Partial Trace Research Articles

On the cardinality of matrices with prescribed rank and partial trace over a finite field

Let F be the finite field of order q and M(n,r,F) be the set of n×n matrices of rank r over the field F. For α∈F and A∈M(n,F), letZA,rα={X∈M(n,r,F)|Tr(AX)=α}. In this article, we solve the problem of determining the cardinality of ZA,rα. We also solve the generalization of the problem to rectangular matrices.

Bundled matrix product states represent low-energy excitations faithfully

Abstract We consider a set of density matrices. All of which are written in the same orbital basis, but the orbital basis size is less than the total Hilbert space size. We ask how each density matrix is related to each of the others by establishing a norm between density matrices based on the truncation error in a partial trace for a small set of orbitals. We find that states with large energy differences must have large differences in their density matrices. Small energy differences are divided into two groups, one where two density matrices have small differences and another where they are very different, as is the case of symmetry. We extend these ideas to a bundle of matrix product states and show that bond dimension of the wavefunction ansatz for two states with large energy differences are larger. Meanwhile, low energy differences can have nearly the same bond dimensions for similar states.

Holographic entropy inequalities and multipartite entanglement

We study holographic entropy inequalities and their structural properties by making use of a judicious grouping of terms into certain multipartite information quantities. This allows us to recast cumbersome entropic expressions into much simpler ones which share interestingly rigid structures. By performing a systematic search over some of these structures, we are able to discover more than 1800 novel entropy inequalities for six parties, thereby demonstrating that these recastings provide a fruitful generating technique for uncovering new holographic entropy inequalities. In attempting to interpret the corresponding sign-definite quantities as correlation measures, we also obtain a no-go result: the superbalance property of holographic entropy inequalities turns out to preclude them from being monotonic under partial tracing. In the process, we also comment on the geometrical significance of multipartite information quantities and present various structural relations amongst them.

The Spatiotemporal Doubled‐Density Operator: A Unified Framework for Analyzing Spatial and Temporal Quantum Processes

AbstractThe measurement statistics for spatial and temporal quantum processes are produced through distinct mechanisms. Measurements that are space‐like separated exhibit non‐signaling behavior. However, time‐like separated measurements can only result in one‐way non‐signaling, as the past is independent of the future, but the opposite is not true. This work presents the doubled‐density operator as a comprehensive framework for studying quantum processes in space‐time. It effectively captures all the physical information of the process, with the measurement and Born rule showing uniformity for both spatial and temporal cases. It is demonstrated that the equal‐time density operator can be derived by performing a partial trace operation on the doubled‐density operator. Furthermore, the temporality of the quantum process can be detected by conducting a partial trace operation on either the left or right half of the doubled‐density operator.

The distribution of polynomials in monotone-independent elements

Building on the work of Arizmendi and Celestino (2021), we derive the ∗-distributions of polynomials in monotone independent and infinitesimally monotone independent elements. For non-zero complex numbers [Formula: see text] and [Formula: see text], we derive explicitly the ∗-distribution of [Formula: see text] whenever [Formula: see text] and [Formula: see text] are monotone or infinitesimally monotone independent elements. This encompasses both cases of the commutator and anti-commutator. This approach can be pushed to study more general polynomials. As applications, we derive the limiting distribution with respect to the partial trace of polynomials in a certain class of random matrices.

Temporal entropy and the complexity of computing the expectation value of local operators after a quench

We study the computational complexity of simulating the time-dependent expectation value of a local operator in a one-dimensional quantum system by using temporal matrix product states(MPSs) and argue that it is intimately related to that of encoding temporal transition matrices and their partial traces. We show that we can upper-bound the rank of these reduced transition matrices by one of the Heisenberg evolution of local operators, thus making aconnection between two apparently different quantities, the temporal entanglement and the local operator entanglement(OE). As a result, whenever the local OE grows slower than linearly in time, we show that computing time-dependent expectation values of local operators using temporal MPSs is likely advantageous with respect to computing the same quantities using standard MPS techniques. Published by the American Physical Society 2024

Quantum state engineering by steering in the presence of errors

Quantum state engineering plays a vital role in various applications in the field of quantum information. Different strategies, including drive-and-dissipation, adiabatic cooling, and measurement-based steering, have been proposed for state generation and manipulation, each with its upsides and downsides. Here, we address a class of measurement-based state engineering protocols where a sequence of generalized measurements is employed to steer a quantum system toward a desired (pure or mixed) target state. Previously studied measurement-based protocols relied on idealized procedures and avoided exploration of the effects of various errors stemming from imperfections of experimental realizations and external noise. We employ the quantum trajectory formalism to provide a detailed analysis of the robustness of these steering protocols against multiple classes of errors. We study a set of realistic errors that can be classified as dynamic or static, depending on whether they remain unchanged while running the protocol. More specifically, we investigate the impact of the erroneous choice of detector-system coupling, erroneous re-initialization of the detector state following a measurement step, fluctuating steering directions, and environmentally induced errors in the detector-system interaction. We show that the protocol remains fully robust against the erroneous choice of detector-system coupling parameters and presents reasonable robustness against other types of errors. Our analysis employs various quantifiers such as fidelity, trace distance, and linear entropy to characterize the protocol's robustness and provide analytical results for these quantifiers against various errors. We introduce averaging hierarchies of stochastic equations describing individual quantum trajectories associated with detector readouts. Subsequently, we demonstrate the commutation between the classical expectation value and the time-ordering operator of the exponential of a Hamiltonian with multiplicative white noise, as well as the commutation of the expectation value and the partial trace with respect to detector outcomes. Our ideas are implemented and demonstrated for a specific class of steering platforms, addressing a single qubit. Published by the American Physical Society 2024

Spin textures of coherent photons with SU(4) symmetry

We show coherent photons with spin and orbital angular momentum have symmetry of special unitary group of degree four, SU(4), both theoretically and experimentally. We show the degree of polarisation could be a measure to characterise the quantum coherency of a spin state, which was significantly affected by the partial trace out of orbital degree of freedom. By changing the amplitudes of the classically entangled state, we observed a crossover from quantum Heisenberg spin to classical Ising spin. We also demonstrated projections of coherent states, which recover the coherency of the spin state, while the SU(4) state is projected to be a simple product state of SU(2)×SU(2). Based on the developed platform, we have explored spin texture of coherent photons, and found a unique spin profile reminiscent of the Bengal cat upon wrapping to a three-dimensional sphere. We have controlled the phase between singlet and triplet states and found the spin texture rotated upon increasing the phase.

Resumable probabilistic teleportation of a three qubit state using different quantum channels

In this paper we describe two different probabilistic teleportation protocols for teleporting the same unknown 3-qubit state from the sender Alice to the receiver Bob. The speciality of the state is that it remains a 2-qubit entangled state after any of the three involved qubits is eliminated by partial trace operation. One of the protocols utilizes a two and a three-qubit separate channels while in the other protocol, a five-qubit channel is utilized. One of the channels in the first protocol is a non-maximally entangled Bell state while the 5-qubit channel used in the case of the second protocol is non-maximally entangled. Both the teleportation schemes are probabilistic with the feature that in the case of failure of the teleportation, the state to be teleported is not lost to the sender. So the process can be repeated after failure till success is achieved. Moreover, one of the channels is recovered in the failure case. Both protocols are resumable in the above sense. The above is the speciality of the present work. We analyze the efficiency of our protocol which shows that the present two protocols are better performing than some existing ones.

Partial trace ideals, torsion and canonical module

For any finitely generated module M with non-zero rank over a commutative one dimensional Noetherian local domain, the numerical invariant h(M) was introduced and studied in [25]. We establish a bound on it which helps capture information about the torsion submodule of M when M has rank one and generalizes the discussion in [25]. We further study bounds and properties of h(M) in the case when M is the canonical module ωR. This in turn helps in answering a question of S. Greco and then provides classifications in the Gorenstein, almost Gorenstein and far-flung Gorenstein setups.

Exact exponent for atypicality of random quantum states

We study the properties of the random quantum states induced from the uniformly random pure states on a bipartite quantum system by taking the partial trace over the larger subsystem. Most of the previous studies have adopted a viewpoint of ‘concentration of measure’ and have focused on the behavior of the states close to the average. In contrast, we investigate the large deviation regime, where the states may be far from the average. We prove the following results: first, the probability that the induced random state is within a given set decreases no slower or faster than exponential in the dimension of the subsystem traced out. Second, the exponent is equal to the quantum relative entropy of the maximally mixed state and the given set, multiplied by the dimension of the remaining subsystem. Third, the total probability of a given set strongly concentrates around the element closest to the maximally mixed state, a property that we call conditional concentration. Along the same line, we also investigate an asymptotic behavior of coherence of random pure states in a single system with large dimensions.

Partial trace inequalities for partial transpose of positive semidefinite block matrices

Partial trace inequalities for partial transpose of positive semidefinite block matrices

Positive maps from the walled Brauer algebra

We present positive maps and matrix inequalities for variables from the positive cone. These inequalities contain partial transpose and reshuffling operations, and can be understood as positive multilinear maps that are in one-to-one correspondence with elements from the walled Brauer algebra. Using our formalism, these maps can be obtained in a systematic and clear way by manipulating partially transposed permutation operators under a partial trace. Additionally, these maps are reasonably easy in construction by combining an algorithmic approach with graphical calculus.

Finite-Dimensional Stinespring Curves Can Approximate Any Dynamics

We generalize a recent result stating that all analytic quantum dynamics can be represented exactly as the reduction of unitary dynamics generated by a time- dependent Hamiltonian. More precisely, we prove that the partial trace over analytic paths of unitaries can approximate any Lipschitz-continuous quantum dynamics arbitrarily well. Equivalently, all such dynamics can be approximated by analytic Kraus operators. We conclude by discussing potential improvements and generalizations of these results, their limitations, and the general challenges one has to overcome when trying to relate dynamics to quantities on the system–environment level.

Stabilizer Subsystem Decompositions for Single- and Multimode Gottesman-Kitaev-Preskill Codes

The Gottesman-Kitaev-Preskill (GKP) error-correcting code encodes a finite-dimensional logical space in one or more bosonic modes, and has recently been demonstrated in trapped ions and superconducting microwave cavities. In this work we introduce a new subsystem decomposition for GKP codes that we call the stabilizer subsystem decomposition, analogous to the usual approach to quantum stabilizer codes. The decomposition has the defining property that a partial trace over the nonlogical stabilizer subsystem is equivalent to an ideal decoding of the logical state, distinguishing it from previous GKP subsystem decompositions. We describe how to decompose arbitrary states across the subsystem decomposition using a set of transformations that move between the decompositions of different GKP codes. Besides providing a convenient theoretical view on GKP codes, such a decomposition is also of practical use. We use the stabilizer subsystem decomposition to efficiently simulate noise acting on single-mode GKP codes, and in contrast to more conventional Fock basis simulations, we are able to consider essentially arbitrarily large photon numbers for realistic noise channels, such as loss and dephasing. Published by the American Physical Society 2024

Block determinants, partial determinants and the exponential map

The block trace and partial trace are closely related. However, the block determinant and partial determinant are quite different. We introduce a new determinant-like operation, the determinant-root, which has a partial operation that is more natural in the algebraic sense. The block determinant retains some of the algebraic properties of determinants, while the partial determinant-root retains the same algebraic properties for matrices which are Kronecker products. We investigate the equations det ( D ( A ) ) = det ( A ) , D ( A B ) = D ( A ) D ( B ) and D ( e A ) = e T ( A ) , where D is the block/partial/partial-root determinant and T is the corresponding block/partial/partial-normalized trace. These results yield a characterization of non-linear preservers of determinants of Kronecker products.

TRANSLATION OF REALIA ON THE MATERIAL OF THE RUSSIAN FOLK TALE «KOSCHEY THE IMMORTAL»

Statement of the problem. The article discusses the interaction of linguistics and cultural studies in translation. The concepts of lacunarity, ethnolinguocultural asymmetry, sociocultural adaptation, significative connotation are revealed. Results. The article describes the difficulties that arise when translating works that are distinguished by a bright national coloring. Particular attention is paid to the problems associated with the difficulties of translation, in particular the translation of non-equivalent vocabulary. Various interpretations of the concept of reality are given. It is explained why it is necessary to distinguish between the concepts of "realia" and "non-equivalent vocabulary". With the help of the theory of renomination, the difficulties associated with the translation of realia from one language to another are analyzed and a comparative analysis of various translation options for realia taken from the Russian folk tale "Koschey the Immortal" of A. S. Afanasyev and its translations into Italian by Rosella Morini (2011) and Sophia de Figner (1903). Conclusion. It is concluded that the translation of folklore is always difficult, since its quality depends very much on how rich the translator's stock of thematic and folk-poetic vocabulary of both languages ​​is. The analysis showed that the language in the translations is modern, neutral vocabulary prevails in it, in contrast to the Russian fairy tale, where there is a lot of stylistically colored colorful vocabulary. To convey the cultural features of the Russian language, transcriptions, transliterations, partial tracing with explanatory notes are used.

Deriving mixed state of light field by partial tracing pure state in higher dimension

In this paper, we show that by partial tracing the pure state in higher dimension one can deduce the mixed state of light field. In this paper the pure state in higher dimension is a three-mode squeezed state S23σS31τ000000S31−1τS23−1σ, while the mixed state’s density operator is sech2σsech2τea2†a3†tanhσsech2σtanh2τa3†a30220ea2a3tanhσ, which involves both squeezing and chaotic properties of optical field. We further explain how such kind of light field can be generated by passing a two-mode squeezed light though a single-mode dissipative channel, its Wigner function is discussed by using the coherent state representation. Our work provides an important method, which can be used to theoretically find more optical fields in quantum communication and quantum optics.

Moments of Random Quantum Marginals via Weingarten Calculus

Abstract The randomized quantum marginal problem asks about the joint distribution of the partial traces (“marginals”) of a uniform random Hermitian operator with fixed spectrum acting on a space of tensors. We introduce a new approach to this problem based on studying the mixed moments of the entries of the marginals. For randomized quantum marginal problems that describe systems of distinguishable particles, bosons, or fermions, we prove formulae for these mixed moments, which determine the joint distribution of the marginals completely. Our main tool is Weingarten calculus, which provides a method for computing integrals of polynomial functions with respect to Haar measure on the unitary group. As an application, in the case of two distinguishable particles, we prove some results on the asymptotic behavior of the marginals as the dimension of one or both Hilbert spaces goes to infinity.

A refinement of Reznick's Positivstellensatz with applications to quantum information theory

In his solution of Hilbert's 17th problem Artin showed that any positive definite polynomial in several variables can be written as the quotient of two sums of squares. Later Reznick showed that the denominator in Artin's result can always be chosen as an N-th power of the squared norm of the variables and gave explicit bounds on N. By using concepts from quantum information theory (such as partial traces, optimal cloning maps, and an identity due to Chiribella) we give simpler proofs and minor improvements of both real and complex versions of this result. Moreover, we discuss constructions of Hilbert identities using Gaussian integrals and we review an elementary method to construct complex spherical designs. Finally, we apply our results to give improved bounds for exponential quantum de Finetti theorems in the real and in the complex setting.