Nonclassical features in higher-dimensional systems through logical qudits

Ever since the inception of quantum mechanics, there has been a burgeoning interest in the studies of quantum foundations [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. This is driven mainly by two reasons: (i) the study of quantum foundations has unraveled many features that are at variance with their classical counterparts or they do not have any classical counterpart, (ii) resource-theoretic importance of different nonclassical features has been recognized in several quantum communication protocols, quantum computation, and quantum search algorithms [12, 13, 14, 15], to name a few.

The prime examples of nonclassical correlations are quantum nonlocality (NL) and quantum entanglement [1, 16]. The resource-theoretic importance of nonclassical features has led to several algebraic approaches for finding conditions for them. For example, many NL inequalities have been derived– for multiqubit and multi-qudit systems (see, for example, [17] and references therein). Their derivations are based on different local-hidden-variable (LHV) models [5, 7, 18, 19], violations of classical probability rules [20], and group-theoretic approaches (for example, stabiliser groups) [21, 22, 23]. Similarly, the resource theory of entanglement has also been developed (see, for example, [24] and references therein). Sufficiency conditions for entanglement, known as entanglement witnesses, have been derived through various approaches (see, for example, [25]). In fact, the detection of different types of nonclassical correlations is ongoing research.

In parallel, the resource theory of quantum coherence has gained a lot of significance [26]. The interrelation of quantum coherence with different nonclassical correlations, viz., entanglement, nonlocality, and quantum discord has been studied from different approaches [27, 28, 29, 30]. In all these works, interrelations of nonclassical features have been studied in the quantum systems belonging to Hilbert spaces of identical dimensions. This prompts us to ask the question: how are the nonclassical features of a mono-party lower-dimensional system related to that of a multi-party higher-dimensional system and vice-versa? This question is not without physical interest because, in fault-tolerant quantum computation, a single logical qubit (or qudit) is composed of physical multiqubit (or multi-qudit) systems. In fact, different types of logical qubits and qudits have been realized experimentally in the context of fault-tolerant quantum computation (such as superconducting qubits) and fault-tolerant quantum communication [31, 32, 33, 34].

To answer this question, we have recently developed a methodology that employs homomorphic maps between stabilizer groups of a logical qubit and a physical multiqubit system as a tool. The tools required to establish this mapping are fairly simple and not complicated. Logical bits have been used in classical error-correcting codes (for example, in parity codes) [35]. In contrast, quantum mechanics allows for many different types of logical qubits in terms of physical qubits, thanks to the principle of superposition [36]. For this reason, coherence in a logical qudit is a manifestation of different types of nonclassical correlations in the underlying physical multi-qudit systems (depicted in figure (1) and illustrated in section (3)). Hence, there should be a mapping between the criteria for quantum coherence of logical qubits and those for nonclassical correlations (be it nonlocality or entanglement or quantum discord) of its physical constituent qubits and vice-versa. In [37], we have shown that this mapping may be established by using the equivalence between logical qubits and different multiqubit physical systems. Logical qudits and stabilizers have already been used extensively in the context of quantum error-correcting codes [38]. To the best of our knowledge, they have not yet been employed to study the interrelations of different nonclassical features in different dimensions.

In this paper, we generalize the methodology proposed in [37] to multi-qudit systems and to continuous variable systems. The paper is structured as follows. Sections (1-4) consist of introduction and tools that we shall use in the rest of the paper. Sections (5-9) consist of applications, i.e., reciprocity among sufficiency conditions for different kinds of nonclassicalities and those for coherence. To elaborate, we show that a single logical operator maps to many different operators acting on physical multi-qudit states (sections (4) and (5)). Thereafter, we show reciprocity among sufficiency conditions for coherence in logical qudits and sufficiency conditions for nonclassical correlations in multi-qudit systems (sections (6, 7, 8, 9)). Section (10) concludes the paper.

In this section, we set up the notations to be used throughout the paper.

1. 1.

   The subscript ‘$L$’ is used for representing logical states or logical operators acting on the Hilbert space of logical qudits. For example, the symbols $|\psi\rangle_{L}$ and $A_{L}$ represent a logical state and a logical operator respectively.

2. 2.

   The numeral subscript of an observable represents the party on which the observable acts. For example, the observables $A_{1},\cdots,A_{N}$ act over the Hilbert spaces of the first, $\cdots$, the $N^{\rm th}$ qudit.

3. 3.

   For qudit systems, the symbol $X_{i}$ represents the cyclic translation operator by one unit acting on the $i^{\rm th}$ qudit. The symbol $Y_{i}$ represents the translation operator followed by a phase shift acting on the $i^{\rm th}$ qudit. The dimension in which these operators are defined will be clear from the context.

We start with some simple observations to elucidate the interrelation between quantum coherence of logical quantum systems and underlying quantum correlations in its constituent physical subsystems:
Observation 1: Suppose that we map multiqudit factorizable states (e.g., $|i\rangle^{\otimes N}$) to mono party logical qudit states (e.g., $|i\rangle_{L}$). In this case, the superposition of such multiqudit states maps to a coherent state in the computational basis of logical states¹¹1At this juncture, we wish to stress that this clubbing is not just a mathematical artifice, its physical instances are manifest in atomic physics through the coupling of angular momenta, i.e., $L-S$ coupling and $jj$ coupling..
Example: The three-qudit entangled GHZ state, $|\psi_{\rm GHZ}\rangle\equiv\frac{1}{\sqrt{d}}\sum_{i=0}^{d-1}|iii\rangle$ maps to the maximally coherent logical state, $|\psi_{\rm GHZ}\rangle_{L}\equiv\frac{1}{\sqrt{d}}\sum_{i=0}^{d-1}|i\rangle_{L}$, if $|iii\rangle\equiv|i\rangle_{L}$.

This observation is valid for mixed states as well. It can be seen by invoking the convexity argument. Consider a mixed bipartite entangled state, $\rho=\sum_{i}p_{i}|\psi_{i}\rangle\langle\psi_{i}|$, in its eigenbasis. Since the state $\rho$ is entangled, there exists at least one value of $i$ (say, $i=1$) for which the state $|\psi_{1}\rangle$ has a Schmidt rank $r>1$, i.e., $|\psi_{1}\rangle\equiv\sum_{j=1}^{r}\sqrt{\lambda_{j}}|jj\rangle$. For the choice, $|jj\rangle\equiv|j\rangle_{L}$, $|\psi_{1}\rangle\equiv\sum_{j=1}^{r}\sqrt{\lambda_{j}}|j\rangle_{L}$. So, the state $\rho$ will exhibit coherence in the logical basis. This argument can be straightforwardly extended to multipartite states.

These observations have been compactly shown in table (1). Hence, in this language, all the sufficiency conditions for nonlocality and entanglement (in general, for any nonclassical correlation) in multi-qudit physical systems (and multimode cv states) should emerge from those for coherence in logical qudits (resp., cv single-mode logical states) and vice-versa.

With this preface, we first present homomorphic maps between the stabilizer groups of physical multi-qudit states and that of a single logical qudit state. Thereafter, we lay down a procedure for obtaining sufficiency conditions for nonclassical correlations in physical multi-qudit systems from that for quantum coherence in a single logical qudit.

In this section, we present homomorphic mappings among the stabilizer groups of a single logical qudit and entangled physical multi-qudit systems. This homomorphic map will later be used as a tool to study reciprocity between sufficiency conditions for different nonclassical correlations and a sufficiency condition for coherence in logical qudits.

From the preceding section, it is clear that there exist different operators acting on physical multi-qudit systems which map to a single logical operator via a homomorphic map. Suppose that a logical qudit state, expressed in the computational basis $\{|i\rangle_{L}\}$, is a direct product of states of $N$ physical qudits, i.e.,

For this choice of logical qudit, which is of main interest to us in this paper, we lay down a procedure to identify different operators acting on physical multi-qudit states as follows (the proof of the existence of such operators has been given in A).

In the next section, we present the methodology to obtain sufficiency conditions for nonclassical correlations, given a sufficiency condition for quantum coherence.

In this section, we present the methodology to obtain sufficiency conditions for nonclassical correlations, given a sufficiency condition for quantum coherence.

Suppose that a sufficiency condition for quantum coherence in the basis, $\{|i\rangle_{L}\}$ is given as

with the stipulation that it is maximally obeyed by the normalized state $|\psi\rangle_{L}\equiv\frac{1}{\sqrt{d}}\sum_{i=0}^{d-1}|i\rangle_{L}$. $b_{a}$ are nonnegative integers and $c_{a}$ are real numbers. The value of $c$ is set in such a way that the ensuing inequality is violated by all the incoherent states in the computational basis of logical states. The task is to infer the underlying quantum correlation in the physical multiqudit states. That is to say, we examine what happens to the correlation in the physical multi-qudit systems as we vary the value of $c$. This would require distinct sets of operators acting over physical qudits that give rise to the witness ${\cal C}_{L}$. We use a two-pronged approach for obtaining sufficiency conditions for nonclassical correlations. That is to say, these operators can be identified in one of the following two ways:

1. 1.

   We identify the operators acting on multiqudit systems through homomorphism between the stabilizer groups. Suppose that $g^{(a)}$ represents an element belonging to the coset $(X_{1}\cdots X_{N})^{a}H_{N}$ (given in equation (8)). We use the homomorphic maps in the reverse direction for each of the operators $X_{L}^{a}$, i.e.,

   $\displaystyle X_{L}^{a}$

   $\displaystyle\to\sum_{g^{(a)}\in(X_{1}\cdots X_{N})^{a}H_{N}}w^{(a)}_{g}g^{(a)},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\sum_{g^{(a)}\in(X_{1}\cdots X_{N})^{a}H_{N}}w^{(a)}_{g}=1.$

   (19)

2. 2.

   We identify the operators acting on multiqudit systems by using the procedure given in section (5) without using the homomorphic map. We may employ the resolutions of these operators, i.e.,

   $\displaystyle{X}^{a}_{L}\to\sum_{k,i}w^{(a)}_{k}\prod_{\alpha=1}^{N}d_{i}^{(k)}\Big{(}X_{\alpha i}^{(k)}\Big{)}^{a},$

   (20)

   where the subscript $\alpha$ labels the qudit over which the operator acts and the superscript $k$ labels the resolution of the operator $X_{L}$. $w_{k}^{(a)}$ are the weights and $d_{i}^{(k)}$ are real numbers.

The operator $\sum_{a=0}^{d-1}c_{a}{X}^{a}_{L}$ can be variously expressed as,

The choices of the weights $w_{k}^{(a)}$ or $w_{g}^{(a)}$ and the bound on the ensuing inequality depend on the notion of classicality under consideration. At this juncture, some comments are in order:

1. 1.

   Of all the resolutions, those resolutions are of particular interest for us in which the operators acting over the same qudit are noncommuting. These resolutions bring out nonclassical correlations in the underlying physical system.

2. 2.

   We note that nonlocality is a nonclassical feature not restricted to quantum mechanics, whereas quantum coherence (in probability amplitudes) is a nonclassical feature restricted to quantum mechanics. A sufficiency condition for quantum coherence provides us with appropriate observables. The bound on their combinations is set after ensuring that the combination is obeyed by all the LHV models.

3. 3.

   We have taken but a simple choice of logical qudits $|i\rangle_{L}\equiv|i\rangle^{\otimes N}$. The procedure, however, is amenable to other choices of logical states discussed at the beginning of the section (6).

We now employ this methodology to identify observables for constructions of entanglement inequalities and nonlocality inequalities for multiqudit as well as infinite-dimensional systems.

We start with a brief recapitulation of GHZ nonlocality in higher dimensions for pedagogic purposes. For details, see, for example, [39] and B. The observables $X$ and $Y$ can be written in the computational basis set, $\{|n\rangle\}$, as,

The nonlocality inequality for a tripartite system is given by,

It is maximally satisfied by the three-qudit GHZ state:

The nonlocality inequality for tripartite $d$–diemsnional system, derived in [19], is given as,

This procedure admits a straightforward generalization to $N$ qudits and for arbitrary $d$.

In this section, we show that the operators employed in GHZ nonlocality emerge as different resolutions of shift operators acting on a single logical qudit system. We first start with a three-party system.