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arXiv:2508.12256 · Physical Review A

Partial transpose as a space-time swap

James Fullwood, Junxian Li

Published Dec 11, 2025• 0 citations
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Abstract

While the partial transpose operation appears in many fundamental results in quantum theory -- such as the Peres-Horodecki criterion for entanglement detection -- a physical interpretation of the partial transpose is lacking. In this work, we show that a partial transpose of a bipartite density operator is a two-time pseudo-density operator, which by definition encodes temporal correlations associated with two-point sequential measurement scenarios. As such, it follows that partial transposition admits a precise physical interpretation as mapping spatial correlations to temporal correlations, thus swapping the roles of space and time for bipartite quantum systems. For maximally entangled qubits, we show that partial transposition maps spatial correlations which violate Bell inequalities to causal correlations which cannot be replicated by spacelike separated systems, thus further solidifying the interpretation of partial transpose as a space-time swap. As it is known that gravitational effects swap the roles of space and time inside a black hole, our results suggest that at a quantum mechanical level, a traversal of a black hole's event horizon by a bipartite quantum system may be described by a partial transpose.

DOI
10.1103/glvk-ns5c

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While the partial transpose operation appears in many fundamental results in quantum theory -- such as the Peres-Horodecki criterion for entanglement detection -- a physical interpretation of the partial transpose is lacking. In this work, we show that a partial transpose of a bipartite density operator is a two-time pseudo-density operator, which by definition encodes temporal correlations associated with two-point sequential measurement scenarios. As such, it follows that partial transposition admits a precise physical interpretation as mapping spatial correlations to temporal correlations, thus swapping the roles of space and time for bipartite quantum systems. For maximally entangled qubits, we show that partial transposition maps spatial correlations which violate Bell inequalities to causal correlations which cannot be replicated by spacelike separated systems, thus further solidifying the interpretation of partial transpose as a space-time swap. As it is known that gravitational effects swap the roles of space and time inside a black hole, our results suggest that at a quantum mechanical level, a traversal of a black hole's event horizon by a bipartite quantum system may be described by a partial transpose.
Recent works suggest that quantum theory may support a unification of the notions of space and time, thus treating the spatial and temporal quantum correlations equally. Specifically, the partial transposition of the maximally entangled state of two quantum systems at one time exactly matches the temporal quantum correlations of one quantum system that evolves unitary between two distinct moments of time. In this work we consider this equivalence of spatial and temporal quantum correlations in the context of AdS/CFT duality. We show that in the high temperature limit the thermofield double state is equivalent to the temporal quantum correlations of a quantum theory evolving unitary between two times. Thus, on the gravity side, we argue that the temporal correlations correspond to a black hole at one time connected behind the horizon by an Einstein-Rosen bridge to the same black hole at another time. Moreover, we construct the spacetime corresponding to this temporal wormholes and find that the spacetime is consistent with the interior solution of the BTZ black hole. We conclude suggesting that the BTZ interior solution could have an interpretation in the dS/CFT correspondence framework.
The statistics of local measurements of joint quantum systems can sometimes be used to distinguish the spatiotemporal structure in which they were measured. We first prove that every bipartite separable density matrix is temporally compatible with direct causal influence for arbitrary finite-dimensional quantum systems and measurements of a tomographically complete class of observables, which includes all Pauli observables in the case of multi-qubit systems. Equivalently, if a bipartite density matrix is not temporally compatible with direct causal influence, then it must be entangled. We also provide an operational meaning for the two temporal evolutions consistent with such correlations in terms of generalized dephasing channels and pretty good measurements. The two temporal evolutions are Bayesian inverses of each other, which is different from them being Petz recovery maps of each other. Finally, we prove necessary and sufficient conditions for an arbitrary bipartite quantum state to be temporally compatible, thereby providing a temporal analogue of the positive partial transpose criterion valid for quantum systems of any dimension.
We first show that partial transposition for pure and mixed two-particle states in a discrete NNN-dimensional Hilbert space is equivalent to a change in sign of the momentum of one of the particles in the Wigner function for the state. We then show that it is possible to formulate an uncertainty relation for two-particle Hermitian operators constructed in terms of Schwinger operators, and study its role in detecting entanglement in a two-particle state: the violation of the uncertainty relation for a partially transposed state implies that the original state is entangled. This generalizes a result obtained for continuous-variable systems to the discrete-variable-system case. This is significant because testing entanglement in terms of an uncertainty relation has a physically appealing interpretation. We study the case of a Werner state, which is a mixed state constructed as a convex combination with a parameter rrr of a Bell state ∣Φ+⟩|Φ^{+} \rangle∣Φ+⟩ and the completely incoherent state, ρ^r=r∣Φ+⟩⟨Φ+∣+(1−r)I^N2\hatρ_r = r |Φ^{+} \rangle \langle Φ^{+}| + (1-r)\frac{\hat{\mathbb{I}}}{N^2}ρ^​r​=r∣Φ+⟩⟨Φ+∣+(1−r)N2I^​: we find that for r0<r<1r_0 < r < 1r0​<r<1, where r0r_0r0​ is obtained as a function of the dimensionality NNN, the uncertainty relation for the partially transposed Werner state is violated and the original Werner state is entangled.
We study entanglement in mixed bipartite quantum states which are invariant under simultaneous SU(2) transformations in both subsystems. Previous results on the behavior of such states under partial transposition are substantially extended. The spectrum of the partial transpose of a given SU(2)-invariant density matrix $\ensuremath{\rho}$ is entirely determined by the diagonal elements of $\ensuremath{\rho}$ in a basis of tensor-product states of both spins with respect to a common quantization axis. We construct a set of operators which act as entanglement witnesses on SU(2)-invariant states. A sufficient criterion for $\ensuremath{\rho}$ having a negative partial transpose is derived in terms of a simple spin correlator. The same condition is a necessary criterion for the partial transpose to have the maximum number of negative eigenvalues. Moreover, we derive a series of sum rules which uniquely determine the eigenvalues of the partial transpose in terms of a system of linear equations. Finally we compare our findings with other entanglement criteria including the reduction criterion, the majorization criterion, and the recently proposed local uncertainty relations.
We show that, by using temporal quantum correlations as expressed by pseudo-density operators (PDOs), it is possible to recover formally the standard quantum dynamical evolution as a sequence of teleportations in time. We demonstrate that any completely positive evolution can be formally reconstructed by teleportation with different temporally correlated states. This provides a different interpretation of maximally correlated PDOs, as resources to induce quantum time evolution. Furthermore, we note that the possibility of this protocol stems from the strict formal correspondence between spatial and temporal entanglement in quantum theory. We proceed to demonstrate experimentally this correspondence, by showing a multipartite violation of generalized temporal and spatial Bell inequalities and verifying agreement with theoretical predictions to a high degree of accuracy, in high-quality photon qubits.