Perfect tensors

The tn that we focus on is shown in Fig. 1b, where each hexagon represents a special six-qubit state |ψ〉. |ψ〉 is called a PT, if and only if that any three-qubit subsystem out of six is maximally entangled with the rest. It is shown that, for a TN made by the PT, its entanglement entropy is holographic and gives the discrete rt formula on the lattice. Actually, the entanglement entropy of such TN equals the minimal number of links cut by the virtual surfaces anchored to the boundary, as illustrated in Fig. 1b.

To prove the above statement, we first introduce the form of the rank-6 PT, which is the building block of the TN. Given the single-qubit Hilbert space \({\cal{H}} \simeq {\Bbb C}^2\), a rank-6 PT |ψ〉 is a state in \({\cal{H}}^{ \otimes 6}\), such that for any bipartition of qubits m + k = 6, the entropy of the reduced density matrix is maximal. Assuming m ≥ k, and labeling the orthonormal basis in \({\cal{H}}^{ \otimes m}\) and \({\cal{H}}^{ \otimes k}\) by |α〉 and |i〉 respectively, a PT \(|\psi \rangle = \mathop {\sum}\limits_{\alpha ,i} {\psi _{\alpha i}} |\alpha \rangle \otimes |i\rangle\) satisfies

$$\mathop {\sum}\limits_\alpha {\psi ^\dagger _{i\alpha }} \psi _{\alpha j} = \frac{1}{{2^k}}\delta _{ij}.$$ (2)

In other words, the reduced density matrix ρ(k) by tracing out m qubits is an identity matrix, whose entanglement entropy

$$S_{EE}^{(k)} = - {\mathrm{tr}}\left( {\rho ^{(k)}\log _2\rho ^{(k)}} \right)$$ (3)

is simply k, the number of remained qubits. In this Letter, we use the superscript (k) to represent the k-qubit subsystem.

With the rank-6 PT (explicit form in appendix A (see Supplemental Information for a detailed description of the theory and experiment)) in hand, the TN state illustrated in Fig. 1b is constructed as follows. Each internal link \(\ell\) represents a two-qubit maximally entangled state \(|\ell \rangle = \left( {|00\rangle + |11\rangle } \right)/\sqrt 2\), where two qubits associate respectively to the two end points of \(\ell\). If we denote by |ψ(n)〉 the pt associated to the hexagon node n, the total tn state |Ψ〉 in Fig. 1b is written as a (partial) inner product form

$$|\Psi \rangle = \mathop { \otimes }\limits_\ell \langle \ell |\mathop { \otimes }\limits_n |\psi (n)\rangle .$$ (4)

The inner product takes place at the end points of each internal link \(\ell\), between one qubit in \(|\ell \rangle\) and the other in |ψ(n)〉. The qubits in |ψ(n)〉 not participating the inner product are boundary qubits corresponding to the dangling legs, and these boundary ones are actually physical qubits, indicating that |Ψ〉 is a state on the boundary.

We then pick a boundary region \({\cal{A}}\) which collects a subset of the boundary qubits, as shown in Fig. 1b. The reduced density matrix \(\rho _{\cal{A}} = {\mathrm{tr}}_{\overline {\cal{A}} }\left( {|\Psi \rangle \langle \Psi |} \right)\) is computed by tracing out all boundary qubits outside \({\cal{A}}\). Initially, this partial trace boils down to computing the reduced density matrix of individual tensors closest to the boundary. By applying Eq. (2) and noticing that \(|\ell \rangle\) is maximally entangled, the trace computation can be effectively pushed from the boundary into the bulk, meaning that the partial trace on the boundary is now equivalent to computing the reduced density matrix of the pt inside the bulk (see Supplemental Information for a detailed description of the theory and experiment). Once again, we can apply Eq. (2) and push the trace further inside. This iteration procedure is repeated until the trace reaches \({\cal{S}}\) in Fig. 1b, where Eq. (2) is not anymore valid, as the number of qubits participating the trace (number of links cut by \({\cal{S}}\)) is less than three for each tensor.

Now we have presented a sketch about how to calculate the entanglement entropy of \(\rho _{\cal{A}}\) via Eq. (3), and direct readers to Appendix B (see Supplemental Information for a detailed description of the theory and experiment) for a concise proof using the graphical computation of tn. Firstly, \({\mathrm{tr}}\left( {\rho _{\cal{A}}} \right)\) is found to be equal to the number of qubits on \({\cal{S}}\), i.e., the same as the number of links cut by \({\cal{S}}\). Moreover, the product \(\rho _{\cal{A}}^2\), involving the inner product of boundary qubits in \({\cal{A}}\), gives that \(\rho _{\cal{A}}^2 \propto \rho _{\cal{A}}\). Note that we have ignored all numerical prefactors but they all cancel when calculating \(\frac{{{\mathrm{tr}}\rho _{\cal{A}}^n}}{{({\mathrm{tr}}\rho _{\cal{A}})^n}}\) in the entanglement entropy. As a result, the Von Neumann entropy gives (see Supplemental Information for a detailed description of the theory and experiment)

$$\begin{array}{c}S_{EE}({\cal{A}}) = \mathop {{\lim }}\limits_{n \to 1} \frac{1}{{1 - n}}\log _2\frac{{{\mathrm{tr}}\rho _{\cal{A}}^n}}{{({\mathrm{tr}}\rho _{\cal{A}})^n}}\\ = {\mathrm{minimal}}\,{\mathrm{number}}\,{\mathrm{of}}\,{\mathrm{cuts}}\,{\mathrm{by}}\,{\cal{S}}.\end{array}$$ (5)

The above result is a discrete version of the rt formula in Eq. (1). The “minimal number of cuts” represents the minimal area Ar min (in the unit of Planck scale) in the RT formula. The bulk surface \({\cal{S}}\) with minimal area emerges effectively from the entanglement entropy of the tn state. Equation (5) demonstrates explicitly that the bulk geometry are created holographically by the entangled qubits of the boundary many-body system.

It is worth emphasizing that, all descriptions about constructing the TN originate from the PT in Eq. (2). Therefore, this rank-6 PT plays the fundamental role in holographic entanglement entropy, and is a key of emerging bulk gravity from TN states. If we choose \({\cal{S}}\) as shown in Fig. 1c by which the minimal number of cuts is three, a rank-6 PT is generated where the boundary and bulk qubits are both three. Here, we demonstrate the emergent gravity program in AdS/CFT for the first time in a six-qubit nmr quantum simulator, by creating the rank-6 pt in Fig. 1c and measuring the relevant entanglement entropies.

Experiment implementation of a rank-6 perfect tensor

The six qubits in the nmr quantum register are denoted by the spin-1/2 13C nuclear spins, labeled as 1 to 6 as shown in Fig. 2a, in 13C-labeled Dichloro-cyclobutanone dissolved in d 6 -acetone. All experiments were carried out on a Bruker drx 700 MHZ spectrometer at room temperature. The internal Hamiltonian of this system is

$${\cal{H}}_{\mathrm{int}} = \mathop {\sum}\limits_{j = 1}^6 {\pi

u _j} \sigma _z^j + \mathop {\sum}\limits_{j < k, = 1}^6 {\frac{\pi }{2}} J_{jk}\sigma _z^j\sigma _z^k,$$ (6)

where ν j is the resonance frequency of the jth spin and J jk is the J-coupling strength between spins j and k. All parameters including the relaxation times for each spin are listed in appendix C (see Supplemental Information for a detailed description of the theory and experiment). To control system dynamics, we have external control pulses with four adjustable parameters: the amplitude, frequency, phase, and duration, based on which arbitrary single-qubit rotations can be realized with simulated fidelities over 99.5% (see Supplemental Information for a detailed description of the theory and experiment).

Fig. 2 a Molecular structure of the 13C-labeled six-qubit quantum processor. The six qubits of the rank-6 pt are mapped to 1 to 6, respectively. b Quantum circuit that evolves the system from |0〉⊗6 to the pt, constructed by several Hadamard gates (blocks) and controlled-Z operations (lines connecting two dots) Full size image

A rank-6 PT can be created from |0〉⊗n through the circuit as illustrated in Fig. 2b, which involves only Hadamard gates and controlled-Z gates. Experimentally, this requires an initialization of the system onto |0〉⊗n. However, initializing an NMR processor to |0〉⊗n is based upon the pseudo-pure state technique, which leads to an exponential signal attenuation. Here, we adopt a temporal averaging approach that enables the pt preparation directly from the thermal equilibrium of nmr, while skipping the intermediate pseudo-pure state stage to avoid the above problem, as shown in Appendix C (see Supplemental Information for a detailed description of the theory and experiment)).

After the creation, we conducted k-qubit (1 ≤ k ≤ 5) quantum state tomography in the corresponding subspace of the whole system, respectively. For simplicity, the cutting of the links was chosen to be continuous in experiment, i.e., in a cyclic manner. It means that six state tomographies for any given k are performed, e.g., when k = 2, we reconstructed \(\rho _{12}^{(2)},\rho _{23}^{(2)}, \cdots ,\rho _{61}^{(2)}\).

Subsequently, Von Neumann entropies of these k-qubit subsystems were calculated by Eq. (3). In theory, each S(k) equals to the minimal number of cuts according to Eq. (5) when k ≤ 3. Combined with the fact that S(k) = S(6−k) for a six-qubit pure state, we have S(k) = min{k, 6 − k} for the theoretical pt, as shown by the orange dashed line in Fig. 3a. In experiment however, inevitable errors lead to imperfection and hence impurity in the truly prepared state, so we cannot just measure k ≤ 3 cases to deduce other k's. Therefore, we measured and compare the experimental S(k) for each 1 ≤ k ≤ 5 (red circles) with their theoretical predictions in Fig. 3a. For each k, the mean and error bar of the experimental S(k) value are calculated from the six cyclic tomographic results. When k ≤ 3, the measured entanglement entropies match extremely well with the theory; when k > 3, there are notable discrepancies between theory and experiment, which should be primarily attributed to decoherence errors, as discussed in the following.

Fig. 3 a Entanglement entropy S(k) of the k-qubit subsystem of the rank-6 pt. In theory, S(k) = min{k, 6 − k} as shown by the orange dashed line. Experimental results are represented by the red circles, where S(4) and S(5) do not fit very well. If the signal’s decay due to decoherence is taken into account, the experimental results are rescaled to the blue squares, which fit much better. As a upper-bound reference, the maximal entropy of a k-qubit subsystem is also plotted (green dotted line) by assuming a six-qubit identity. b Density matrices of the theoretical rank-6 pt ρ pt (left) and the experimentally reconstructed state ρ e (right) on a two-dimensional plane. The rows and columns are labeled by the six-qubit computational basis from |0〉⊗6 to |1〉⊗6, respectively. c Direct observation of ρ e in the nmr spectra (red), with probe qubits C 1 (top) and C 4 (bottom), respectively. The simulated spectra of the pt are also shown in blue. For a better visualization, experimental signals are rescaled by 1.25 times to neutralize the decoherence error Full size image

The pulse sequence that creates the pt is around 60 ms; this is not a negligible length compared to the \(T_2^ \ast\) time (~400 ms) of the molecule, meaning that decoherence will induce substantial errors during experiments(see Method). As \(T_2^ \ast\) relaxation is the dominating factor, the off-diagonal terms in the pt density matrix are mainly affected. To estimate this imperfection, we performed full state tomography23 on the prepared state and got ρ e . The real part of ρ e is depicted in the right panel of Fig. 3b, by projecting each element onto a two-dimensional plane. As a comparison, the figure of the theoretical pt ρ pt = |ψ〉〈ψ| is placed in the left panel of Fig. 3b. In fact, the diagonal elements of ρ e are almost the same as that of ρ pt , but the off-diagonal are lower due to the \(T_2^ \ast\) errors. The state fidelity between ρ e and ρ pt , defined as

$$F\left( {\rho _{pt},\rho _e} \right) = {\mathrm{tr}}\left[ {\sqrt {\sqrt {\rho _{pt}} \rho _e\sqrt {\rho _{pt}} } } \right],$$ (7)

is about 85.0%. Direct observations of ρ e in terms of nmr spectra are also shown in Fig. 3c, where experimental and simulated spectra highly match if the experimental signal is rescaled by 1.25 times to compensate for the decoherence effect.