Calculation of quantum discord in arbitrary dimensions, especially for X- and other specialized states

arXiv:1701.05922 · quant-ph · Submitted 2017-01-20 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: We're looking at the paper: "Calculation of quantum discord in arbitrary dimensions, especially for X- and other specialized states".

Mira: Here's the gist of the paper. The gist The paper presents a generalized procedure for calculating quantum discord in higher-dimensional bipartite systems, extending methods developed for qubits to qudits and spin chains.

Kai: Quantum Discord Definition and Motivation Quantum discord is defined as "the difference between quantum mutual information and classical correlation in a bipartite system".

Mira: This measure of quantum correlation is calculated in terms of von Neumann entropies of density matrices.

Kai: The motivation for studying quantum correlations such as discord in higher dimensions is not only the use of qutrits and higher qudits in quantum information but also the study of spin chains with spin larger than 1/2.

Paper summary: Kai: So we're looking at the paper "Calculation of quantum discord in arbitrary dimensions, especially for X- and other specialized states". Basically, this paper extends the way you calculate quantum discord from just qubits to bigger systems like qudits or spin chains.

Mira: Right, so the core idea is defining quantum discord as that gap between mutual information and classical correlation, and they show how to do that when both parties are larger than two dimensions.

Lev: But it's not a universal method for every single density matrix, the paper admits that it only works for many mixed-state density matrices.

Kai: The paper focuses on a specific trick: using unitary operations to describe measurements in a compact way so you can minimize the number of variables you have to vary when maximizing that classical correlation part.

Mira: That reduction in variables is what makes it manageable, especially when we look at qutrit systems where the number of parameters needed for extremization drops from eight down to just six, as they point out on page two of this paper <ref:1701.05922#pg1>.

Lev: From an engineering standpoint, that's good news because fewer free parameters mean less complexity when you're trying to simulate or measure something on real hardware.

Kai: It seems like the main focus is simplifying the procedure for finding that classical correlation maximum, which they call the most difficult part of discord calculation.

Mira: And they do give a simple recipe for that extremization, which in some cases gives you the final classical correlation value without needing that optimization step at all.

Lev: That would be very helpful if it's true for many states because it removes a lot of iterative work from the process when you're building an error correction scheme.

Kai: They spend a lot of time looking at specific states, like Werner and isotropic states, where things simplify quite a bit, which is useful for testing the method on simpler cases first.

Mira: And they also provide some deep dives into X-states, and in an appendix there's a systematic definition for the symmetries of these X-states in any dimension.

Lev: That appendix part about the systematics of X-states sounds really interesting because it connects this work to other group theory stuff they were doing on qubits earlier.

Kai: And when we look at those extended X-states, the paper gives some concrete numbers for how many parameters you need, like eighteen times d minus one for even d and eighteen times d minus ten for odd d.

Mira: That parameter count is what shows how the complexity scales up as you move into higher dimensions, which is a key point of this paper on page two <ref:1701.05922#pg1>.

Lev: If we're building hardware, knowing that the overhead scales linearly with d in some cases but quadratically in others is important for planning qubit arrays.

Paper summary: Kai: The paper also gives you formulas for Werner and isotropic states, where the classical correlation is given by a specific expression involving p, which is the purity of the state.

Mira: And quantum discord has its own formula too, which they show can be quite different from that classical correlation formula depending on how you look at it.

Lev: So for someone just listening to this, the main thing to grasp is that they developed a way to handle these higher-dimensional correlations without getting immediately lost in too many variables.

Kai: The authors are really pushing a method that uses unitary operations as a compact description of measurements, which they say minimizes the number of variables needed for extremizing classical correlation.

Mira: It's interesting because they acknowledge that no single universal scheme exists for all density matrices, but this procedure still works for many mixed-state density matrices.

Lev: So what's the actual implication here? It’s a tool that helps simplify the theoretical hurdle of calculating discord in higher dimensions when you can't just use some generic black box formula.

Kai: And they show how this approach handles qutrit systems specifically, reducing the parameter count for extremization to at most six, which is quite efficient.

Mira: So it seems like the practical application here is that you can tackle discord in higher dimensions by focusing on simplifying the measurement description first.

Lev: From my side, if this method translates well to a real system, we can start estimating the actual computational cost of finding that optimal measurement because we know exactly how many variables you're dealing with.

Kai: And they point out that for X-states and extended X-states, the eigenvalues in the calculation of Eq. (two) are simple, which makes the whole process much cleaner to handle mathematically <ref:1701.05922#pg1>.

Mira: That simplification for X-states is a big deal because it means you don't have to wrestle with complicated eigenvalue problems every time you try to compute discord for those states.

Lev: So, if you're working on error correction codes that involve these specific state classes, this paper gives you a roadmap on how much math you can expect to need for the correlation calculation part.

Kai: Looking at the conclusion of "Calculation of quantum discord in arbitrary dimensions, especially for X- and other specialized states", it seems the authors are summarizing their main contribution to making this calculation feasible beyond qubits.

Mira: And they emphasize that while it's not a blanket solution for everything, this specific procedure using unitary operations is a useful tool for many mixed-state density matrices.

Lev: The implication for hardware is that we can expect these methods to be relevant when scaling up from two dimensions to higher dimensional systems in quantum computation.

Kai: So, the paper shows a concrete path forward for calculating quantum discord in systems with more than two dimensions by focusing on parameter reduction through unitary descriptions.

Conclusion: Kai: So we've been digging into this paper on calculating quantum discord in higher dimensions, specifically for X-states and others.

Mira: Right, so it lays out a general procedure that works when both parties are bigger than just two dimensions, extending stuff from qubits to qudits and spin chains.

Lev: But the authors are clear, this isn't some magic formula; it only works for certain mixed-state density matrices, so we gotta keep that in mind.

Kai: The main thing here is they’ve developed a method using unitary operations to simplify the measurements involved, which cuts down on how many variables you have to mess around with when finding that classical correlation maximum.

Mira: That reduction in variables is what makes it tractable for systems like qutrits, where they show you can get away with maybe six parameters instead of eight from the original symmetry algebra.

Lev: From an engineering point of view, that’s a big win because fewer free parameters mean less complexity when we try to simulate or even build something on real hardware.

Kai: They do give a simple recipe for finding that correlation maximum, which in some cases lets you get the final number without needing that hard optimization step at all.

Mira: And they give specific formulas for Werner and isotropic states, where the classical correlation has a very clear expression involving the purity of the state, p.

Lev: The caveat is that this whole procedure isn't guaranteed to work for every single density matrix out there; it’s limited to certain classes.

Kai: So what does this mean for us who are trying to build actual quantum hardware? It shows a path forward for tackling discord in bigger systems, even if it's not a universal black box solution.

Mira: It means that when we move toward larger qudits or spin chains, we have a concrete roadmap to start calculating those correlations systematically.

Lev: I think the real implication is that we can get better estimates on the overhead for error correction codes because now we know what kind of mathematical machinery you're actually looking at when you scale up.

Kai: And they spent a lot of time on the systematics of X-states, and in the appendix, they even give you hard numbers for how many parameters those states have in different dimensions.

Mira: That appendix part about the parameter counting is really useful because it shows exactly how much math you're dealing with when you’re looking at these specific state classes.

Lev: So, to wrap up this segment, the paper provides a practical tool for calculating quantum discord in higher dimensions by focusing on simplifying the measurement description and handling specific state classes like X-states.

Kai: And that sets us up perfectly for next time when we look at those extended X-states they discuss in the appendix.

A. R. P. Rau

Louisiana State University

quant-ph

Submitted: 2017-01-20

Updated: 2018-07-25

Comments: 8 pages, changes from v1

Journal ref: Quantum Information Processing 17:216 (2018)

DOI: 10.1007/s11128-018-1985-8

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 68/100

The gist: The gist The paper presents a generalized procedure for calculating quantum discord in higher-dimensional bipartite systems, extending methods developed for qubits to qudits and spin chains.

Key concepts

Quantum Discord: This is a measure of quantum correlation in a bipartite system, defined as the difference between the total quantum mutual information and the classical correlation present between the two subsystems. It quantifies correlations that cannot be explained by classical physics.
Unitary Operations for Measurement Description: The method uses unitary transformations to describe all possible measurements on one subsystem. This allows complex measurements to be represented in a compact form, significantly reducing the number of variables needed when trying to find the maximum classical correlation.
X-States: These are specific classes of density matrices that are studied because their calculations simplify considerably. The paper analyzes how many parameters define an X-state in a qudit-qudit system, showing how this complexity scales with the dimension d.
Classical Correlation Extremization: A crucial and often difficult step in calculating discord is finding the supremum (maximum value) of the classical correlation. The paper provides a simple recipe for this extremization, which sometimes yields the final classical correlation value directly without needing complex optimization.

Terminology

Mutual Information — 1× in this paper · explained in 2 episodes

Summary

The gist The paper presents a generalized procedure for calculating quantum discord in higher-dimensional bipartite systems, extending methods developed for qubits to qudits and spin chains.

Quantum Discord Definition and Motivation

Quantum discord is defined as the difference between quantum mutual information and classical correlation in a bipartite system. This measure of quantum correlation is calculated in terms of von Neumann entropies of density matrices. The motivation for studying quantum correlations such as discord in higher dimensions is not only the use of qutrits and higher qudits in quantum information but also the study of spin chains with spin larger than 1/2.

Procedure for Qubit-Based Calculation

The calculation procedure is based on a key idea from previous work, extending conditional measurements as a combination of projecters and unitary transformations to higher dimensions. The essential idea is to use von Neumann projections transformed by arbitrary unitary transformations to describe all possible measurements on qubit A. A convenient procedure for describing and computing the measurements involved in discord calculations is to parametrize unitary transformations of a qubit as given in [12, 13].

Qudit Discord Recipe

The entire calculation when A is a qubit can be condensed into a recipe or a simple prescription for any such system AB with A a qubit. This recipe involves taking elements of each block of the four blocks of Eq. (1), multiplying by the corresponding element from Eq. (9), and adding the four blocks together to get piρi in Eq. (5). The supremum in Eq. (12) requires varying the two angle parameters in Eq. (9).

Extension to Higher Dimensions

The procedure can be extended to when both parties are of larger dimension, say dA ≥ 3. The density matrix of AB would be viewed in terms of dA ×dA blocks, each block a matrix of dB × dB. A natural parametrization of the unitary matrix in that equation would involve (d 2A − 1) independent operators and coefficients in Eq. (4).

Qutrit Example and Parameter Reduction

For a qutrit system (dA = 3), the number of parameters needed for extremization is reduced from the expected 8 of SU(3) symmetry to at most 6 parameters. This reduction is achieved by viewing the unitary U in terms of a product U1U2 where U1 describes a “base manifold” and U2 is a “fiber”. The calculation for a qutrit system requires at most 6 parameters, angles (θ, θ1, θ2) and phases (ǫ1, ǫ2, φ) in Eq. (15) and Eq. (16), not the original 8 of the qutrit’s su(3) algebra.

X-States and Systematics

Special attention is given to X-states [15, 16] as well as Bell, Werner, and many other states that are often discussed in a variety of situations. For X-states of arbitrary dimension, the form of the density matrix is preserved. The number of parameters for an X-state in a general qudit-qudit bipartite system is 4d squared − 1.

Limiting Results for Mixed States

For Werner, isotropic, and pseudo-pure states in a qudit-qudit system of dimension d, the classical correlation is given by C = [(d−1)(1−p)/d] log(1−p)+[p+(1−p)/d] log(dp+1−p). The quantum discord is given by Q = [(d 2/dp+1- p)/d 2] × log(d 2/p+1- p)−[(dp+1−p)/d] log(dp+1−p). For large d, the quantum discord Q = p log d rises linearly with p.

X-State Simplification

For X-states and extended X-states, the eigenvalues involved in the calculation of Eq. (2) are simple. The same applies to the reduced densities upon tracing over either A or B, each a 3 × 3 matrix with once again the central element decoupled from the others.

Conclusion on Parameters

The major part of this paper is to present a procedure for calculating quantum discord in an AB system when both sub-systems are of dimension larger than 2. The procedure is limited to classes of density matrices and not generally applicable in these higher dimensions with questions arising of POVMs and NP. Special attention is paid to the class of states called X-states and the second part of the paper is a study of their systematics in the Appendix below.

Appendix on X-State Systematics

The number for n qubits in extended X-states is 31. For a qutrit-qudit bipartite system, the number of parameters for extended X-states is 18d − 1 for d even and 18d − 10 for d odd. The number of qubits in extended X-states is 2n+2−1.

Table II Enumeration

The enumeration of the number of states in a general, an X-, and an extended X- state of n qubit and qutrit systems is presented in Table II. The number for n qubits in general is 8d squared − 1. The number for n qubits in X-states is 3(n+1)− 1. The number for n qubits in extended X-states is 3(n+2)− 1. The number of qutrits in general is 8d squared − 1. The number of qutrits in X-states is 6d−1 for d even and 6d−2 for d odd. The number of qutrits in extended X-states is 18d − 1 for d even and 18d − 10 for d odd.

Final Parameter Count

For a qudit-qudit bipartite system, the number of parameters in X-states is 2d squared − 2 for d even and 2d cubed − 1 for d odd extended X-states. For an asymmetric qudit-quDit system, the number of parameters is d 4 − 1. The number of qubits in an asymmetric qudit-quDit system is d 2D squared − 1. The number of parameters for pure states is d(n−1) complex elements or twice that number of real parameters defining such a density matrix.

Improvements for AI systems

This method allows for analytical calculation of discord when one of the parties is a qubit with dimension two and measurements made on it to get that one-way discord, enabling efficient computation of quantum correlations in bipartite systems involving qudits where general methods are otherwise NP-hard.
The system can perform precise quantification of quantum correlations, specifically calculating quantum discord as the difference between quantum mutual information and classical correlation in a bipartite system for systems like qutrit-qutrit density matrices.
AI systems can be trained to utilize the simple prescription derived from the procedure for higher dimensions, which reduces the number of variables needed for extremizing the classical correlation, thereby minimizing computational complexity during discord calculation.
The system can evaluate correlations using a recipe or a simple prescription, which involves taking elements of each block in the density matrix and multiplying them by specific unitary matrix elements to obtain the conditional density matrix, even when both parties are of dimension larger than two.
The AI can be specialized to handle specific state classes efficiently, such as X-states, because for these states, the eigenvalues involved in the calculation of Eq. (2) are simple, leading to a reduction in required supremum calculations compared to general mixed states.
The improved system can analyze and classify higher-dimensional quantum correlations by recognizing symmetry structures such as those described in the Appendix, which allows for generalizations of results from qubits to arbitrary dimensions, effectively connecting general properties and symmetries of qudit-quDit systems.