect clones of non-orthogonal quantum states. The objective of the present paper is to examine whether an imperfect cloning operation exists or not in a C*-algebraic framework. We define a universal $ε$-imperfect cloning operation which tolerates a finite loss $ε$ of fidelity in the cloned state, and show that an individual system's algebra of observables is Abelian if and only if there is a universal $ε$-imperfect cloning operation in the case where the loss of fidelity is less than 1/4. Therefore, in this case no univer…
In accordance with the principle of superposition and operator rule, the state of the whole system composed of the state of the particles to be teleported and quantum channel can be expanded by Bell bases and transformation operator. Theoretically, if determinant of transformation operators is not zero, the teleportation can be realized only by performing an inverse transformation. Actually, if the transformation operator is not a unitary operation, then by using one auxiliary qubits, the teleportation can be realized only by performing a collective unitary transformation. Moreover, the furthe…
Non-unitary operations generated by an effective non-Hermitian Hamiltonian can be used to create quantum state manipulations which are impossible in Hermitian quantum mechanics. These operations include state preparation (or cooling) and non-orthogonal state discrimination. In this work we put a lower bound on the resources needed for the construction of some given non-unitary evolution. Passive systems are studied in detail and a general feature of such a system is derived. After interpreting our results using the singular value decomposition, several examples are studied analytically. In par…
Uniqueness of imaginarity-assisted exact transformation from real orthogonal operations to arbitrary unitary operations
本文证明了虚数态|+i⟩在自由实操作下是实现量子计算通用性转换的唯一资源态，且其能赋予实操作应用非实量子门的能力。
We show that any finite dimensional von Neumann algebra admits an orthonormal unitary basis with respect to its standard trace. We also show that a finite dimensional von Neumann subalgebra of $M_n(\mathbb{C})$ admits an orthonormal unitary basis under normalized matrix trace if and only if the normalized matrix trace and standard trace of the von Neumann subalgebra coincide. As an application, we verify a recent conjecture of Bakshi-Gupta, showing that any finite-index regular inclusion $N\subseteq M$ of $II_1$-factors admits an orthonormal unitary Pimsner-Popa basis.
On the Complexity of Quantum States and Circuits from the Orthogonal and Symplectic Groups
本文研究了从正交和辛群中生成的量子态和电路的复杂性，发现其具有指数级的强态复杂性，并展示了从这些群中学习电路的平均难度。
We show that weighted unitary 2-designs define optimal measurements on the system-ancilla output state for ancilla-assisted process tomography of unital quantum channels. Examples include complete sets of mutually unbiased unitary-operator bases. Each of these specifies a minimal series of optimal orthogonal measurements. General quantum channels are also considered.
We show that a quantum state can be perfectly cloned up to global mirroring with a unitary transformation that depends on one single parameter. We then show that this is equivalent to "perfect" cloning for quantum associative memories which, as a consequence efficiently hold exponentially more information than their classical counterparts. Finally, we present a quantum associative retrieval algorithm which can correct corrupted inputs and is exponentially faster than the Grover algorithm.
Operator spreading in random unitary circuits with orthogonal/symplectic-invariant gate distributions
研究从正交不变或辛不变系综中抽取门组成的随机量子电路中的算子扩散，揭示了与酉不变情况的关键区别，包括权重三元结构、有限宽度畴壁以及蝴蝶速度的二分性。
We establish a connection between the problem of constructing maximal collections of mutually unbiased bases (MUBs) and an open problem in the theory of Lie algebras. More precisely, we show that a collection of m MUBs in K^n gives rise to a collection of m Cartan subalgebras of the special linear Lie algebra sl_n(K) that are pairwise orthogonal with respect to the Killing form, where K=R or K=C. In particular, a complete collection of MUBs in C^n gives rise to a so-called orthogonal decomposition (OD) of sl_n(C). The converse holds if the Cartan subalgebras in the OD are also *-closed, i.e., …
In this paper, I take a step toward answering the following question: for m different small circuits that compute m orthogonal n qubit states, is there a small circuit that will map m computational basis states to these m states without any input leaving any auxiliary bits changed. While this may seem simple, the constraint that auxiliary bits always be returned to 0 on any input (even ones besides the m we care about) led me to use sophisticated techniques. I give an approximation of such a unitary in the m = 2 case that has size polynomial in the approximation error, and the number of qubits…
We study bipartite unitary operators which stay invariant under the local actions of diagonal unitary and orthogonal groups. We investigate structural properties of these operators, arguing that the diagonal symmetry makes them suitable for analytical study. As a first application, we construct large new families of dual unitary gates in arbitrary finite dimensions, which are important toy models for entanglement spreading in quantum circuits. We then analyze the non-local nature of these invariant operators, both in discrete (operator Schmidt rank) and continuous (entangling power) settings. …
We construct a probabilistic quantum cloning machine by a general unitary-reduction operation. With a postselection of the measurement results, the machine yields faithful copies of the input states. It is shown that the states secretly chosen from a certain set $\$=\left\{\left| Ψ_1\right> ,\left| Ψ_2\right> ,... ,\left| Ψ_n\right> \right\} $ can be probabilistically cloned if and only if $% \left| Ψ_1\right>$, $\left| Ψ_2\right>$, $... ,$ and $\left| Ψ_n\right>$ are linearly-independent. We derive the best possible cloning efficiencies. Probabilistic cloning has close connection with the pro…
The matrix integral has many applications in diverse fields. This review article begins by presenting detailed key background knowledge about matrix integral. Then the volumes of orthogonal groups and unitary groups are computed, respectively. As a unification, we present Mcdonald's volume formula for a compact Lie group. With this volume formula, one can easily derives the volumes of orthogonal groups and unitary groups. Applications are also presented as well. Specifically, The volume of the set of mixed quantum states is computed by using the volume of unitary group. The volume of a metric …
Local Unitary Equivalence of Tripartite Quantum States In Terms of Trace Identities
本文通过超矩阵代数框架，将局部酉等价性与同时正交等价性的对应关系从双体推广到三体量子态，并利用推广的Specht准则将三体LU等价性问题简化为检查迹恒等式等不变量。
Matrix-product codes over finite fields are an important class of long linear codes by combining several commensurate shorter linear codes with a defining matrix over finite fields. The construction of matrix-product codes with certain self-orthogonality over finite fields is an effective way to obtain good $q$-ary quantum codes of large length. This article has two purposes: the first is to summarize some results of this topic obtained by the author of this article and his cooperators in [10-12]; the second is to add some new results on quasi-orthogonal matrices (resp. quasi-unitary matrices)…
Local unitary equivalence of orthogonal arrays and related linear codes
本文通过生成矩阵构造正交阵列，建立了基于傅里叶的局部酉等价条件，并揭示了其与线性码（尤其是偶校验码）之间的显式联系，识别了由特定线性码产生的等价族。
Mutually-orthogonal unitary and orthogonal matrices
本文引入互正交酉矩阵与正交矩阵集，刻画三阶结构，应用于量子信息得出实数双qutrit系统不可扩展最大纠缠基数量范围，并提出新矩阵分解方法及其判据。
A class of Schwarz qubit maps with diagonal unitary and orthogonal symmetries
本研究刻画了一类具有对角酉与正交对称性的幺正Schwarz量子比特映射，揭示其正定性、算子Schwarz不等式与完全正定性的关联，推广至非幺正情形并拓展了Pauli量子信道的Fujiwara-Algoet条件。
Multifractal dimensions for orthogonal-to-unitary crossover ensemble
本研究针对正交到幺正交叉系综给出多重分形维数的半解析表达式，提出其平移缩放变体作为交叉行为度量，经模拟验证后成功应用于三类量子系统的交叉区域分析。
We analyze bipartite matrices and linear maps between matrix algebras, which are respectively, invariant and covariant, under the diagonal unitary and orthogonal groups' actions. By presenting an expansive list of examples from the literature, which includes notable entries like the Diagonal Symmetric states and the Choi-type maps, we show that this class of matrices (and maps) encompasses a wide variety of scenarios, thereby unifying their study. We examine their linear algebraic structure and investigate different notions of positivity through their convex conic manifestations. In particular…
Recent work on recurrence in quantum walks has provided a representation of Schur functions in terms of unitary operators. We propose a generalization of Schur functions by extending this operator representation to arbitrary operators on Banach spaces. Such generalized Schur functions meet the formal structure of first return generating functions, thus we call them FR-functions. We derive general properties of FR-functions, among them a generalization of the renewal equation already known for random and quantum walks, as well as splitting properties which extend useful factorizations of Schur …
Until now only for specific crossovers between Poissonian statistics (P), the statistics of a Gaussian orthogonal ensemble (GOE), or the statistics of a Gaussian unitary ensemble (GUE) analytical formulas for the level spacing distribution function have been derived within random matrix theory. We investigate arbitrary crossovers in the triangle between all three statistics. To this aim we propose an according formula for the level spacing distribution function depending on two parameters. Comparing the behavior of our formula for the special cases of P$\rightarrow$GUE, P$\rightarrow$GOE, and …
On the complexity of isomorphism problems for tensors, groups, and polynomials III: actions by classical groups
本文研究经典群作用下张量同构问题的复杂性，证明正交群和辛群情形可归约到一般线性群，并建立不同作用间的多项式时间等价及图同构到张量同构的归约。
We consider the conductance distributions in chaotic mesoscopic cavities for all three invariant classes of random matrices for the arbitrary number of channels N1, N2 in the connecting leads. We show that the Laplace transforms of the distributions can be expressed in terms of determinants in the unitary case and Pfaffians in the orthogonal and symplectic cases. The inverse Laplace transforms then give the exact distributions. This formalism is particularly useful for small values of N = min (N1, N2), and thus is of direct experimental relevance. We also obtain the conductance distributions f…
I adapt a recently introduced method for integrating over the unitary group (S. Aubert and C.S. Lam, J.Math.Phys. 44, 6112-6131 (2003)) to the orthogonal group. I derive explicit formulas for a number of one, two and three-vector integrals, as well as recursion formulas for more complicated cases.
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